Atomic physics, Quantum optics, Condensed matter physics ... In these lecture
series, we will often work in “natural units”: ... Tutorial on Topological Insulators.
Lecture Series Winter Term 2012/13 Innsbruck University
UNIVERSITY OF INNSBRUCK
Many-Body Physics with Cold Atoms IQOQI AUSTRIAN ACADEMY OF SCIENCES
SFB Coherent Control of Quantum Systems
Sebastian Diehl Institute for theoretical physics, Innsbruck University, and IQOQI Innsbruck
Introduction: Cold Atoms vs. Condensed Matter Q: What can cold atoms add to many body physics?
•
New models of own interest - Bose-Hubbard model - Strongly interacting continuum systems: BCS-BEC Crossover; Efimov effect - long range interactions other than Coulomb; SU(N) Heisenberg models with variable N...
•
Quantum Simulation: clean/ controllable realization of model Hamiltonians which are - less clear to what extent realized in condensed matter - extremely hard to analyze theoretically
•
e.g. 2d FermiHubbard model
Nonequilibrium Physics, time dependence: - Condensed matter: thermodynamic equilibrium and ground state physics. - Cold atoms: e.g. creation of excited many-body states, study their dynamic behavior
•
Scalability: - Study crossover from systems with few- to (thermodynamically) many degrees of freedom: Change of concepts?
•
Integrate quantum optics concepts and many-body theory, coherent-dissipative manipulation, non-equilibrium stationary states
Introduction: Cold Atoms vs. Condensed Matter Q: What can cold atoms add to many body physics?
•
New models of own interest
- Bose-Hubbard model - Strongly interacting continuum systems: BCS-BEC Crossover; Efimov effect - long range interactions other than Coulomb; SU(N) Heisenberg models with variable N...
•
Quantum Simulation: clean/ controllable realization of model Hamiltonians which are
- less clear to what extent realized in condensed matter - extremely hard to analyze theoretically
•
e.g. 2d FermiHubbard model
Nonequilibrium Physics, time dependence:
- Condensed matter: thermodynamic equilibrium and ground state physics. - Cold atoms: e.g. creation of excited many-body states, study their dynamic behavior
•
Scalability:
- Study crossover from systems with few- to (thermodynamically) many degrees of freedom: Change of concepts?
•
Integrate quantum optics concepts and many-body theory, coherent-dissipative manipulation, non-equilibrium stationary states
A Brief History of cold atoms and BEC research
A Brief History of cold atoms and BEC research 4He
liquified by K. Onnes
1908
Bose studies statistical Properties of photons
1924
Kapitza / Allen & Misener discover superfluidity of 4He
1938
Allen & Jones discover fountain effect
1925
Bose/Einstein predict condensation of a noninteracting atomic Bose gas
1938
London makes first connection between 4He superfluidity and BEC
1941
Landau produces first phenomenological theory of superfluids
1947
Bogoliubov produces microscopic theory of weakly interacting Bose gas
1949
Onsager predicts quantized vortices in liquid 4He (+ Feynman 1955)
1951
Landau & Lifshitz / Penrose link BEC and off-diagonal long range order.
A Brief History of cold atoms and BEC research 1976 Neutral Atom traps:
Gaithersburg (Magnetic)
Stwalley and Nosanow (following Hecht, 1959) argue spin polarised 1H would be Superfluid BEC (Realised MIT 1998).
1985
AT&T Bell Labs
W. D. Phillips, S. Chu, and C. Cohen-Tannoudji: Nobel Prize for Laser Cooling
1997
E. Cornell, C. Wieman, W. Ketterle: Nobel Prize for BEC in dilute gases
2001
1987
Simultaneous Cooling and Trapping in Magneto-Optical Trap (AT&T Bell labs)
1995
BEC in dilute gases: JILA (87Rb), MIT (23Na), Rice (7Li)
A Brief History of cold atoms and BEC research
+ Yb (2004), Cr (2005) Molecules: Li2 (2003), K2 (2003) Earth alkaline atoms: Ca (2009), Sr (2009) Strongly dipolar atoms: Dy (2011), Er (2012)
A Brief History of cold atoms and BEC research
• Further milestone experiments: Tunable interations: Feshbach resonance Bosonic Sodium
Degenerate and strongly interacting fermions
1999 Innsbruck, 2004
pairing gap
Ketterle Group, MIT MIT, 2005
2002 Optical Lattices: Mott insulator-superfluid quantum phase transition
2004 JILA, 2005
Haensch Group, Munich
pair correlations
vortices
Absolute Scales in Condensed Matter vs. Cold Atoms
• Compare the absolute scales in of liquid
4He
and cold atom BEC
• Liquid 4He “BEC” • Typical density 1022 cm-3 • Condensation temperature ~ 2.17 K • Strongly Interacting • Small Condensate Fraction • BEC in dilute gases • Typical density 1013 – 1015 cm-3 (cf. density of air: 3x1019 cm-3) • Condensation temperature < 10-5 K (cf. CMB radiation: ~ 2.7K) • Weakly interacting • Much larger condensate fraction
Modelling in Condensed Matter vs. Cold Atoms
• Compare the way of constructing microscopic models in these systems •
Condensed Matter
- given chunk of matter - have to guess the microscopic Hamiltonian - ex: Fermi-Hubbard model
•
Cold Atoms
- microscopic parameters from scattering physics (n=T=0) - many body physics in separate experiments
➡ Direct experimental access to both micro- and macrophysics ➡ Challenge to theorists to make the connection ➡ Many disciplines and theoretical techniques are involved:
Atomic physics, Quantum optics, Condensed matter physics
•
Further general properties and features:
Clean system:
Control:
Measurements:
• No (uncontrollable) disorder • tune microscopic parameters, • (quasi-)momentum distribution, noise correlations by releasing atoms. • Weak dissipation (>1s) (cf. • modify lattice structure phonons in solid state) • choose effective dimensionality • Spectroscopy (e.g., lattice
modulations / Bragg scattering).
Units
• In these lecture series, we will often work in “natural units”: • Quantum mechanical problem: ~=1
➡ Measure energy and frequency in the same unit,
•
E = ~!
Many-Body problem:
kB = 1 ➡ Measure energy and temperature in the same unit,
•
E = 12 kB T
Nonrelativistic problem: (we often indicate the mass explicitly)
2M = 1
➡
•
p~ 2 Measure energy and momentum square in the same unit, E = 2M
Optical problem:
c=1 ➡ Measure frequency an wave number in the same unit,
! = ck
Lecture Outline The lecture series consists of two parts:
Part I: Introduction to the theory of ultracold atomic quantum gases
• Physics: • Continuum systems: • Scales and interactions, Effective theories • Weakly interacting Bosons, Bose-Einstein condensation • Weakly interacting Fermions, BCS superfluidity • Strong interactions, the BCS-BEC crossover • Lattice systems: • Quantum phase transitions • Microscopic derivation of the Bose-Hubbard model • Phase Diagram of the Bose-Hubbard model, Mott insulator -
Bose-Einstein Condensation
MIT, 2005
vortices
Fermion Superfluidity
superfluid transition
• Methods: • Second quantization vs. functional integral techniques
Mott insulator - superfluid transition
Lecture Outline The lecture series consists of two parts:
Part II: Selected Topics
1 @t Uk [⇢] = STr 2
• Advanced field theoretical techniques • (Functional) Renormalization Group • Application to ultracold atoms: e.g. Efimov effect, quantum criticality in Bose-Hubbard model, Tan effect, ...
• Many-Body Physics with Open Atomic Systems • Dissipation engineering, driven-dissipative BEC • Nonequilibrium phase transitions: open Dicke model, competing unitary and dissipative dynamics, ...
• Tutorial on Topological Insulators • Suggestions welcome!
1 (2) k [
] + Rk
@ t Rk
Literature
• General BEC/BCS theory: - C. J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press (2002) - L. Pitaevski and S. Stringari, Bose-Einstein Condensation, Oxford University Press (2003)
• Recent research review articles: - I. Bloch, J. Dalibard and W. Zwerger, Many-Body Physics with Cold Atoms, Rev. Mod. Phys. 80, 885 (2008) - S. Giorgini, L. Pitaevski and S. Stringari, Theory of ultracold atomic Fermi gases, Rev. Mod. Phys. 80, 1215 (2008) - I. Boettcher, J. M. Pawlowski, S. Diehl, Ultracold Atoms and the Functional Renormalization Group, Nucl. Phys. Proc. Suppl. 228, 63 (2012) arxiv:1204.4394
• Many-Body Physics (with and without cold atoms) - J. Negele, H. Orland, Quantum Many-particle Systems, Westview Press (1998) - A. Altland and B. Simons, Condensed Matter Field Theory, Cambridge University Press (2006, second edition 2010) - S. Sachdev, Quantum Phase Transitions, Cambridge University Press (1999) - K. B. Gubbels and H. C. T. Stoof, Ultracold Quantum Fields, Springer Verlag (2009)
Basic BEC theory -- scales in ultracold quantum systems Eq
q
• Microscopic scattering physics: Lennard-Jones (LJ) potential -Statistical 1/r12 hard core repulsion: repulsion ofNoninteracting electron clouds rrep = O(a B) Mechanics of Bosons - 1/r6 attraction: van der Waals (induced dipole-dipole interaction) rvdW = (50...200)aB for alkalis: typ. order of magnitude for interaction length scale
Hamiltonian for free particles in occupation number representation (“second •• The Simplifications:
√ quantization”) - T = 0, no interactions: whole qdensity sits in the condensate: ψ0 = N 2 √ † H = (a/d "a1): - T = 0, weak interactions N q aqexpect ψ0 ≈ 2m q - we can formalize this: p † †N - The operator aq creates a particle in the momentum mode q : aq |vac! = (N + 1)!|nq = N !. Similarly, aq annihilates a particle in the mode q.
-
The operators satisfy bosonic commutation relations, [aq , a†q! ] = δ(q−q! ) (cf. harmonic oscillator ladder operators, but with an index label q).
-
The operator nq = a†q aq measures the particle number in the momentum mode q. The total particle P † ˆ number is N = aq aq . q
-
The dispersion relation is Eq = !ω =
q2 . 2m
This is the energy of a single particle in the mode q.
The Sodium atom has one valence electron outside a closed shell. Eq 2 2 6 The ground state has the configuration 1s 2s 2p 3s. = Free particles in occupation number representation
•
We order the interactions according to the importance: collection of independent harmonic oscillators with energy
q2 Eq = 2m
q
Statistical Mechanics of Noninteracting Bosons
•
Statistical properties described by the Free Energy:
U = kB T log Z,
ˆ) µN
1
Z = tr exp
kB T (H
temperature
•
Trace easily carried out in occupation number basis: H is diagonal:
U = kB T log
1 Y X
1
e
2
q kB T ( 2m
µ)nq
= kB T log
q nq =0
•
chemical potential
Y
1
1
e
2
q kB T ( 2m
µ)
1
=
kB T
q
X
1
log 1
e
2
q kB T ( 2m
q
Consider low temperature behavior of equation of state:
N=
@U =h @µ
X q
a†q aq i
=
X q
ha†q aq i
=
X q
1 1
e kB T
q2 ( 2m
µ)
1
!V
Z
dd q (2⇡)d
1 1
2
q ( 2m e kB T
continuum limit Bose-Einstein distribution
µ)
1
µ)
√ † The operator n = a a measures the particle number in the momentum mode q. The total particle q q q - T = 0, weak interactions (a/d " 1): expect ψ ≈ N 0 ˆ = P a†q aq . number is N -Bose-Einstein we can formalizeq this: Condensation, 3D p † †N q2 - - The operator aq creates aE particle momentum |vac! = (N + 1)!|n The dispersion relation is . This is the mode energyqof: aaqsingle particle in the modeq q.= N !. q = !ωin=the 2m Similarly, aq annihilates a particle in the mode q.
d=3 • Lower T and study the behavior of µ at fixed n (3D): † - The operators satisfy bosonic commutation relations, [aq , aq! ] = δ(q−q! ) (cf. harmonic oscillator ladder • Atoperators, a finite Tbut , µwith hits below equation dofd qstate has no anzero: index label T q). this Tc theN 1 solution † n = number =needs The operator n = a aq measures the particle q. The total particle q q2 due q •T =Bose and Einstein (1925): Equation below T modification V c in the (2 momentum )d 1 ( mode 0 µ) to macroscopic
-
•
e kB T
ˆ = P a†q aq . number is N q
The dispersion relation is Eq = !ω =
2m
1
q2 . 2m
Polygamma
This is the energy3of a single particle = dB (T )g3/2 (eµ/kBinT )the mode q. function Lower T and study the behavior of µ at fixed n (3D): g (z) = • Lower T and study the behavior of µ at fixed n (3D):
µ
zn n n=1
• At a finite T , µ hits zero: below this Tc the equation of state has no solution • At a finite T , µ hits zero: below this Tc the equation of state has no solution • Bose and Einstein (1925): Equation below Tc needs modification due to macroscopic • Bose and Einstein (1925): Equation below Tc needs modification due to macroscopic occupation of zero mode: Eq occupation of zero mode: d d q dd q 1 1 † † n =n!a=0 a!a 0" + q^2 q^2 0 a0 " +(2π)d d (2π) T −T1 e 2mkeB2mk B −1
- plausible: Bosons can populate single quantum state with arbitrary number • macroscopic: N0 = !a†0 a0 "† = O(N/V ), i.e. extensive - macroscopic: N0 = !a0 a0 " = O(N ) ∝ V , i.e. extensive • plausible: Bosons can populate quantum state with arbitrary number - critical temperature: determinedsingle by • critical temperature: determined bynλ3dB = g3/2 (1) = ζ(3/2) ≈ 2.612
q
=1 above - zero order O(N3): homogenous mean field reproduced √ = g{3/2} (1) = ζ(3/2) ≈ 2.612 or λdB /d ! O(1) nλ{dB} interparticle 2 proper choice1/2 1/3(equilibrium - linear terms O( N ): vanishes upon of the chemical potential condition) de Broglie wavelength = (2⇥ /mk T ) d = n dB B spacing µ = gn0 \m - quadratic part O(1):
-
The operator nq = a†q aq measures the particle number in the momentum mode q. The total particle P † ˆ number is N = aq aq . q
-
Role of Dimension: No qBEC in 2D The dispersion relation is Eq = !ω =
2
2m
. This is the energy of a single particle in the mode q.
• Lower T and study the behavior of µ at fixed n (2D): • Lower T and study the behavior of µ at fixed n (3D):
T Tc the equation of state has no solution • At a finite T , µ hits zero: below this ➡ EoS without condensate can be = 0 and Einstein (1925): Equation below T needs modification due to macroscopic • TBose c fulfilled for all T: d = 3 ➡ No BEC in (homogeneous) 2d d=2 space
µ
•
Reason: Infrared (low momentum) divergence due to smaller phase space
d2 q d2 q ⇤ˆ nq ⌅ = q2 2 (2⇥) (2⇥)2 e( 2m
1 µ)/kB T
1
⇥
low momenta, small mu
•
dqq q2
µ logarithmic divergence
Remark: More general result
-
No spontaneous breaking of continuous symmetries at finite temperature (Mermin-Wagner Theorem)
-
quasi long range order possible: Kosterlitz-Thouless transition
Summary
• •
BEC is statistical effect, no interaction needed
• •
quantum degeneracy condition:
BEC is macroscopic population of a single quantum state, the q=0 mode
d
dB
True BEC at finite temperature only in d=3
dB
•
Eq
= (2⇥
2
/mkB T )1/2
d=n
q
1/3
Now: more realistic description of ultracold Bose systems - Trapping potential - Interactions
Trapping Potential and Weak Interactions
•
So far: free particles in homogeneous space, continuum limit : ⇤ ⇤ 2 ⇥ ⇥ ⇥ † q † ˆ H0 = H µN = aq µ aq = ax µ ax 2m 2m q x after Fourier transform
aq =
eiqx ax ; x
•
[ax , a†y ] = (x
y)
Now we aim at a more realistic description of cold atomic gases:
-
Trapping potential: the local density experiences a local potential energy
Htrap =
V (x)ˆ nx , x
V (x) = 12 m
x
2 2
Remark: this is just a collection of local harmonic oscillators: exactly solvable
-
Local two-body interactions:
Hint =
g (x x,y
•
V (x)
y)ˆ nx n ˆy = g x
n ˆ 2x
d
Our workhorse Hamiltonian is
H = H0 + Htrap + Hint
contact interaction
Microscopic Origin of the Interaction Term model potential with same scattering length
Example: Na atom (Alkali atoms)
• Microscopic scattering physics: Lennard-Jones (LJ) potential - 1/r12 hard core repulsion: repulsion of electron clouds rrep = O(aB ) - 1/r6 attraction: van der Waals (induced dipole-dipole interaction)
true interatomic potential U(x) x
rvdW = (50...200)aB for alkalis: typ. order of magnitude for interaction length scale
The Sodium atom has one valence electron outside a closed shell.
2 2 of the LJ type potentials at • General The groundproperties state has configuration 1slow 2senergies: 2p6 3s.
- isotropic s-wave scattering dominates; the scattered wave function behaves asymptotically as (x)
We order the interactions according to the importance:
- a is the scattering length. Knowledge of this single parameter is sufficient to describe low energy scattering!
- within Born approximation, it can be calculated as Electron - nucleus and electron - electron interaction
interatomic potential
U (x) system: Wave function of the atom as a aBorn many electron x
➡ very
un!same (r) scattering length! different interaction potentialsΨmay have the =Φ Y (θ, φ) core
r
!m
a/x
The Model Hamiltonian as an Effective Theory H=
⇧
x
• •
⇤ † ax
⇥ 2m
⇥ ⌅ 2 µ + V (x) ax + gˆ nx
Efficient description by an effective Hamiltonian with few parameters. For ultracold bosonic alkali gases, a single parameter, the scattering length a, is sufficient to characterize low energy scattering physics of indistinguishable particles : Effective interaction 2
8⇡~ g= a m
•
A typical order of magnitude for the scattering length is
•
The validity of the model Hamiltonian is restricted to length scales
•
For bosons, we must restrict to repulsive interactions a > 0 (else: bosons seek solid ground state, collapse in real space)
•
So far: microscopic description; now: many body scales!
a = O(rvdW ), rvdW (50...200)aB
l & rvdW
The antisymetrization of the wave function can be ignored (core wave function and valence electron wave function do not overlap).The description of the valence electron becomes a Validity of our Hamiltonian: Scales in Cold Dilute Bose Gases single electron problem. • The effective Hamiltonian is valid because none of many-body length scales can resolve interaction length scale : • Many-body scales: density and temperature in terms of length scales. - diluteness a/d ! 1 (d = n−1/3 ) ∗ dilute means weakly interacting: interaction energy gn ∼ a/d · d−2 ∗ clear: three-body interaction terms irrelevant - quantum degeneracy: d/λdB ! 1 (λdB =! (2π!2 /mkB T )1/2 ) - trap frequencies: λdB /losc ! 1 (losc = 1/ m/2ω) dB
= (2⇥
2
/mkB T )1/2
d=n
1/3
a
d
dB
2a Summary of length scales length
phys. meaning of the ratio:
aB = 5.3 ⇥ 10
scattering length
interparticle sep.
de Broglie w.l.
(0.05 ... 0.2)10^3
(0.8 ... 3)10^3
(10 ... 40)10^3
a/aB
weak interactions/ dilute gases
d/aB
dB /aB
quantum degeneracy
2
nm
Bohr radius
trap size
losc /aB
(3 ... 300)10^3
local density approximation valid
Violations of the scale hierarchy
• Generic sequence of scales and possible violations: interaction scale
interparticle spacing
de Broglie wavelength
(Oscillator length of trap)
Feshbach resonance or optical lattice spacing
• With Feshbach resonances, violation of a/d richer than Schrödinger equation
Interplay of quantum mechanics and nonlinearity: quantized vortex solutions - uniform case V(x) = 0, search static cylinder symmetric symmetric solutions with no z dependence:
'(x, t) = '(r, ) = f (r)ei`
-
GP equation:
0= -
2
f f” + 2m r
⇥2 f ⇥ r2
integer, such that phase returns after 2 pi: unique Wave function
f (r)
µf + gf 3
large distances: constant solution, determine chemical pot. short distances: condensate amplitude must vanish due to centrifugal barrier, in turn rooted in the quantization of the phase
=
2mgn
coherence length
vortex solution
r
Bose and Einstein (1925): Equation below Tc needs modification due to macroscop occupation of zero mode:
Quantum Fluctuations: Bogoliubov Theory n=
•
† !a0 a0 "
+
dd q (2π)d
homogeneous case: plane wave expansion of field
1
q^2 operator e 2mkB T(
p p p 0 (x)a − 1 0 = 1/ V N0 = n0 )
X p iqx - plausible: Bosons can populate state with ax =singlenquantum + e aqarbitrary number 0
-
•
† macroscopic: N = !a a0 " = O(N ) ∝ V , i.e. q extensive 0 phase 0 without choice of zero critical temperature: loss of generalitydetermined by
3 Grand canonical Hamiltonian. Take fluctuations into account to leading order: nλthe dB = g3/2 (1) = ζ(3/2) ≈ 2.612
-
zero order: homogenous mean field reproduced linear terms: vanishes upon proper choice of the chemical potential (equilibrium condition) µ = gn0 quadratic part: (2) HBog
=
X
† q2 aq ( 2m
µ + 2gn0 )aq +
gn0 † † (a qa q 2
+ aq a
lectron - nucleus and electron electron interaction q6=0
q)
!✓ ◆ q X ave function of the electron system: aq 1 atom† as a many + gn0 gn0 2m = (aq , a q ) + const. 2 † q a q 2 gn0 + gn0 2m q6=0 un! (r) Y!m (θ, φ) Ψ = Φcore µ = gn0 r 2
th core wave function (filled shell) and u(r) the radial wave function of the valence ele
Quantum Fluctuations: Bogoliubov Theory
•
quadratic Bogoliubov Hamiltonian: (2) HBog
=
X
† q2 aq ( 2m
µ + 2gn0 )aq +
Eq
gn0 † † (a qa q 2
+ aq a
q)
q6=0
1X † = (aq , a 2 q6=0
•
q)
q 2m
+ gn0 gn0
gn0 q2 2m + gn0
!✓
aq a† q
◆
+ const.
Discussion:
-
•
2
q
Off-diagonal terms: pairwise creation/annihilation out of the condensate Coupling of modes with opposite momenta Hamiltonian not diagonal in operator space: diagonalize to find spectrum and elementary excitations
Remarks:
-
An equivalent approach linearizes Heisenberg EoM around homog. GP mean field Validity: The ordering principle is given by the power of the condensate amplitude. Bogoliubov theory is not a systematic perturbation theory in U but becomes good at weak coupling
-
The operator aq creates a particle in the momentum mode q : aq |vac! = The operators satisfy Similarly, aq annihilates a-particle in the mode q. bosonic commutation relation but with an index label q). † ! ) (cf. ha ] = δ(q−q - The operators satisfy bosonicoperators, commutation relations, [a , a q ! q † The operator n = a a measures the particle n q q q operators, but with an index label q). P † ˆ † number the is Nparticle = qa q aq . - The operator number in the momentum mo Diagonalization in operator space: nBogoliubov transformation q = aq aq measures P † q2 ˆ number is N = q aq aq .- The dispersion relation is Eq = !ω = 2m . This is
Bogoliubov Quasiparticles
•
q † q
•
⇥
⇥
2 † ! q a u v q q q The trafo be canonical, [α , α ]= δ( −particle ), - ⇥ The⇥dispersion relation is Eq = !ω = 2m . This is the energy !of a single † vq uq a q † ! 2 2
• [α Lower The trafo be canonical, , α !T ] =and δ(
-
⇠
• At a finiteofT ,µµathits zero: belowqthis Tc the e • Lower T and study the behavior fixed n (3D): 2 q 2 1 ⇠q 2 2 ⇠ = ✏ + gn , ✏ = + 1), v = 1) ( q q 0 q E = ⇠ ✏ q 2M q 2 E q has q no • At a finite T q, µ hits zero: below this T the equation of state c
Rewrite the Hamiltonian with new operators and do normal ordering
H=V
gn 2
2
1 2
X
(✏q + gn0
Eq ) +
q6=0
•
study − ),the thusbehavior uq − vq =of 1. µ at fixed
The transformation coefficients are chosen make Tany • Lower T and study the behavior ofoff-diagonal µ study at fixed (2D): • toLower and thencontribution behavior ofinµ at fixed terms of new operators vanish. They can be chosen real and evaluate to
u2q = 12 ( Eqq
•
=
⇥
X
Eq ↵q† ↵q
q6=0
quasiparticle dispersion
Result: Collection of harmonic oscillators. New operators: elementary excitations on Bogoliubov ground state: quasiparticles † ↵ - q creates a quasiparticle qexcitation and ↵q |0Bog i =q0
-
Their dispersion is Eq =
q!0
2gn0 ✏q + ✏2q ! c|q|,
c=
gn0 m
At low momenta, this is linear and gapless
At high momenta, like free particles: quadratic
speed of sound
phonons
particles
Energy Correction and Density Depletion
•
Physical effects due to quantum fluctuations:
•
Correction to the total energy:
E/V = h0Bog |H|0Bog i/V = interpretation:
-
•
gn2 2
1 2
Z
interaction energy
d3 q (2⇡)3 (✏q
+ gn0
Eq )
kinetic energy from modes outside the condensate
Problem: linear high momentum UV divergence Reason: too naive treatment of interactions: formally assumed to be constant up to arbitrarily large momenta Cure: UV renormalization of interaction (will perform such program explicitly for fermions later) gn2 8⇡a 3 1/2 128 Result: E/V = 2 (1 + 15⇡ g = (na ) ) 1/2 M Depletion of the condensate (via normal ordering of the particle number operator):
n = n0 +
Z
q
ha†q aq i
= n0 +
Z
q
vq2
!
n
n0 n
8 ⇡ p (na3 )1/2 3 ⇡
expand in quasiparticle operators
na3 = (a/d)3
interpretation: particles kicked out of condensate smallness parameter recovered
Phonon Mode and Superfluidity
•
Landau criterion of superfluidity: frictionless flow - Gedankenexperiment: move an object through a liquid with velocity v. - Landau: the creation of an excitation with momentum p and energy ✏p is energetically unfavorable if
✏p v < vc = p ➡ in
•
this case, the flow is frictionless, i.e. superfluidity is present
Weakly interacting Bose gas: Superfluidity through linear phonon excitation
✏p = c|p|, c =
•
q
gn0 m
! vc = c
Free Bose gas: No superfluidity due to soft particle excitations
phonons
p2 ✏p = ! vc = 0 2m
➡ Superfluidity is due to linear spectrum of quasiparticle excitations
particles
Idea of Landau Criterion
• •
moving object
v
⌃0
Consider moving object in the liquid ground state of a system Question: When is it favorable to create excitations?
•
ground state system
General transformation of energy and momentum under Galilean boost with velocity v
•
⌃: w/o moving object
E,
p
0 with moving ⌃ object :
E0 = E
⌃
total system mass
pv + 12 M v2 ,
p0 = p
Mv
Energy and momentum of the ground state w/o moving object
E0 ,
with moving object ⌃0 :
E00 = E0 + 12 M v2 ,
•
p0 = 0
p00 =
Mv
Energy and momentum of the ground state plus an excitation with momentum, energy w/o moving object ⌃:
Eex = E0 + ✏p ,
with moving object ⌃0 :
0 Eex = E0 + ✏ p
•
pex = p pv + 12 M v2 ,
p0ex = p
Mv
Creation of excitation unfavorable if 0 Eex
E00
= ✏p
pv
✏p
|p||v| > 0
✏p ) v < vc = p
p, ✏p
Summary
•
The basic phenomenon of Bose-Einstein condensation is a statistical effect, not driven by interactions.
•
Ultracold bosonic quantum gases realize a situation close to that: model Hamiltonians with weak, local, repulsive interactions.
• •
Important scale hierarchy:
•
GP mean field equation = nonlinear Schrödinger equation for macroscopic wave function (hallmark: quantized vortices).
•
Bogoliubov theory encompasses quadratic fluctuations around GP mean field and explains superfluidity through existence of phonon mode.
a
d
dB
In consequence, such systems are well described by relatively simple approximations: at T=0, the physics is well understood in terms of GP equation + quadratic fluctuations.
Tutorial: Functional Integrals and Effective Action
Z = tre
ˆ H
=
Z
⇤
D(' , ')e
⇤
S[' ,']
Feynman’s formulation of quantum mechanics
Formula Summary: Functional Integrals I Functional Integrals for Statistical
Mechanics
• Some familiarity with functional integrals is assumed but we refresh our knowledge:
• Given a grand canonical Hamiltonian , e.g. with general two-body interactions, normally ordered ! ! † ˆ = (hij − µδij )ai aj + Vijkl a†i a†j ak al H − µN ij
ijkl
• Quantum partition function Z = tre
ˆ) −β(H−µN
!
=
ˆ
"n|e−β(H−µN ) |n#,
β = 1/kB T
{n}∈Fock space
• Here the partition function is represented in Fock space. We now perform a basis change to coherent states leading to the functional integral • Coherent states (bosons) – eigenstates to the annihilation operators ai : ai |φ#
•
properties:
|φ# "θ|φ# 1Fock
= φi |φ#, "φ|a†i = "φ|φ∗i =e =e =
!
i
!
∗ i θ i φi
φi a†i
|vac# , "φ|φ# = e
dφ∗ i dφi − e i π
´"
explicit form
• Note: The creation operators do not have eigenstates
!
i
!
i
φ∗ i φi
φ∗ i φi
|φ#"φ|
overlap and normalization completeness
Functional IntegralsIntegrals II Formula Summary: Functional • Insert identity into quantum partition function ˆ) −β(H−µN
ˆ
= tre = {n}∈Fock space !n|e−β(H−µN ) |n" ! ´ " ˆ) ∗ − i φ∗ φ −β(H−µ N ∗ i i = D(φ , φ)e !φ|e |φ", D(φ , φ) ≡ i
Z
!
dφ∗ i dφi π
ˆ (a† , ai ), we can apply the Feynman strategy of dividing • For a normal ordered Hamiltonian H(a†i , ai )−µN i the (imaginary) time interval β into N segments ∆β = β/N . Note |φn=0 " = |φn=N " – periodic boundary conditions ! ´ − i φ∗ ∗ i,n φi,n |φ "!φ | after each time step, the respective • Inserting the identity 1Fock = D(φn , φn )e n n matrix elements can be calculated explicitly due to the smallness of ∆β with result Z=
ˆ # N
D(φ∗n , φn )e−
!N
n=0 [
!
i
∗ ∗ φ∗ i,n (φi,n −φi,n−1 )+∆β(H(φi,n ,φi,n−1 )−µN (φi,n ,φi,n−1 ))
n=0
- Here, H − µN is a function of classical, but fluctuating variable. - The additional index n labels the (imaginary) time evolution. Continuum notation for N → ∞: ∆β
N $
n=0
→
ˆ
0
β
dτ,
φi,n → φi (τ ),
φi,n − φi,n−1 → ∂τ φi (τ ), ∆β
N #
n=0
D(φ∗n , φn ) → D(φ∗ , φ)
Functional Functional Integrals Integrals III • Continuum notation for the imaginary time dependence ´ ∗ −S[φ∗ ,φ] Z = D(φ , φ)e S[φ , φ]
=
´β
H[φ∗ , φ]
=
"
∗
0
dτ
!"
∗ φ i i (τ )(∂τ
∗ ij hij φi φj +
• S is the “classical” or “microscopic” action
"
∗
− µ)φi (τ ) + H[φ , φ]
ijkl
Vijkl φ∗i φ∗j φk φl
#
• Continuum notation for the discrete indices: - interpret indices as spatial indices - consider local density-density interactions Vijkl = vδik δjl δil and hij such that it describes kinetic energy (1D) $ $ ∗ ∗ ∗ (φ∗i φi )2 h(φi+1 − φi )(φi+1 − φi ) + v H[φ , φ] = i
ij
- the sites i, i + 1 be separated by a distance a. The continuum limit a → 0 obtains for fixed ϕ(τ, x) = a−1/2 φi (τ ), 1/2M = ha2 , g/2 = va , such that ∗
S[ϕ , ϕ]
=
´β 0
dτ dx[ϕ∗ (τ, x)(∂τ −
# 2M
− µ)ϕ(τ, x) + g2 (ϕ∗ (τ, x)ϕ(τ, x))2 ]
• S is the “classical” or “microscopic” action of nonrelativistic continuum bosons with local interaction in one spatial dimension
temperatures andfrequencies Matsubara FiniteFinite temperatures and Matsubara
frequencies
• Unlike the spatial integrations, the imaginary time integrations are restricted to a finite interval
• For bosons, we have used periodic boundary conditions, i.e. ϕ(τ = β, x) = ϕ(τ = 0, x)
• This gives rise to a discreteness in the frequency domain (cf. particle in a box problem) for finite β < ∞ (T > 0): Matsubara modes ´ ! iωn τ ϕ(τ, x) = T n ϕ(ωn , x)e , ϕ(ωn , x) = dτ ϕ(τ, x)e−iωn τ , ωn
= 2πnT
• E.g. the free part of the action reads (β S[ϕ∗ , ϕ] =
" n
T
−1
´β 0
ˆ
dτ ei(ωn −ωm )τ = δωn ,ωm )
dx[ϕ∗ (ωn , x)(iωn −
# 2M
− µ)ϕ(ωn , x)]
• In the zero temperature limit, the frequency spacing ∆ω = 2πT → 0 and the Matsubara summation ´ dω ! goes over into a continuum Riemann integral, n T → 2π
'(⌧ = , x) = '(⌧ = 0, x)
= 1/T
Matsubara torus
Formula Summarysummary Formula • Functional integral representation of the quantum partition function for ultracold bosons (3D): ´ ∗ −S[ϕ∗ ,ϕ] Z = D(ϕ , ϕ)e S[ϕ , ϕ] =
´β
=
´β
∗
0
0
dτ dτ
!"´ ´
3
# $ ∗ d xϕ (τ, x)(∂τ − µ)ϕ(τ, x) + H[ϕ (τ, x), ϕ(τ, x)]
3
%
∗
∗
d x ϕ (τ, x)(∂τ −
# 2M
− µ)ϕ(τ, x) +
g ∗ 2 (ϕ (τ, x)ϕ(τ, x)) 2
&
• Discussion: - The validity of the microscopic action is based on the validity of the microscopic Hamiltonian - The boson fields ϕ∗ , ϕ are commuting but temporally and spatially fluctuating complex numbers • Fermions: The same form of the microscopic action is obtained when representing the partition function for continuum fermion fields ψ. Important differences are: - Additional spin index, for two-component fermions: ψ(τ, x) = (ψ↑ (τ, x), ψ↓ (τ, x))T - Due to the anticommutation relations of the fermion operators, the fermionic fields are anticommuting Grassmann numbers, {ψσ (τ, x), ψσ∗ " (τ, x)} = {ψσ (τ, x), ψσ" (τ, x)} = {ψσ∗ (τ, x), ψσ∗ " (τ, x)} = 0 - They obey antiperiodic boundary conditions in imaginary time ψσ (τ = β, x) = −ψσ (τ = 0, x), thus ωn = (2n + 1)πT - Obviously, these properties will strongly affect the concrete evaluation of the functional integrals
From the Partition Function Function to the Effective From the Partition to Action the Effective
Action
• We have stored the information on the many-body system in the partition function ´ ∗ −S[ϕ∗ ,ϕ] Z = D(ϕ , ϕ)e • Of interest are the low order correlation functions such as e.g. !ϕ(τ, x)ϕ∗ (0, 0)"
• They can be calculated from introducing (imaginary time and space dependent) artificial sources into the partition function: ˆ ´ −S[ϕ∗ ,ϕ]+ τ,x (j ∗ (τ,x)ϕ(τ,x)+j(τ,x)ϕ∗ (τ,x)) ∗ ∗ Z → Z[j (τ, x), j(τ, x)] = D(ϕ , ϕ)e • and taking (functional) derivatives evaluated at vanishing sources, e.g. ! 2 δ Z ! !ϕ(τ, x)ϕ∗ (0, 0)" = ∗ ! ∗ δj (τ.x)δj(0, 0) j=j =0
• Remarks: - functional derivatives are defined with δf (x)/δf (y) = δ(x − y) plus the standard algebraic rules for differentiation - successive differentiation generates the “disconnected correlation functions”. The free energy W [j ∗ , j] = log Z[j ∗ , j] generates the “connected correlation functions”
Le
From the Partition Function to the Effective Action From the Partition Function to the Effective Action II • The information accessible from Z[j ∗ , j] or W [j ∗ , j] via derivatives wrt an unphysical source (j ∗ , j)
• There is a way to store the information in a more intuitive way: Legendre transform of W (cf. classical mechanics, or thermodynamics) ˆ δW [j ∗ , j] ∗ ∗ ∗ ∗ Γ[φ , φ] = −W [j , j] + j φ + jφ , φ(τ, x) = = "ϕ(τ, x)# δj(τ, x) • Properties and Discussion: - The Legendre transform implements a change of the active variable (j ∗ , j) → (φ∗ , φ) - The expectation value φ = "ϕ# is called the classical field. For ultracold bosons, it has a direct physical interpretation in terms of the condensate mean field - The effective action carries the same information as Z and W , only organized differently. It generates the one-particle irreducible correlation functions
From the Partition Function to the Effective Action From the Partition Function to the Effective Action III • The effective action has a functional integral representation ˆ exp −Γ[φ∗ , φ] = D(δϕ∗ , δϕ) exp −S[φ∗ + δϕ∗ , φ + δϕ],
δΓ[φ∗ , φ] = j(τ, x) = 0 δφ(τ, x)
• Discussion - NB: Action principle is leveraged over to full quantum status (see below) - The effective action can be understood “classical action plus fluctuations”. It lends itself to semiclassical approximations (small fluctuations around a mean field) - Symmetry principles are leveraged over from the classical action to full quantum status
additional material
Effective Action and Propagator
Effective Action and Propagator
• A useful property: The second derivative of the effective action is the inverse propagator
• Consider a propagation amplitude (in matrix notation)
δ2W (2) = Wij Gij = !δϕi δϕj " = δji δjj the indices could stand for e.g. !δϕ(τ, x)δϕ∗ (0, 0)"
• The above statement is then poperly formulated as the identity !
(2)
Γik Gkj = δij
k
• Proof:
! k
(2) Γik Gkj
=
! k
! δ 2 Γ δφk δ2Γ δji δ2Γ δ2W = = = = δij δφi δφk δjk δjj δφi δφk δjj δφi δjj δjj k
we have used the chain rule and the field equation for the effective action δΓ/δφi = ji
Functional Integral for Weakly Interacting Bosons
Some Introductory Remarks
•
This section develops the theory of low temperature weakly interacting boson degrees of freedom from the functional integral point of view. The aim is threefold:
(1) Reproduce the results of Gross-Pitaevski and Bogoliubov theory (2) Get some familiarity with the functional integral: symmetries, approximations, etc. (3) The emergence of bosonic quasiparticles at low energies is ubiquitous in a large variety of low temperature quantum systems: - The concept of quasiparticles/ effective low energy theories is particularly accessible in the functional integral formulation - Examples of emergent bosonic low energy theories:
• • •
BCS theory (Cooper pairs) Even more explicitly, BCS-BEC crossover (bound states of fermions: molecules) spin waves in magnetic lattice systems
- A rather universal understanding of such theories is crucial, and accessible within the framework developed here
Recovering the Gross-Pitaevski Equation
•
Quantum equation of motion: action principle for the Effective Action
0=
•
⇤ (X)
/
Z
D( '⇤ , ')
h
S i e ⇤ (X)
S[ + ']
X = (⌧, x)
explicitly:
0 = (@⌧
4 2M
µ)h'(X)i + gh'⇤ (X)'2 (X)i
= (X)
➡ in general, needs information on 3-point correlation, which needs information on 5-
point correlation...: infinite hierarchy
•
however, at very low temperature and weak interaction:
➡ GP Approximation: neglect O(V
0 = (@⌧
4 2M
➡ “imaginary time” GP equation
1/2
'(X) = (X) + '(X) O(V 0 ) O(V
)
µ) (X) + g
⇤
(X)
2
(X)
1/2
)
Recovering the Gross-Pitaevski Equation
Recovering the Gross-Pitaevski Equation
•
Indeed, time was introduced rather formally. But the connection to real times is readily made:
• We consider the imaginary time classical action S and and view it as being analytically continued from the real axis (at T = 0 or β → ∞): τ S[φ∗ , φ]
˜ x) ⇒ φ(τ, x) → φ(t, ! ´ ´ ˜ = i dt d3 x φ˜∗ (t, x)(−i∂t − → iS[φ˜∗ , φ] → it,
" 2M
˜ x) + g (φ˜∗ (t, x)φ(t, ˜ x))2 − µ)φ(t, 2
"
• We derive the field equation of motion for the real time classical action δS/δ φ˜∗ (t, x) = 0 # $ ∗ 1 ˜ ˜ ˜ ˜ x) i∂t φ(t, x) = − 2M % − µ + g φ (t, x)φ(t, x) φ(t,
• This is precisely the Gross-Pitaevski equation for a weakly interacting Bose-Einstein condensate
• Remark: “classical” refers to the absence of fluctuations. Physically, the global phase coherence implied in this equations is a quantum mechanical effect, with observable consequences: cf. discussion of quantized vortices (see below for more discussion)
⌧ = it
Wick rotations
Im
!E = i!M
!M
t Complex time plane
Re
Im
note
!M · t =
Complex energy plane
!E · ⌧
Re
Symmetries of the Microscopic Action Symmetries of the microscopic action • Gross-Pitaevski action: ˜∗
˜ = S[φ , φ]
ˆ
dt
ˆ
! d x φ˜∗ (t, x)(−i∂t − 3
" 2M
˜ x) + − µ)φ(t,
g ˜∗ ˜ x))2 ( φ (t, x) φ(t, 2
"
• Symmetries: - Most important is the invariance under global phase rotations U (1). Such transformation is imple˜ x) → eiθ φ(τ, ˜ x), φ˜∗ (τ, x) → e−iθ φ˜∗ (τ, x). mented by φ(τ, - The associated conserved Noether charge (of the full effective action) is the total particle number ´ 3 N = d x#ϕ∗ (0, x)ϕ(0, x)$ (NB: needs U (1) symmetry plus linear time derivative) - Further continuous “equilibrium” symmetries: time and space translation invariance (energy and momentum conservation) and Galilean invariance (center of mass momentum)
Mean Field Action and Spontaneous Symmetry Breaking
•
Z
Homogeneous action: time- and space independent amplitudes ( volume)
S(
• •
•
⇤
, ) = V /T ( µ
⇤
+ g2 (
⇤
d⌧ d3 x = V /T -- quantization
)2 )
homogeneous GPE or equilibrium condition:
@S 0= = @
µ+g
⇤
)
µ=
S(
•
for the ground state, the system chooses spontaneously the direction: spontaneous symmetry breaking (symmetry: global phase rotations U(1))
• •
Radial (amplitude) excitations: cost energy, gapped mode
Next steps:
⇤
0
s, in the classical limit we have ∗ Γ[φnonperturbative , φ] = S[φ∗ , φ] approaches such as (functional) In these cases, immediate need for RG (Castellani& ’04, Wetterich & ‘08,’09; Kopietz & ’08,’10; Dupuis ’09)
•
•
even in d=3, or d=2,T=0, one should be suspicious since in the range of small momenta the power counting is questionable: Z nq = h '⇤Q 'Q i ⇠ 1/Eq ⇠ 1/|q| divergent occupation number !
! ' ⇠ 1/|q|1/2
?!
Validity of Bogoliubov Validity of Bogoliubov Theory
Theory
• We study perturbative corrections to the self-energy for weakly interacting bosons (zero temperature): The full quadratic part of the effective action is (Q = (ω, q) ) 1 Γ= 2
ˆ
Q
! " ∗ ϕ(−Q), ϕ (Q)
#
q2 2M
Σan (Q) −iω + − µ + Σn (Q) q − µ + Σn (Q) Σan (Q) iω + 2M 2
%$ϕ(Q) ϕ∗ (−Q)
&
• We may view Bogoliubov Theory as the zero order self energies Σ(0) n (Q) = 2gρ0 ,
Σ(0) an (Q) = gρ0
⇢0 =
⇤ 0 0
• The leading pertrubative corrections are shown diagrammatically. The second diagram has in IR divergence (log in d=3, poly in d 0 to ensure a fixed particle number • Local s-wave two-body density-density interactions between the two spin states are described by Sint
g = 2
ˆ
β
dτ dx[ψ † (τ, x)ψ(τ, x)]2
0
The precise relation of the coupling constant g < 0 to the scattering length a is discussed below • Validity: as for bosons, in particular, a " d ∼ k −1 F
• Symmetries: as for the boson action, supplemented by global SU (2) spin rotation invariance
Cooper and Hubbard-Stratonovich Cooper pairs andPairing Hubbard-Stratonovich Transformation I
Transformation
• We perform a Hubbard Stratonovich transformation, which makes the expected pairing of the fermions # 0) explict. There are 3 steps: (!ψ↑ ψ↓ " = 12 !ψ T "ψ" =
1. Rearrange the fermion fields into local singlets, to make the pairing explicit (Fierz transformation) ! " 0 1 (ψ † ψ)2 = (ψ † δψ)2 = − 12 (ψ † "ψ ∗ )(ψ T "ψ), ε = −1 0 ´
2. Introduce a properly written factor of unity into the fermionic functional integral Z = Dψ exp −(Skin + ´ ´ ´ # T $† ´ † ∗ Sint ), (note ψ "ψ = −ψ "ψ ; we use X = (τ, x), X = dτ x, Dψ = D(ψ ∗ , ψ)) 1
=N
´
Dϕ exp −
´
X
m2 (ϕ∗ +
h † ∗ 2m2 ψ "ψ )(ϕ
−
h T 2m2 ψ "ψ)
The parameters m2 , h are real and m2 > 0 to make the Gaussian integral convergent but other$ wise arbitrary. The resulting partition function is (D = ∂τ − 2M − µ) ˆ ˆ % † & 2 ∗ † ∗ ∗ T † ∗ T h 1 h2 ψ Dψ + m ϕ ϕ + 2 (ϕψ "ψ − ϕ ψ "ψ) − 4 (g + m2 )(ψ "ψ )(ψ "ψ) Z = N Dψ)Dϕ exp − X
h2 − m2
3. For attractive interaction g < 0, we can now choose the parameters such that g = - Then we get rid of the fermion interaction term - Only this ratio is physical. We can redefine ϕ → hϕ and get ˆ ˆ % † & † ∗ ∗ T 1 ∗ 1 ψ Dψ + |g| ϕ ϕ + 2 (ϕψ "ψ − ϕ ψ "ψ) Z = N DψDϕ exp −SHS , SHS [ψ, ϕ] = X
Cooper Pairing and Hubbard-Stratonovich Transformation Cooper pairs and Hubbard-Stratonovich Transformation II • We discuss the Hubbard-Stratonovich action ˆ ! † ψ Dψ + SHS [ψ, ϕ] = X
1 ∗ |g| ϕ ϕ
+
† ∗ 1 (ϕψ #ψ 2
∗
T
" − ϕ ψ #ψ)
• The interacting fermion theory is mapped into a coupled fermion-boson theory.
• The boson field represents Cooper pairs, φ ∼ ψ↑ ψ↓ . A nonzero expectation value φ = #ϕ$ %= 0 breaks the U (1) symmetry and describes Cooper pair condensation • The theory is quadratic in the fermions. We introduce Nambu-Gorkov fields ΨT = (ψ T , ψ † ) to make this explicit, # $ ˆ † ∗ ! T (2) " 1 −ϕ # −Dδ 1 ∗ SHS = Ψ S [ϕ]Ψ + |g| ϕ ϕ , S (2) [ϕ] = Dδ ϕ# 2 X They can be eliminated by Gaussian integration. However, the result is an interacting boson theory due to the dependence S (2) [ϕ]
• The BCS approximation consists in neglecting the fluctuations of the bosonic Cooper pair field: Mean field approximation for the boson dofs
BCS Effective BCS Effective Action I
Action
• We implement the above strategy on the effective action for homogeneneous classical field configurations, 1 ´ Γ[ψcl , φ]/(V /T ) = − log DδψDδϕ exp −SHS [ψcl + δψ, φ + δϕ] ´ V 1 ∗ = = |g| φ φ − log T
Dδψ exp −SHS [δψ, φ]/(V /T )
• Due to the approximation on ϕ, SHS is now truly quadratic, in the background of a classical field φ. It is useful to switch to momentum space where ´ 1 SHS [δψ, φ] = 2 Q ΨT (−Q)S (2) [φ]Ψ(Q), S
(2)
[φ]
=
!
∗
−φ % (iωn + %q − µ)δ
−(−iωn + %q − µ)δ φ%
"
,
Ψ(Q) =
!
ψ(Q) ψ ∗ (−Q)
"
• NB: the classical field couples φψ↑∗ (Q)ψ↓∗ (−Q). Interpretation similar to Bogoliubov theory: creation of fermions with opposite momenta out of the Cooper pair condensate " (Q)
#(
Q)
⇤
annihilation process of two oppositely directed fermions into the pair condensate
1
- The fermion field expectation takes the physical value ψcl = 0 but serves to carry out successive derivatives to generate the 1PI correlation functions
BCS Effective Action BCS Effective Action II • Evaluation of the Grassmann Gaussian integral (2 log cosh x = Γ[ψcl , φ]/(V /T ) =
1 ∗ |g| φ φ
=
1 ∗ |g| φ φ
Eq
!
2 2 log(1 + x /((n + 1/2)π) )) n
− 21 tr log S (2) [φ]/(V /T ) − 2T
´
d3 q (2π)3
log cosh(Eq /2T ) + const.
" = ($q − µ)2 + φ∗ φ
• NB: The homogeneous BCS effective action depends on the chemical potential µ and on the U (1) invariant combination ρ = φ∗ φ, Γ = Γ[µ, ρ]
• •
we now derive two physical conditions from this effective BCS potential:
• •
the equation of state the condition for the transition into the BCS condensed phase
we will face problems at high momenta and provide the cure: UV renormalization
The Equation of State
The Equation of State
• The explicit expression for the density is obtained from ˆ ∂U d3 q nΛ = − = − (2π)3 tanh(Eq /2T ) ∂µ
U = /(V /T )
Again, the expression is UV divergent, here ∼ Λ3
• Using tanh(x/2) = 1 − 2(exp x + 1)−1 , we split the integral into a physical (depending on µ and ρ) and an unphysical contribution: " !ˆ ˆ 1 d3 q 1 d3 q − nΛ = 2 (2π)3 Eq /T (2π)3 2 e +1 The overall factor of two counts the degenerate spin states • Interpretation: - First term: involves the Fermi-Dirac distribution. The physical density is ˆ 1 d3 q n = 2 (2π)3 E /T e q +1 - Second term: Unobservable zero-point energy shift due to the fact that the functional integral does not respect operator ordering: consider single fermion mode field ψ and associated operator c ({c, c† } = 1), #ψ ∗ ψ$ = 21 #ψ ∗ ψ − ψψ ∗ $ = 12 #c† c − cc† $ = #c† c$ − 21
Spontaneous Spontaneous SymmetrySymmetry Breaking
Breaking
• The phase transition towards a state with macroscopic Cooper pairing at low T can be obtained from analyzing symmetry breaking patterns • Study the field equation/ equilibrium condition for the effective potential U[µ, ρ] = Γ[µ, ρ]/(V /T ), ∂U ! ∂U =φ· =0 ∂φ∗ ∂ρ • There are three possibilities:
1. Spontaneously symmetry broken phase:
φ, φ∗ != 0,
2. “Symmetric” phase:
φ = φ∗ = 0,
3. Critical point:
φ = φ∗ = 0,
∂U ∂ρ = 0 ∂U ∂ρ != 0 ∂U ∂ρ = 0
• NB: in the symmetry broken phase, ∂U ∂ρ = 0 signals the vanishing mass of the Golstone mode Cuts through potential landscape for U(1) symmetric theory
U(
⇤
)
Symmetric phase
U(
⇤
U(
)
Critical point
⇤
)
Symmetry broken phase
Critical Temperature and a Problem at Large Momenta Critical Temperature and a Problem at Large Momenta • We focus on the critical point for the Cooper instability first, i.e. we study 1 1 0=− − g 2
ˆ
3
d q (2π)3
1 tanh !q − µ
"
!q − µ) 2T
#
=
1 1 − 2 |g| 4π
ˆ
2
dq
∂U ! ∂ρ ρ=0
!
= 0. Explicitly:
q tanh 2 q /(2M ) − µ
"
2
q /(2M ) − µ) 2T
#
• The integral is linearly divergent, as the integrand tends to 1 for large momenta q → ∞: “Ultraviolet divergence” • This is because we have assumed spatially local interactions, i.e. constant in momentum space up to arbitrarily large momenta • In reality, this is not the case. There is a (smooth) momentum cutoff at large momenta. Details of the precise cutoff function should not matter (cf. discussion of effective theories above)
g(q)
Naive implementation of interaction True interaction potential: Smooth cutoff
Sharp cutoff
kF
a
1
1 ⇤ ⇠ aBohr
q
Ultraviolet Renormalization
Ultraviolet Renormalization
• The problem is cured by a proper ultraviolet (UV) renormalization procedure: - Introduce a (sharp) UV cutoff Λ in the momentum space integral (UV regularization) - Interpret g as “bare” coupling with cutoff dependence, g = gΛ - Trade the bare coupling for a physical observable: Require that ∂U/∂ρ performed in vacuum (µ = 0 (n = 0), T = 0, ρ = 0) produce the physical inverse scattering length (UV renormalization): 1 1 ∂U !! − 2 ! =− ∂ρ vac gΛ 4π
ˆ
Λ
q2 MΛ ! M 1 1 dq 2 − =− =− =− 2 q /(2M ) gΛ 2π gp 4πa
NB: The relation between gΛ and a is not perturbative; a more systematic treatment uses gΛ (q) = gΛ θ(Λ − q) and interprets the above equation as a resummed loop equation; the relation between a physical fermionic two-body coupling gp and a scattering lenght is a = M gp /(4π) • The renormalized equation for the critical temperature reads " # 2 $ % ˆ 2 2 q q /(2M ) − µ) 1 bosons: 8pi; − 1 bosons indistinguishable, 0=− − dq 2 tanh a π q − 2M µ 2T It will be analyzed below
fermions distinguishable (spin)
BCS Equations for the Critical Temperature BCS Equations for the Critical Temperature I • We have derived two equations governing the thermodynamics of weakly interacting fermions: - The equation of state ˆ 1 d3 q n = 2 (2π)3 (" −µ)/T e q +1 - The equation for the critical temperature 1 2 0=− − a π
ˆ
!
2
q dq 2 tanh q − 2M µ
"
2
q /(2M ) − µ) 2T
#
−1
$
• Given µ we could solve the second equation. For T /TF " 1, the corrections to the chemical potential from "F = µ(T = a = 0) are negligible. We therefore work with µ ≈ "F
The Critical BCS Equations for theTemperature Critical Temperature II ˜ = E/!F . The dimensionless critical temperature is • We rescale momenta and energies as q˜ = q/kF , E determined from ! 2 " 2 # $ ˆ 1 2 q˜ q˜ − 1) −1 0= − d˜ q 2 tanh ˜ |akF | π q˜ − 1 2T • For T˜ → 0, the integral develops a logarithmic divergence at the Fermi surface where q˜ = 1. Thus, the equation has a solution for arbitrarily weak interaction a if T is lowered sufficiently • This establishes the BCS transition with critical temperature π 8γ − 2|ak Tc F| = 2e !F πe
The exponential dependence reflects the log divergence. The prefactor is ≈ 0.61 with the Euler constant γ ≈ 1.78
• For T < Tc a gap ρ > 0 develops to cure the log divergence
divergent part of the integrand, and distribution function (dashed)
0
green:
T˜ = 0.1
blue:
T˜ = 0.01
red:
T˜ = 0.001
BCS Equations for the Gap at Zero Temperature BCS Equations for the Gap at T=0 • We have derived two equations governing the thermodynamics of weakly interacting fermions at T = 0: - The equation of state ˆ 1 d3 q n = 2 (2π)3 E /T e q +1 - The dimensionless equation for the gap ρ = φ∗ φ (from ∂U/∂ρ = 0 in the presence of symmetry breaking ρ != 0 and T = 0) " ! ˆ 2 1 2 q˜ 0= −1 − d˜ q ˜ |akF | π Eq # with Eq = (%q − µ)2 + ρ
• The scale T is exchanged for the scale ρ, wich now regularizes the log divergence. Again, for ρ/TF # 1, the corrections to the chemical potential from %F = µ(T = a = 0) are negligible. We therefore work with µ ≈ %F • This establishes the BCS gap at zero temperature (π/γ ≈ 1.76)
π 8 − 2|ak π Tc ρ(T = 0) | F = 2e = %F e γ %F
Fermion Gap and Superfluidity
• •
Below the critical temperature, a “gap” opens up This means that single fermion excitations are suppressed even very close to the Fermi surface (unlike the behavior above T_c). The spectrum is given by
Eq = 3.0 2.5
⇥
q2 ( 2M
2
µ) + ⇢
⇤1/2
2.0 1.5 1.0
“gap”
0.5 0.0 0.0
0.5
1.0
q⇠
•
p
⇢
1.5
✏F
2.0
q
The suppression of single fermion excitations is responsible for superfluidity:
• •
The low lying modes are bosonic phonons with linear dispersion This ensures superfluidity according to Landau’s criterion
Summary: Weakly Interacting Fermions
•
We have derived BCS theory within the effective action formalism - Condensation of the fermion field is impossible due to Pauli’s principle, but a second order pairing correlation can develop. - The BCS mechanism builds on two cornerstones:
• •
the sharpness of the Fermi surface, guaranteed by akF
⌧1
Cooper pairing of fermions close to the Fermi surface with opposite momenta
- Pairing is introduced in the functional integral by a Hubbard-Stratonovich transformation, mapping the purely fermionic to a fermion-boson theory - BCS theory integrates out the quadratic fermions and treats the bosons classically - The exponential dependence of critical temperature and gap at zero temperature persists to arbitrarily small interactions. It can be traced to a logarithmic divergence at the Fermi surface. - Like the condensation amplitude for bosons, the pairing correlation breaks the U(1) symmetry. The finite gap prevents single fermion excitations and leads to superfluidity
Experimental (Ir)relevance of Weakly Interacting Atomic Fermions
•
We compare the critical temperatures for a noninteracting BEC and weakly attractive fermions 1/2 - Free bosons of mass M undergo condensation at n dB = ⇣(3/2), dB = (2⇡/(M T )) 3 - Rewrite by using definitions from fermions n = kF /(3⇡ 2 ),
(BEC) Tc
✏F
= 4⇡(3⇡ 2 ⇣(3/2))
2/3
✏F = kF2 /(2M )
⇡ 0.69 = O(1)
- In contrast, the BCS critical temperature is exponentially small for akF ⌧ 1 (BCS) Tc
✏F
8 = 2e ⇡e
⇡ 2|akF |
⇡ 0.61e
⇡ 2|akF |
•
additionally, cooling of degenerate fermions is experimentally more challenging due to Pauli blocking
•
On the other hand, note a (formal) exponential increase of T_c for rising towards strong interactions
• •
Q: What is the fate of the exponential increase in T_c for rising akF A: BCS-BEC crossover
akF i.e.
Strong Interactions and the BCS-BEC Crossover
0
(akF )
1
Physical picture: BCS-BEC Crossover
•
We have discussed two cornerstones for quantum condensation phenomena:
• weakly interacting bosons
• fermions with attractive interactions ➡
BCS superfluidity at low T
➡
Bose-Einstein Condensate (BEC) at low T bosons could be realized as tightly bound molecules (“effective theory”)
Physical picture: BCS-BEC Crossover
•
We have discussed two cornerstones for quantum condensation phenomena:
• weakly interacting bosons
• fermions with attractive interactions ➡
BCS superfluidity at low T
➡
Bose-Einstein Condensate (BEC) at low T bosons could be realized as tightly bound molecules (“effective theory”)
0
•
(akF )
1
There is an experimental knob to connect these scenarios: Feshbach resonances 1
• microscopically, the phenomenon is due to a bound state formation at the resonance ak = 0 F • from a many-body perspective, the phenomenon is understood as • • ➡
Localization in position space Delocalization in momentum space
In the strongly interacting regime, no simple ordering principle is known: Challenge for Many-Body methods
Microscopic Origin: Feshbach Resonances
•
Start from fermions: (Euclidean) Action
S [ ]=
•
fermion field: two hyperfine states
Z
†
d⌧ d3 x
(@⌧
4 2M )
+
2
(
†
=
✓
" #
◆
)2
Consider a second interaction channel with bound state close to scattering threshold V=0, detuned by
⌫
V(r) ``closed channel''
⌫
DE ``open channel''
|⌫, kB T | ⌧ | E|
•
Detuning
⌫ can be controlled with magnetic field ⌫(B) = µB (B
magnetic moment
B0 ) resonance position
examples:
r
6
Li,40 K
Microscopic Origin: Feshbach Resonances
•
bosonic molecule field:
S [ ]=
Z
interconversion: SF [ , ] = Feshbach, Yukawa term
"
h
⇤
# molecule formation
•
⌫
Effective Model to describe this situation: Interconversion of two fermions into a molecule
V(r)
``closed channel''
DE ``open channel''
d
d⌧ d x
h
Z
⇤
4 4M
(@⌧
⇤
d⌧ dd x
"
|⌫, kB T | ⌧ | E|
+ ⌫)
#
r
⇤ "
⇤ #
•
NB: cf. BCS Cooper pairing with condensate amplitude: ⇤ 0 = const.
•
Now we allow for dynamic bosonic degrees of freedom
⇤
(⌧, x)
Parameters:
• (background scattering in open channel) • Feshbach coupling: width of resonance h ⌫ • detuning: distance from resonance
or
⇤
(!, q)
Relation to a strongly interacting theory
• •
Total action:
S = S + S + SF [ , ]
Field equations:
S ⇤
•
Formally solve for
•
=0
)
=
+ ⌫) = h
@t
h 4 4M
+⌫
"
" #
, and insert solution into the Feshbach term
Z
3
d⌧ d x
"
#
h2 4 2M +⌫
@t
† "
"
† #
h ! 1, S!S +
Z
3
d⌧ d x
h2 ⌫ † "
† #
! const. "
h
#
take constrained “broad resonance” limit: pointlike interactions
h2 ⌫
#
⇤
,
S=S +
4 4M
) (@t
#
=S
1 h2 2 ⌫
Z
d⌧ d3 x (
h 1 ⌫
†
† " † #
)2
Relation to a strongly interacting theory
•
pointlike/broad resonance limit: The action takes the form
S[ ] =
Z
d
d⌧ d x
e↵
=
†
(@t
h2 ⌫(B)
effective fermionic interaction
•
remember
⌫(B) = µB (B
4 2M )
a(B) eff
+
2
(
)
2
eM B
a=
4⇡ eff M
scattering length a and binding energy
scattering length (nonidentical fermions)
B0 )
➡ resonant (divergent) interaction at B_0
•
†
abg
in the following, we shall ignore the background scattering for simplicity
observation of divergent scattering length Ketterle Group, MIT (1999) bosonic sodium
Regimes in the BCS-BEC Crossover
•
Compare the scattering length to the mean interparticle spacing
d = (3⇡ 2 n)
1/3
➡ three regimes
a < 0, |a/d| ⌧ 1
|a/d| & 1
a > 0, |a/d| ⌧ 1
•
weakly interacting (dilute) fermions strong interactions, dense molecular bound states: dilute bosons ➡ see below!
We identify the inverse scattering length as an adequate “crossover parameter”
a
1
(B) =
M ⌫(B) 4⇡h2
since the Feshbach resonance is located at the zero crossing of the detuning ⌫(B)
•
Cf. microscopic justification: a/d > 1 does not invalidate the microscopic Hamiltonian (as could be suspected from the discussion of weakly interacting gases). The relevant ratio for the validity is rvdW /d, rvdW / dB ⌧ 1. Feshbach resonances violate the generic relation rvdW /a ⇡ 1 : “anomalously large scattering length”
Evaluation Strategy: Effective Action
• •
So far: microscopic physics of few scattering particles Quantize the many-body theory via functional integral for the effective action ➡ Consider S, the classical action:
S[ ]
S[ ]
=0
➡ The effective action is given by
[ ]=
•
log
Z
D
exp S[ +
]
Result:
•
Effective action includes all quantum and statistical fluctuations by integrating over all possible paths
•
Leverages over the intuition from classical physics: symmetries action principle, field equation
•
Lends itself for powerful modern field theory techniques
• •
[ ]
=0
Integrating out the fermions
•
For practical purposes, the Feshbach type action is favorable: - the potential bosonic bound state is already explicit - NB: can be seen as result of Hubbard-Stratonovich transformation of fermionic theory - for simplicity restrict to broad resonance limit, no background coupling
•
This theory is quadratic in the fermions
S[ ] =
Z
1 = 2
Z
(
T
,
(@t
T
]=
4 2M
(2) SF F
)
0
Q
µ) + (⌫ 2µ) !✓ ◆ 0 (2) SBB
✓
⇤
✏↵ h PF (Q) ↵
P F ( Q) ✏↵ h
PF (Q) = i ! +
⇤
h(
q2 2M
↵
µ
◆
(2) SBB
"
⇤ "
#
⇤ #
Nambu-Gorkov ✓ ◆fields
=
=
- Inverse propagator matrix depends on the fluctuating bosonic field
• •
⇤
spin indices
Fourier transform (2) SF F [
†
d⌧ d3 x
=
✓
⇤
✓
⇤
◆
0 ⌫
⌫ 2µ
2µ 0
◆
boson carries two atoms
fermions can be integrated out the result is a (nontrivial) purely bosonic theory, to be analyzed subsequently
Gaussian Integral for Fermions
•
0
Effective Action:
[
0,
0]
= =
log log
Z
Z
D
D
D
exp Sint [
=0
exp S[
0
0
(Pauli principle)
+
+
, ]
0
+
]
composite field expectation value (condensation for 0 6= 0)
with intermediate action
Sint [ ] = S
(cl)
[ ]
1 (2) (cl) log det SF F [ ] = S [ ] 2 Z
•
(⌫
2µ)
1 (2) Tr log SF F [ ]. 2
⇤
The Tr Log term features arbitrary powers of
(compatible with symmetries)
➡ This gives rise to an effective bosonic theory (Hertz-Millis approach)
Effective Quadratic Theory for Bosons
• •
Integrating out the fermions yields bosonic theory: Interpretation? Progress can be made by expansion in fluctuation field powers:
1 Tr log PF + F 2 (2)
=
= SF F [ ].
• •
1 Tr log PF 2
1 1 1 TrPF F + Tr(PF 1 F)2 + ... 2 4
O(
- PF depends on the condensate - F is the “fluctuation matrix” ⇠
)
vanishes due to field equation
0
O(
2
)
Good qualitative understanding at arbitrary temperature: Truncate after quadratic term Yields effective Bogoliubov theory for boson fluctuations SBog
1 1 1 2 = Tr(PF F) = 4 2
Z
( K),
⇤
K
-
(K)
✓
⇤ ⇤ 0 0
(K) P ( K)
P (K) (K) 0
The coefficients are given by the classical + fluctuation contributions, e.g.
P (K) = ⌫
h
2
Z
PF ( Q
K)PF (Q)
|2| |2| PF (Q)PF (Q Q
+ K)
= iZ ! +
A 2 k 4M
2
+ m + ... “mass term”
low energy/derivative expansion
0
◆✓
(K) ⇤ ( K)
K = (!, k) frequency and momentum
◆
Crossover Effective Potential I
•
The steps in summary (and a last one...): Effective Potential integration of fermion fluctuations
[
0]
=
log
Z
D
exp Sint [
0
+
]
quadratic expansion of boson fluctuations
=
Z
(⌫
2µ)
Q
⇤ 0 0
⇤ 1 Tr log P [ F 0 0] 2
log
Z
D
exp SBog [
]
integration of quadratic boson fluctuations
= V /T ( 2µ + ⌫)
⇤ 0 0
⇤ 1 Tr log P [ F 0 0] 2
+ 12 Tr log PB [
⇤ 0 0]
fermionic fluctuations bosonic fluctuations (approximation)
“classical contribution”: condensed bosons important in regime:
BEC
BCS
BEC
⌘ V /T U [
⇤
]
effective potential: homogeneous contribution to effective action
Crossover Effective Potential II
•
The Crossover Physics can now be extracted from the Effective Potential 1 2 Tr log PF [⇢; µ]
U [⇢; µ] = ( 2µ + ⌫)⇢ = (⌫
perform spin and frequency traces
Eq(F) Eq(B)
• •
= =
⇥
q2 ( 2M
⇥
2µ)⇢
⇤1/2 µ) + h ⇢ 2
2
2
2 2
(A q + m ) + 2
2T
Z
3
+ 12 Tr log PB [⇢; µ]
d q (F) log cosh(E /2T ) + T q 3 (2⇡)
Z
⇢=
d3 q (B) log sinh(E /2T ) q 3 (2⇡)
- single fermion excitation energies, coefficients from fermionic action 2
2
⇢(A q + m )
The effective potential depends on
µ
⇤1/2
- effective boson excitation energies
- coefficients: integrals from TrLog + derivative expansion 2 - mu-dependence: mainly m (µ)
and
⇢
These quantities will be determined by:
• •
The gap equation The equation of state
⇤ 0 0
Self-consistency relations for the solution of the crossover thermodynamics
recapitulation
The Gap Equation and Spontaneous Symmetry Breaking
•
field equation for the effective action:
•
simplifications:
• •
⇤ (x) , 0 (x)] 0 0 (y)
due to U(1) (global phase) symmetry, it only depends on the invariant ⇤ 0 0
0]
=
if the symmetry is broken spontaneously,
@U [⇢] @⇢
@U [⇢] @⇢ ⇤ 0
⇤ 0
⇢=
⇤ 0 0
=0
6= 0, then we must have
=0
gap equation (cf. fully analogous treatment in BCS theory)
•
=0
the effective potential depends only on homogeneous field configuration
@U [ @
•
[
p
⇢0
NB: This criterion based on symmetry breaking works throughout the whole crossover. At T=0, the symmetry will be broken for any value of scattering length. Therefore, there is no quantum phase transition, but only a crossover phenomenon
The Equation of State chemical potential
• We would like to consider a situation with fixed density • Our problem is formulated in the grand canonical ensemble, • The density can be obtained from the Zeffective potential: N/V = lim h x!0
=
•
†
(x)
(0)i = lim
x!0
D
[
†
(x)
S3
Z
dtd3 x( µ
(0)] exp S[
0
+
]/N
@U @µ
In our approximation, there are three additive contributions to U. Thus
n=2
=2 classical bosonic condensate contribution
➡
1 ⇤ 1 0 0 + 2 TrPF + ⇤ 0 0 + nF + nB
1 @m2 1 Tr 2 PB @µ
fermionic (quasiparticle) contribution
contribution of fermions bound in bosons
~ integral over Fermi distribution
~ integral over Bose distribution
effective bosonic propagator
in the presence of condensates
definite interpretations only possible in the BCS and BEC limits
†
)
The Extended Mean Field Theory of the BCS-BEC Crossover
•
The two self-consistency equations from the Effective Potential read - The Gap equation
@U 0= =⌫ @⇢
2µ
h2 2
Z
d3 q 1 [ (F) (2⇡)3 Eq
tanh
(F) Eq 2T
1
(F) |
Eq
=µ=0 ]
NB: Bosonic contribution neglected for simplicity
- The Equation of State @U @µ
n=
⇤ 0 0
=2
UV Renormalization: in complete analogy to BCS theory
+ nF + nB
• Solve for µ and ⇢ • Plot as a function of dimensionless crossover parameter µ ✏F
✏F =
2 kF 2M
Fermi energy
(a/d)
1
= (akF )
a
n=
1
2.5
1 2.0
1
M ⌫ 4⇡ h2
= 3 kF 3⇡ 2
Fermi momentum
= h2 ⇢
0
-1
1.5
-2
1.0
-3 0.5
-4
-5 -3
0.0 -3
-2
-1
0
1
2
(akF )
0
➡
-2
1
What do these solutions tell us?
-1
0
0
1
2
3
4
(akF )
1
The Extended Mean Field Theory of the BCS-BEC Crossover
•
The two self-consistency equations from the Effective Potential read - The Gap equation
@U 0= =⌫ @⇢
2µ
h2 2
Z
d3 q 1 [ (F) (2⇡)3 Eq
tanh
(F) Eq 2T
1
(F) |
Eq
=µ=0 ]
NB: Bosonic contribution neglected for simplicity
- The Equation of State @U @µ
n=
⇤ 0 0
=2
UV Renormalization: in complete analogy to BCS theory
+ nF + nB
• Solve for µ and ⇢ • Plot as a function of dimensionless crossover parameter µ ✏F
✏F =
2 kF 2M
1
(a/d)
= (akF )
a
n=
1
2.5
1 2.0
1
M ⌫ 4⇡ h2
= 3 kF 3⇡ 2
Fermi momentum
= h2 ⇢
0
-1
1.5
BCS Fermi energy regime -3
strongly interacting
BCS regime
BEC regime
strongly interacting
-2
1.0
BEC regime
0.5
-4
-5 -3
0.0 -3
-2
-1
0
1
2
(akF )
0
➡
-2
1
Discuss the limiting cases!
-1
0
0
1
2
3
4
(akF )
1
The limiting cases: BCS limit
•
µ ✏F
Solution above:
!1
BCS regime
µ ✏F
➡ Expression of a Fermi surface, weakly
interacting fermion gas is approached
1
0
-1
-2
•
Simplifications
-3
-4
➡ The EoS reduces to ⇤ 0 0
n=2
-5 -3
0=
⌫ 2µ h2
1 2
+ nF + nB ! nF
d3 q 1 [ 3 (F) (2⇡) Eq
tanh
⌫ 2µ h2
1 a
0
1
2
(F) Eq 2T
1 (F)
Eq
single fermion excitation spectrum
Eq(F) 3.0
|
1
=µ=0 ]
2.5
=
⇥
⇤1/2 µ) + h ⇢
q2 ( 2M
2
2
2.0 1.5
h ! 1, ⌫ / h2
➡ NB: For broad resonances:
0=
-1
(akF )
➡ The gap equation can be solved analytically
Z
-2
M 8⇡
!
Z
⌫ h2 3
=
tanh
“gap”
0.5
1 4⇡ a M
d q 1 [ (F) (2⇡)3 Eq
= h2 ⇢
1.0
0.0 0.0
0.5
1.0
q⇠ (F) Eq 2T
1 (F) Eq
|
cf. above: BCS Gap equation reproduced
=µ=0 ]
p
1.5
✏F
q
2.0
The limiting cases: BCS limit
•
nq
Interpretations:
•
Fermi surface
Strongly expressed Fermi surface
Fermi distribution
q
kF
➡ Scattering/Pairing highly local in momentum space
•
The result for the gap: ⇡ 2akF
= 0.61✏F e
Locality in momentum space
➡ Condensation is very weakly expressed: only
Delocalization in position space
Fermions close to Fermi surface contribute 1.0
⇢/n
0.8
slight cheating here, this plots the renormalized condensate (below)
0.6
0.4
0.2
0.0 -2
0
2
4
6
8
(akF )
•
1
2.5
= h2 ⇢
2.0
Comparison to full result
1.5
➡ Strong deviations from BCS result once
(akF )
1
⇠
1
1.0
0.5
0.0 -3
(akF ) -2
-1
0
1
2
3
1 4
BEC regime
The limiting cases: BEC limit µ ✏F
•
Solution above:
µ ✏F
!
1
0
-1
-2
1
-3
-4
-5 -3
-2
-1
0
1
2
(akF )
➡ Strong gap
µ develops on the normal (
†
1
)
sector of the inverse fermion propagator ➡ However, there is a piece from the anomalous part
that is independent of
µ
Eq(F)
➡ The fermion density can be written
nF = 12 Tr PF 1 !
m2
single Fermion excitation spectrum
@ m2 @µ
8
=
⇥
⇤1/2 µ) + h ⇢
q2 ( 2M
2
2
6
⇤ 0 0
is identified as the fluctuation correction to the bosonic mass term
µ/✏F ⇡ 1
4
µ/✏F ⌧ 0 2
0 0.0
0.5
1.0
➡ Interpretation?
= h2 ⇢
1.5
2.0
Renormalized Condensate
•
total mass term: classical + fluctuations
We consider the equation of state in the BEC regime:
n=2
•
⇤ 0 0
+ nF + nB ! (2
For µ
PB =
•
m2 @µ
)
⇤ 0 0
+ nB =
We consider the bosonic density further (T=0):
nB =
•
@
✓
!
@ 1 @µ 2 Tr log PB [⇢; µ]
⇡
@m2 @µ
@m2 @µ
⇤ 0 0
Bosonic propagator
Tr log PB 1 [⇢; µ]
1 the bosonic propagator matrix simplifies to
(K) ⇤0 ⇤0 P ( K)
P (K) (K) 0
0
◆
!Z
We observe that the expression
Z :=
2aZ i! +
q2 4M
⇤ ⇤ 0 0
+ 2aZ
i! + ⇤ 0 0
2 @m 1 2 @µ
is ubiquitous to all the formulas here
•
+ nB
We introduce a renormalized condensate density as
˜⇤ ˜ = Z
⇤ ⇤ 0 0
2
q 4M
+ 2aZ 2aZ 0 0
⇤ 0 0
!
The limiting cases: BEC limit
•
Expressing all quantities in terms of the renormalized condensate gives:
• •
Equation of state
n = 2 ˜⇤0 ˜0 + 2TrP˜B
1
= 2 ˜⇤0 ˜0 + 2˜ nB
with renormalized inverse boson propagator
P˜B = PB /Z =
2a ˜⇤0 ˜⇤0 q2 i! + 4M + 2a ˜⇤0 ˜0
i! +
2
+ 2a ˜⇤0 ˜0 2a ˜0 ˜0
q 4M
➡ Reduction to an effective theory of “renormalized” bosonic bound states
• • •
• • • •
Mass 2M Interaction strength 2a
Local objects in position space
Atom number 2
All reference to the concrete value of Z is gone in the renormalized quantities Macroscopic measurements probe the renormalized quantities Microscopic probes can measure Z, but this is beyond the scope here NB: While boson mass and atom number follow from symmetry, the interaction strength 2a is an approximation. The exact answer is 0.6a (four-body problem)
!
Connection to Scattering Physics: Vacuum limit
•
Heuristics: Study the gap equation (broad resonance) for
1 a
=
p
µ ✏F
!
1
µ · 2M
➡ The density scale k_F (also: temperature) have disappeared from the gap equation: ➡ only two-body physics left! ➡ Very different from BCS limit, cf. ✏F
= 0.61 e
⇡ 2akF
Connection to Scattering Physics: Vacuum limit
• More systematically: Project on physical vacuum by Gk!0 (vak) = lim Gk!0
- Diluting procedure: - Getting cold:
d T ⇠ eF
kF !0 ⇠ kF 1
molecular bound state formation
T /eF >Tc /eF =const.
!•
kF3 n= 2 3p
µ
1
0
-1
•
The chemical potential plays the role of half the binding energy
Ebind =
2
1/(M a )
-2
atom scattering threshold
-4
-5 -3
•
Ebind /2
-3
Smooth crossover terminates in sharp
-2
-1
0
1
2
(akF )
two-body vs. many-body
“second order phase transition” in vacuum
• NB: Using this vacuum procedure, the above functional integral treatment solves the two-body scattering problem -- as quantum mechanics would do
➡ microscopic origin of the crossover is bound state formation
1
Finite Temperatures
• •
So far: Crossover Physics at T=0
•
Finite temperature phase diagram:
The above formalism is readily applied to finite temperature: Frequency integration -> Matsubara sum
Tc ✏F
0.30
BCS limit: BCS theory
Normal state 0.25
BEC limit: Free bosons of atom number 2, mass 2M
0.20
0.15
Superfluid state
0.10
(BEC)
Tc
✏F
0.05
0.00
-2
0
2
4
6
8
(akF )
1
= 2⇡(6⇡ 2 ⇣(3/2))
2/3
⇡ 0.218
Challenges beyond Mean Field Beyond mean field effects and challenges at very different scales: critical behavior: long distance scales
Tc
kld
1/2
n1/3 , T 1/2 , eM
0.30
0.25
0.20
few-body physics of effective dimers: microscopic scales
0.15
Many-Body fermion physics: Thermodynamic scales
kF3 n = 2 ,T 3p
0.10
eM =
0.05
0.00
-2
two-body bound state
0
2
4
zero crossing of fermion chemical potential
6
1 Ma2
8
(akF )
1
Strategy: Find an interpolation scheme which incorporates known physical effects in the limiting cases Methods: t-matrix approaches, 2PI Effective Action, Functional RG, ...
kF2 T, 2M
Results beyond Mean Field
•
Functional RG treatment: connecting micro- and macrophysics, unified treatment of physics on various scales microscopic
eM =
1 Ma2
thermodynamic
TTcc eF
long distance
kF3 n = 2 ,T 3p
1/2
n1/3 , T 1/2 , eM
kld
0.35 0.30
FRG S. Flörchinger, M. Scherer, SD, C. Wetterich
0.25 0.20 0.15
FRG +Rebos. Gorkov
0.10 0.05 0.00 !2
0
2
4
6
(ak c−1 F )
zero crossing of fermionOrange line: FRG withBlack line: FRG including particle-hole fluctuations; chemical potential out particle-hole fluctuations; Green line: BCS result; Red line: Gorkov’s correction; Yellow Free BEC; Blue line: physics Interacting FRG treatment of line: molecular scattering inBEC BECwith regime
1
FIG. 1:
• Accurate • Accurately reproduce Gorkov effect in the BCS regime from rebosonization procedure • Long wavelengths: second order phase transition
Experiments in the BCS-BEC Crossover ENS, 2004
vortices
MIT, 2005
Innsbruck, 2004
release energy pairing gap
JILA, 2005
Duke ´04, Innsbruck 2004 & `06
BEC
pair correlations
collective modes
BCS
Summary: BCS-BEC Crossover
•
We have discussed a simple approximation scheme based on the effective action capturing the qualitative features of the BCS-BEC crossover at zero and finite temperatures - The BCS-BEC crossover interpolates between the two cornerstones for quantum condensation phenomena, BCS and BEC type superfluidity - The microscopic origin is a Feshbach resonance. The divergence of the scattering length is accompanied by the formation of a bosonic bound state. There are three regimes for the many-body physics
a < 0, |akF | ⌧ 1 |akF | & 1 a > 0, |akF | ⌧ 1
weakly interacting (dilute) fermions dense regime, strong interactions molecular bound states: dilute bosons
- The crossover from BCS to BEC regimes may be viewed as a localization process in real space, or a delocalization in momentum space - Though there are profound quantitiative changes in the thermodynamics, the ground state symmetry properties are unchanged: Crossover instead of quantum phase transition
Lattice Systems: Outline Quantum Phase transitions: General Overview
T quantum critical region
• What is a Quantum Phase transition? • Example: Mott Insulator -- Superfluid transition
gc
g
Microscopic derivation of the Bose-Hubbard model • Atoms in optical potentials • Periodic potentials, Bloch theorem • Bose-Hubbard model Phase diagram of the Bose-Hubbard model
• Basic mean field theory: phase border and limiting cases • Path integral formulation: excitation spectrum in the Mott phase
3.0
MI 2.0
and bicritical point
SF
MI 1.0
0.0 0.0
MI 0.010
0.020
0.030
Quantum Phase Transitions: General Overview
T quantum critical region
gc
g
What is a quantum phase transition?
•
Consider a Hamiltonian of the form:
H = H1 + gH2 dimensionless parameter
• •
Study the ground state behavior of the energy
•
Two possibilities:
E(g) = hG|H|Gi
Quantum phase transition: Nonanalytic dependence of the ground state energy on coupling parameter g
E
• •
parts commute,
[H1 , H2 ] = 0
but eigenvalues have crossing
E
• •
parts do not commute, [H1 , H2 ] 6= 0 eigenvalues crossing develops in the thermodynamic limit
gap
EG
EG
gc
g low eigenvalues of H
gc
Literature: Subir Sachdev, Quantum Phase Transitions, Cambridge University Press (1999)
g
What is a quantum phase transition? H = H1 + gH2 E
E
gap
EG
EG
gc
g low eigenvalues of H
gc
g
•
The second possibility is more common and closer to the situation in conventional classical phase transitions in the thermodynamic limit
•
The first possibility often occurs only in conjunction with the second (ex: BoseHubbard phase diagram)
•
The phase transition is usually accompanied by qualitative change in the correlations in the ground state
What is a quantum phase transition?
• •
We concentrate on second order transitions (as those above) characteristic features: - vanishing of the energy scale separating ground from excited states (gap) at the transition point critical exponent
- universal scaling close to criticality,
⇠ J|g
gc |⌫zd
typical microscopic energy scale (in H)
- diverging length scale describing the decay of spatial correlations at the transition point ⇠ 1 ⇠ ⇤|g gc |⌫ typical microscopic length scale (e.g. lattice spacing)
- the ratio defines the dynamic critical exponent,
⇠⇠
zd
Quantum vs. Classical Phase Transition
• • •
A quantum phase transition strictly occurs only at zero temperature T=0 Temperature always sets a minimal energy scale, preventing scaling of Generic quantum phase diagram: T
- classical description of critical behavior applies if
Disordered
~!typ ⌧ kB T
quantum critical region (no gap) line of second order phase transitions
- This is always violated at low enough T: classical-quantum crossover
Ordered with symmetry breaking
(possibly) ordered without symmetry breaking (gapped)
gc
g
•
Phase transitions in classical models are driven by statistical (thermal) fluctuations. They freeze to fluctuationless ground state at T=0
•
Quantum models have fluctuations driven by Heisenberg uncertainty principle
➡ Quantum critical region features interplay of quantum (temporal) and statistical (spatial) fluctuations
Microscopic Physics of the Bose Hubbard Model (see appendix for more details)
Bosons in the Optical Lattice Bose Hubbard Model System: We consider N bosonic particles moving on a lattice (“lattice gas”) consisting of M lattice sites. The essential ingredients of the dynamics are • hopping of the bosonic particles between lattice sites (kinetic energy)
• repulsive / attractive interaction between the particles (interaction energy) • Bose statistics
Bose Hubbard (BH) Hamiltonian: ! † ! †2 ˆ = −J bi bj + 12 U bi b2i H
=
i
Tˆ + Vˆ
• Second quantized notation:1 b and b†i denote destruction and creation " #op-
erators for bosons at lattice site i obeying commutation relations bi , b†j =
δij . The occupation number operator for site i is n ˆ i ≡ b†i bi . $ ˆ • Kinetic energy: the first term is the kinetic energy T = −J b†i bj . It describes the hopping of electrons between adjacent lattice sites (notation
Cold Atoms in Optical Lattices Atoms in Optical Lattices Optical Lattices
⇠ aB
extent of atom much smaller than laser wavelength
aB ⌧
• AC-Stark shift - Consider an atom in its electronic ground state exposed to laser light at fixed position !x. - The light be far detuned from excited state resonances: ground state experiences a secondoder AC-Stark shift
δEg = α(ω)I
-
with α(ω) - dynamic polarizability of the atom for laser frequency ω, I ∝ E!2 - light intensity. Example: two-level atom {|g" , |e"}.
! x, t) = !eE sin kx e−iωt + h.c., AC• For standing wave laser configuration E(! StarkRabishift is a function of position: It generates an optical potential frequency
⇥ Eg = 4
2
Ω2 (!x) Vopt (!x) ≡ δEg (!x) = ! !eg 4∆
Example: for a two-level atom {|g" , |e"} in the RWA the AC-Starkshift is given detuning from =x) eg resonance Ω2 (" by δEg (!x) = ! 4∆ with Rabi frequeny Ω and detuning ∆ 0 g ⇥ Note that for red detuning (∆ < 0) the ground state shifts down δEg < 0, while for blue detuning (∆ > 0) we have δEg > 0. Optical Lattice (here 1D):
laser light at fixed position x. If the light is far detuned from excited state reso-
for blue detuning (⇥state > 0)will weexperience have ⇥Eg > 0. nances, the ground a second-oder AC-Starkshift shift⇥E⇥E (x) = nances, the ground state will experience a second-oder AC-Stark g (gx) = (⌥)I( x)Lattice with (⌥) (⌥) the1D): dynamicpolarizability polarizabilityofofthe theatom atomfor forfrequency frequency⌥,⌥,and and (⌥)I( x) with the dynamic Optical (here Atoms inintensity. Optical Lattices I( x) the light I(x) the light intensity. i t • standing wave laser configuration: E( x, t) = eE sin kx e + h.c. is given Example: for a two-level atom {|g⌥ , |e⌥} in the RWA the AC-Starkshift
Example: for a two-level atom {|g⌥ , |e⌥} in the RWA the AC-Starkshift is given 2 ⇥2 (⇥ (⇥ x)) a function of position ⇥ xas •by ⇥E AC Starkshift with Rabi Rabi frequeny frequeny⇤⇤and anddetuning detuning⇥⇥==⌥⌥ ⌥eg ⌥Bloch ⇧⇥). ⇥). by ( x) = eg(⇤(⇤ ⇧ ⇥Egg (x) = 44 with bands Note that forStarkshift red detuning detuning (⇥ 0.0. > of the (⇥ atom
⇥ Optical⌃( Lattice (here 1D): 2 Lattice (here 1D): laser laser ⇤2 (x) x, t) 2 (Vopt (x)i ⇥t ⇥Eg (x) = i = + Vopt (x) ⌃(x, t) ) i t t wave 2mconfiguration: 4⇥ • standing wave laser laser configuration:E( E(x,x,t)t)==eEeEsin sinkx kxe e ++h.c. h.c.
hopping • where AC Starkshift Starkshift as a function of position as a function of position the position dependent AC-Stark shift appears as a (conservative) • “optical The AC potential” Starkshift appears appearsas asaaconservative conservativepotential potentialfor forthe thecenter-of-mass center-of-mass Starkshift motion of the the atom atom 2 V (x) = V sin kx (k = 2⇧/⇤) ⇥ ⇥ opt 0 22 2 2 ⇤ ⌃(x, t) ⇤(x) (x) 22 (V(V (x) = + ) ) ⌃(x,x,t)t) (x)⇥⇥⇥E ⇥E (x)== i = +VVopt x) ⌃( opt g (gx) opt opt((x) 2m 4⇥ 2m motion of the atom. 4⇥ for the tcenter-of-mass
• The potential is periodic with AC-Stark lattice period andas thus supports a band where the dependent shift appears the position position dependent AC-Stark shift⇤/2, appears asa a(conservative) (conservative) structure (Bloch bands). The depth of the potential is proportional to the “optical potential” potential” laser intensity 22 VVopt (x) = V sin kx (k(k==2⇧/⇤) 0 2⇧/⇤) opt (x) = V0 sin kx Single Atom moving in an Optical Potential: If we include the motion of the forwe the center-of-mass motion thederive center-of-mass motionof ofthe theatom. atom. atomfor the Schrödinger Equation The potential with ⇥ period •• The potential is is periodic periodic with lattice lattice period⇤/2, ⇤/2,and andthus thussupports supportsa aband band 2 2 ⌃( x, t) ⇤ (the x) structure (Bloch bands). The depth of the potential is proportional to the 2 structure (Bloch bands). The depth of the potential is proportional to i = + Vopt (x) ⌃(x, t) (Vopt (x) ⇥ ⇥Eg (x) = ), laser intensity 4⇥ lasert intensity 2m
Boson Gas in Periodic Optical Potentials
•
Starting point: workhorse Hamiltonian for weakly interacting ultracold bosons
H=
• •
Z
x
⇥
⇤ 4 † 2 + V (x) ˆx + g( ˆx ˆx ) 2m
see above: trapping potential can be treated classically due to scale separation instead, now we are interested in a periodic potential of wavelength comparable to the typical interparticle distance: light in with optical wavelength, as
⇠ 500nm = 5 · 10
typical wavelength of light
•
ˆ† x
5
cm
d . 10
4
cm (n & 1012 cm
)
typical interparticle separation
create such conservative potential by weakly coupling the atoms in their ground state ($ auxiliary internal level by above means: position dependent second order AC Stark shift Rabi coupling to aux. level
X ⌦2 (x) V (x) = ~ ⌘ V0 sin2 (ki xi ), 4 i
ki = 2⇡/
laser detuning from aux. level
•
3
atomic fermions can be treated analogously: Fermi Hubbard model
i
ˆx ) to
• The gas is sufficiently dilute so that only two body interactions are important, we can treat the composite atoms as bosons • The enery / temperature are sufficiently that two-body interactions reBose Hubbard Hamiltonian (Jaksch small et al. ’98) spatially localized duce to s-wave scattering, parametrized by the scattering length as .
Wannier functions
•
For dominant optical potential V0 (other scales), we expect localized single particle wavefunctions to provide a useful Bose Hubbard Hamiltonian description of the system.
•
A suitable complete set of basis functions areGardiner, the Wannier Lit.: D. Jaksch, C. Bruder, J. I. Cirac, C. W. and functions P. Zoller, Phys Rev. Lett. 81, 3108 (1998)
•
We W expand the field operators in Wannier functions of the lowest band ⌅ ˆ(⇤x) = w(⇤x ⇤xi )bi i
to obtain the Bose Hubbard model ⌅ ⌅ †2 † 1 ˆ H = Jij bi bj + 2 U bi b2i ij
U J
i
⇧
⌃ 2 ⇤ + V0 (⇤x) w(⇤x 2m 2
with hopping Jij = d3 xw(⇤x ⇤xi ) ⇤xj ) and interaction ´ 3 4 1 U = 2 g d x |w(⇤x)| valid for J, U, kB T ⇥ ⇥Bloch . (tight binding lowest band approximation) ´
additionally, we are bound to interactions (scattering lengths) extent of Wannier function
g ⌧ a0 , a
here, it means lattice spacing
This is not true close to Feshbach resonances!
Bose Hubbard Parameters Parameters as function of laser intensity
H=
J
X
hi,ji
•
† bi bj
+
X i
✏i n ˆ i + 12 U
X
n ˆ i (ˆ ni
i
J/U mainly controlled by extraction of kinetic energy via deepening the lattice
1)
Summary: Bose-Hubbard Model H=
† bi bj
J i,j⇥
kinetic energy
1 n ˆ + i i 2U
n ˆi +
µ i
i
n ˆ i (ˆ ni
1)
i interaction energy
trapping potential U
• •
Achieved via coherent manipulation of ultracold atoms.
•
Possible to penetrate high density regime⇥ˆ ni ⇤ continuum.
•
The Bose-Hubbard model is an exemplary model for strongly correlated bosons. It is not realized in condensed matter.
•
Ratio of kinetic and interaction energy tunable via lattice parameters (and Feshbach resonances). In particular, reach interaction dominated regime.
= O(1). Not possible in the
Remark: strong interactions and high density not in contradiction to earlier scale considerations:
• • •
strong interactions: J/U
J
1mainly from reduction of kinetic energy via lattice depth.
High density due to strong localization of onsite wave function. For validity of lowest band approximation, it is however important that
a
Phase Diagram of the Bose Hubbard Model 3.0
MI 2.0
SF
MI 1.0
0.0 0.0
MI 0.010
0.020
0.030
Kinetic vs. Interaction Domination - Limiting Cases
• • •
Goal: Find the ground state (gs) phase diagram for Bose-Hubbard model (T=0)
•
Interaction dominated regime: set J = 0
Strategy: (i) analyze limiting cases, (ii) find interpolation scheme Restrict to the homogeneous system
H=
n ˆ i + 12 U
µ i
-
n ˆ i (ˆ ni
1) =
hi
number eigenstates n=1
i
i
Purely local Hamiltonian: gs many-body wavefunction takes product form
|
=
➡ Remains to analyze onsite problem only. -
=0
i
i
|
i
n ˆ i |n
i
= n|n
i
Only onsite density operators occur, with (real space) occupation number eigenstates
➡ Onsite Hamiltonian also diagonal in this basis: Thus, minimize onsite energy and find the optimal n for given mu:
for µ/U < 0 n=0 for 0 < µ/U < 1 n = 1| for 1 < µ/U < 2 n = 2 and so on
particle number quantization: for ranges of the chemical potential (particle reservoir), the system draws an integer number out or it: “Mott states”
lattice direction
Kinetic vs. Interaction Domination - Limiting Cases
•
lattice dispersion ✏q =
2J
Kinetically dominated regime: set U=0
H=
b†i bj
J
X
cos qe
n ˆi
µ i
i,j⇥
- Free bosons at T=0: Bose Einstein condensation! - See that: Diagonalize with Fourier transformation
H=
•
X
(✏q
µ)b†q bq
momentum eigenstate q=0
q
Ground state wave function: fixed particle number N (M - no. of lattice sites) N b†q=0 |vac = (M
1/2 i
b†i )N |vac
- product state in momentum space, not in position space
•
Work in grand canonical ensemble: coherent state with av. density N 1/2 b†q=0
e
(N/M )1/2
|vac = e
ensures av. particle no.
P
† b i i
|vac =
⇤ i
((N/M )1/2 b†i )
e
n ˆ i ⇥ = N/M |vac
i
⇥
➡ grand canonical ground state can be written as a product of onsite coherent states
Intermediate Overview: Competition
•
Hopping J favors delocalization in real space:
•
Interaction U favors localization in real space for integer particle numbers:
• •
Condensate (local in momentum space!)
• •
Mott state with quantized particle no.
Fixed condensate phase: Breaking of phase rotation symmetry i ⇥bi ⇤ e
no expectation value: phase symmetry intact (unbroken)
➡ Competition gives rise to a quantum phase transition as a function of
U/J ➡ Link between extremes: position space product ground states, respectively
Mean Field Theory
• •
Interpolation scheme encompassing the full range J/U . Main ingredient: Based on above discussion, construct local mean field Hamiltonian
H
(MF)
=
X
hi
H=
µˆ ni +
b†i bj
•
• •
X
n ˆi +
1 2U
i
1 ˆ i (ˆ ni 2Un
1)
Jz(
⇤
bi +
† bi )
+ Jz
n ˆ i (ˆ ni
i
Discussion: Derivation: Decompose bi
=
⇤
coordination number
z = 2d
(cubic lattice)
+ bi, neglect b†i bj terms, rewrite in terms of bi
The problem is reduced to an onsite problem is the “mean field”: - information on other other sites only via averages - if nonzero, assumes translation invariance but spontaneous phase symmetry breaking
•
X
full Bose Hubbard Hamiltonian
n ˆ i = b†i bi
•
µ
hi,ji
i
hi =
J
X
Validity: approximation neglects spatial correlations via local form - becomes exact in infinite dimensions (Metzner and Vollhardt ’89)
- reasonable in d=2,3 (T=0)
1)
Phase Diagram: Derivation
•
Assume second order phase transition and follow Landau procedure:
• •
•
Study ground state energy
E( )
E( ) = const. + m2 | |2 + O(| |4 )
Determine zero crossing of mass term
unstable towards SSB
Calculate E in second order perturbation theory
hi = (0) hi
=
(0) hi
+ Vi smallness parameter close to phase transition
µˆ ni + 21 U n ˆ i (ˆ ni
Vi = Jz(bi +
† bi )
1) + Jz
⇤
Phase Outline
Diagram: Derivation (0)
• Zero order Hamiltonian hi (0)
: diagonal in Fock basis {|n!}, n = 0, 1, 2, ...
• The eigenvalues are En = −µn + 12 U n(n − 1) + Jzψ 2
• The ground state energies for given µ are ! 0 (0) En¯ = −µ¯ n + 21 U n ¯ (¯ n − 1) + Jzψ 2
for µ < 0 for U (¯ n − 1) < µ < U n ¯
• The second order correction to the energy is (2) En¯
=ψ
2 " |#¯ n |V |n!| i 2 (0) E ¯ n!=g n
−
(0) En
2
= (Jzψ)
#
n ¯+1 n ¯ + U (¯ n − 1) − µ µ − U n ¯
$
¯ = U/Jz) • For E = const. + m2 ψ 2 + ... the phase transition happens at (¯ µ = µ/Jz, U n ¯ n ¯+1 m2 =1+ ¯ + =0 ¯ Jz U (¯ n − 1) − µ ¯ µ ¯ − Un ¯ •
Bose-Hubbard phase border
• A quantum computer with N qubits can simulate such a problem. This is why a quantum can easily outperform a classical computer in this task. Thus it is of interest to build quantum
• if ⇤ = 0 we have a Mott phase. The transition between these phases will occur Diagram: for a certain critical Uc /J. Shape It is a second order transition. Phase Overall This gives the phase diagram as a function of µ/U and J/U . Finally, we plot ⇥ˆ ni ⇤ = n as a function of µ to determine the chemical potential 3.0N/M . for a given density n =
MI
n ¯ 1+ ¯ U (¯ n 1)
2.0
SF
MI 1.0
“Mott Lobe”
0.0 0.0
n ¯+1 + =0 ¯ µ ¯ µ ¯ Un ¯
NB: for non-commensurate (= integer) fillings, superfluidity persists for U -> 0: excess particles condense.
MI 0.010
0.020
0.030
Simple picture: MI: Quantization of particle number
ˆ , '] ”[N ˆ = i”
SF: Quantization of phase - conjugate variables
Limiting cases: Weak Limiting cases: Weak coupling
coupling, Superfluid
• Consider site dependent mean fields, hi = −µˆ ni +
1 ˆ i (ˆ ni 2Un
• Consider the equation of motion for the order parameter: - Heisenberg equation of motion for onsite Hamiltonian: ∂t ρ = −i[hi , ρ],
ρ = |ψ"#ψ|,
− 1) − J
|ψ" =
!
" i
† ∗ (ψ b + ψ b i j j i ) + const. !j|i"
|ψ"i
- Equation of motion for the order parameter: i∂t ψi ≡ i∂t tr(bi ρ) = −J
#
!j|i"
ψi − µ#ˆ bni " + U #ˆni bi " i
2
1 X
n i
- Weak coupling: assume coherent states bi |ψ"i = ψi |ψ"i | i i = e | i | /2 p |ni n! n=0 # ψi − µψi∗ ψi + U ψi∗ ψi2 = −J%ψi − µ$ ψi∗ ψi + U ψi∗ ψi2 i∂t ψi = −J !j|i"
X • At weak coupling, the (lattice) Gross-Pitaevski equation is reproduced 4fi = fj zfi lattice Laplacian hj|ii
µ0 = µ + Jz
shift: defines zero of kinetic energy
Consider a system of N spin-1/2 particles or qubits on a lattice. Its state vector lives in the product Hilber ➡ At weak coupling, the (lattice) Gross-Pitaevski equation is reproduced space H➡ = {|↑" , |↓"}⊗N ≡ {|0" , |1"}⊗N , and is in general a superposition certain spatial fluctuations are included: scattering off the condensate #q ➡ dispersion: X cσ1 σ2 ...cN⇤ |σ1 , σ2 , . . . , σN " |Ψ" = ✏q = 2J (1 cos aq ) !q = ✏q (2U + ✏q ) {σi =0,1}
Limiting Strong Limiting cases:cases: Strong coupling • Mean field Mott state : |¯ n! =
!
coupling, Mott Insulator
ni ! = n ¯ i |¯
−M/2
!
†¯ n b i i |vac!: Quantization of particle number
• Discussion: - Within mean field, Mott-ness follows as a consequence of purity:
∗ assume mechanism that suppresses SF off-diagonal order: ρ diagonal, 1 = trρ = " ( pm )M ! " 2 M 2 2 hom. ∗ Zero temperature: pure state, 1 = trρ = i tri ρi = ( pm ) ∗ only solution is pm = δn,¯n
- Quantization of particle number within MI is an exact result in the sense #b†i bi ! = n ¯
!
hom.
i tri ρi =
" ˆ ∗ at J = 0, Mott state |¯ n! is (i) exact ground state, (ii) eigenstate to particle number N = i n ˆi, (iii) separated from other states by gap ∼ U " ∗ kinetic perturbation Hkin = −J "i,j# b†i bj ˆ , [Hkin , N ˆ] = 0 ∗ commutes with N ➡ switching on J adiabatically, the ground state remains exact eigenstate to
number operator. Assuming translation invariance gives exact result † hbi bi i = n ¯
-
Implication: the Mott insulator is an incompressible state,
ˆi @hN =0 @µ
Excitation spectrum in the Mott phase
• • •
!q
We are looking for the single particle dispersion relation
This is a dynamical quantity: hard to get within Hamiltonian framework above Path integral formulation of the Bose-Hubbard model Z ˆ H
Z = tre
Da exp SBH [a]
=
SBH [a] = Sloc [a] + Skin [a] Sloc [a] = Skin [a] =
•
Z Z
/2
d⌧ /2
X⇥
a⇤i (@⌧
µ)ai +
⇤2 2 1 U a i ai 2
i
/2
d⌧ /2
X
tij a⇤i aj
i,j
⇤
local contribution
bi-local contribution; for nearest-neighbour hopping
tij =
J
Discussion:
-
Nonrelativistic action (on the lattice)
-
Note symmetry (T=0): temporally local gauge invariance
-
so far: weak coupling problems J>>U, decoupling in the interaction U
-
now: Mott physics, i.e. strong coupling problem U>>J: decoupling in J
ai ! ai ei
X
(⌧ )
i,j+ e
= ±1
= x, y, z
spatial directions
, µ ! µ + i@⌧ (⌧ )
Decoupling in J: Hopping Expansion SBH [a] = Sloc [a] + Skin [a] ⇠U
⇠J
Decoupling in J: Hopping Expansion Goal: treat the strong coupling problem U>>J via decoupling in J • • Hubbard-Stratonovich transformation: ´ ´ ´ ! Z = Da exp −SBH [a] = N DaDψ exp −SBH [a] + dτ ij tij (ψi∗ − a∗i )(ψj − aj ) =N
Seff [ψ] "O#Sloc
´
DaDψ exp −Sloc [a] + ∗ t ψ ij i ψj ij
=
´
dτ
=
´
DaO exp −Sloc [a]
!
´
dτ
− log"exp −
´
!
∗ ij tij (ψi ψj
dτ
−
ψi∗ aj
−
ψj a∗i )=
N
´
Dψ exp −Seff [ψ]
∗ ∗ (ψ a + ψ a j j i i #Sloc ij
!
• Discussion - Form of intermediate action identical to mean field decoupling above (for site dependent mean fields) - The effective action Seff [ψ] can now be calculated perturbatively
Decoupling in J: Hopping Expansion Decoupling in J: Hopping Expansion • expansion in powers of the hopping !exp −
ˆ
dτ
!
tij (ψi∗ aj + ψj a∗i )#Sloc
ij
ˆ ∞ ! ! tij (ψi∗ aj + ψj a∗i )]2m #Sloc =1+ ![ dτ m=1
ij
→ note: averages of odd powers of ai or a∗i vanish in the Mott state
• To lowest order, the effective action thus reads ˆ ˆ ! ! ∗ # tij ψi (τ )ψj (τ ) + dτ dτ tij ti! j ! ψj∗ (τ )ψj ! (τ # )!ai (τ )a∗i (τ # )#Sloc Seff [ψ] = dτ ij
iji! j !
• Discussion - the approach is inherently perturbative: no known closed form expression for above average ´ - the hopping expansion does not lead to an exact solution of the problem: Z = Dψ exp −Seff [ψ] would need to be calculated for this purpose - the hopping expansion is closely related to the mean field approximation (see below)
Quadratic Effective Action Quadratic Effective Action • The correlation functions !ai (τ )a∗i! (τ " )"Sloc can be evaluated explicitly since the local onsite problem is solved exactly • The effective action is then, in frequency and momentum space and for nearest neighbour hopping (2) Seff [ψ] −1
G
=
´
dω dd q ∗ −1 (ω, q)ψq (ω) 2π (2π)d ψq (ω)G
(ω, q) = $q −
$2q
!
n ¯ +1 −iω−µ+¯ nU
+
n ¯ iω+µ−(¯ n−1)U
"
,
$q = 2J
#
λ
cos qeλ
• Evaluating G−1 (ω = 0, q = 0) reproduces the above mean field result ($q = Jz): The fluctuations included here are the same, but their spatial and temporal dependence is resolved within the functional integral formulation • The frequency dependence is dictated by the temporally local gauge invariance to −iω − µ
• The quasiparticle spectrum obtains from the poles of the Green’s function analytically continued to real frequencies, ! G−1 (ω → iω, q) = 0
Excitation Spectrum: Particles and Holes
3.0
MI 2.0
Excitation Spectrum
SF
MI −1
• Solving G
1.0
!
(ω → iω, q) = 0 yields the quasiparticle dispersion ! ! n + 1)U "q + U 2 ωq± = −µ + U2 (2¯ n − 1) − 2q ± 12 "2q − 2(2¯
0.0 0.0
MI 0.010
0.020
0.030
• For µ within the Mott phase, ωq+ ≥ 0 and ωq− ≤ 0. They correspond to quasiparticle and quasihole excitations. • If we are interested in the true excitation spectrum, we need to consider that the quasiholes are propagating backward in time. The true particle and hole excitation energies are therefore ! " # ! n + 1)U "q + U 2 Eq± = ±ωq± = ± − µ + U2 (2¯ n − 1) − 2q + 12 "2q − 2(2¯ ± • Generically, both branches of the spectrum are gapped: Eq=0 = O(U ) > 0 !
• Study the phase border for Jz % U , defined with G−1 (ω = 0, q = 0) = 0 −→ µcrit (U ) in the upper branch of the lobe: (¯ n+1)U+Jz # + - there δ"q , δ"q = ∆ $ is a gapless particle2 with quadratic dispersion for q → 0, Eq ≈ (1 + 2J λ (1 − cos qeλ ) ≈ Jq % n + 1)U Jz + U 2 - there is a gapped hole with gap ∆ = Eq=0 = Jz 2 − 2(2¯ • For the lower branch of the lobe and Jz % U , the holes disperse Eq− ≈ Jq2 and the particles are gapped with ∆ ≈ U
3.0
Bicritical point: Change of Universality Class The Tip of the Lobe: Bicritical Point • The interpretation in terms of particle and hole excitation only holds for Jz ! U
MI 2.0
SF
MI 1.0
0.0 0.0
MI 0.010
0.020
0.030
• In general, at the phase transition there is one gapless mode. Choosing it to set the zero of energy, the gap of the other mode is given by ! + − n + 1)U Jz + U 2 ∆ = Eq=0 + Eq=0 = Jz 2 − 2(2¯ ! • At the tip of the lobe, U/Jz = 2¯ n + 1 + (2¯ n + 1)2 − 1 and thus ∆ = 0: there are two gapless modes (particle-hole symmetry) • The excitation spectrum at this point is dominanted by the square root for q → 0 and reads " n + 1)U + Jz)δ"q ∼ |q| Eq± = 12 (2(2¯
• The spectrum changes form a nonrelativistic spectrum E ∼ q2 (dynamic exponent zd = 2) to a relativistic spectrum E ∼ |q| (dynamic exponent zd = 1)
➡ At a generic point on the phase border, the system is in the z_d = 2 O(2) universality class ➡ At the tip of the lobe, the system is in the z_d = 1 O(2) universality class
The Tip of the Lobe: Bicritical Point
Bicritical point: Symmetry Argument
• We show that the change in universality class at the tip of the lobe is not an artifact of mean field theory
• The full effective action (including fluctuations) at low energies has a derivative expansion ˆ Γ[ψ] = ψ ∗ [Z∂τ + Y ∂τ2 + m2 + ...]ψ + λ(ψ ∗ ψ)2 + ...
• At the phase transition, we have m2 = 0. At the tip of the lobe, we have additionally(vertical tangent) ∂m2 =0 ∂µ • Using the invariance under temporally the local symmetryψ → ψeiθ(τ ), µ → µ + i∂τ θ(τ ), we find the Ward identity (q = (ω, q)) ! ! ∂m2 ∂ δ2Γ δ2Γ ∂ ! ! − =− = =Z ! ! ∗ ∗ ∂µ ∂µ δψ (q)δψ(q) ψ=0;q=0 ∂(iω) δψ (q)δψ(q) ψ=0;q=0
• Thus, there cannot be a linear time derivative ath the tip of the lobe, Z = 0. The leading frequency dependence is quadratic 3.0
2.0
MI
MI
MI 0.010 0.020
moving in positive mu direction enhances SF 2
SF
0.030
@m2 ! >0 @µ
1.0
0.0 0.0
moving in positive mu direction suppresses SF
µ/U
!
@m J • The competition of kinetic and interaction energy g = J/U gives rise to a quantum phase transition for commensurate fillings n=1,2,...
• The strong coupling “Mott phase” is an ordered phase (quantized particle number) without symmetry breaking
• One characteristic property is incompressibility, which has been observed in experiments
Excursion: 3-Body Interactions from 3-Body Loss: 2 Applications
Outline (1) Dissipative generation of a 3-body hardcore interaction
• Mechanism and experimental perspectives A. J. Daley, J. Taylor, SD, M. Baranov, P. Zoller, Phys. Rev. Lett. 102, 040402 (2009)
(2) Phase diagram for 3-body hardcore bosons
• Unique effects of interactions and the constraint
0
2
SD, M. Baranov, A. J. Daley, P. Zoller, Phys. Rev. Lett, 104, 165301(2010); SD, M. Baranov, A. J. Daley, P. Zoller , Phys. Rev. B 82, 064509 (2010); Phys. Rev. B 82, 064510 (2010)
4
6
8 0.0
0.5
1.0
1.5
2.0
Motivation • 3-body loss processes (-) • ubiquitous, but typically undesirable • inelastic 3 atom collision • molecule + atom ejected from lattice
• 3-body interactions (+) • Stabilize bosonic system with attractive interactions • interesting many-body consequences • 3-component fermion system: stabilize atomic color superfluidity
➡ We make use of strong 3-body loss to generate a 3-body hardcore constraint
ig3 ! g3
3-body interactions via 3-body loss
A. J. Daley, J. Taylor, SD, M. Baranov, P. Zoller, Phys. Rev. Lett. 102, 040402 (2009)
An analogy: optical pumping master equation in Lindblad form
|e
d = dt
Γ |0
⇥ H = (|0⇤⇥e| + |e⇤⇥0|) 2
with
0 1 eff
|1 pumping rate
i[H, ] + L[ ]
L[ ] = 0 1 eff
⇤2 = 4⇥2 +
(for
2
⇥ 1 † (J J + J † J) 2
J J†
⇤
|e⇤⇥e|
J = |1⇥ e|
, ⇥) 0 1 eff
0.01
0.0075
0.005
0 1 eff
⇤2 4⇥2
0 1 eff
0.0025
0
0
0
large
:
⇥2
2
4
6
5
8
10
12
14
/⇥
16
18
15
20
20
Zeno regime: system frozen in |0
• system “freezes” in |0> • leading correction is effective small loss rate 0 -> 1
0!1 e↵
⇡ ⌦2 /
Analogy to three-body loss 2U
|e Γ e↵
|1
J
3
|0
J e↵
2U J
detuning Rabi frequency
Γ
decay rate
3
e↵ 0.01
onsite interaction energy tunnel coupling
0.0075
e↵
three-body recombination rate
0.005
• system “freezes” in subspace with fewer 3 particles per site: 3-body constraint • stabilizes against particle loss: effective loss rate ➡ For
3
e↵
2
⇡J /
⇡
J2 3
0.0025
0
3
0
0
2
4
5
6
8
10
12
14
3 /2U 15
U, J, realization of a Bose-Hubbard-Hamiltonian with three-body hard-core constraint on time scales t < 1/ e↵
16
18
20
20
Physical Realization in Cold Atomic Gases •Estimate loss rate for Cesium close to a zero crossing of the scattering length (e.g. Naegerl et al.)
g3 |U| J
lattice depth ➡ 3-body loss can be made the dominant energy scale
• Preparation of the ground state of constrained Hamiltonian: • role of heating effects due to residual particle loss • Approach: Exact numerical time evolution of full Master Equation in 1D; combine DMRG method with stochastic simulation of ME
• Find optimal experimental sequence to avoid heating
Ground State Preparation
Ramp: Superlattice, V/ J=30 to V/J=0, N=M=20
•Start from Mott ground state in superlattice •Loss causes heating, inhibits reaching ground state “lucky” trajectory
“unlucky” trajectory
➡ “Probabilistic” ground state preparation
Phase Diagram for Three-Body Hardcore Bosons 0
2
4
6
8 0.0
0.5
1.0
microscopic
thermodynamic
interactions
interactions condensation
1.5
2.0
long distance
interactions condensation spin waves
SD, M. Baranov, A. J. Daley, P. Zoller, Phys. Rev. Lett, 104, 165301(2010); SD, M. Baranov, A. J. Daley, P. Zoller , Phys. Rev. B 82, 064509 (2010); Phys. Rev. B 82, 064510 (2010)
Physics of the constrained Hamiltonian • The constrained Bose-Hubbard Hamiltonian stabilizes attractive two-body interactions PHP =
U †ˆ ˆ J  bi b j + 2 hi, ji
 nˆi (nˆi i
1) & b†i 3 ⌘ 0
U quantum field theory • exact mapping of the constrained Hamiltonian to a coupled boson theory with polynomial interactions • the bosonic operators find natural interpretation in terms of “atoms” and “dimers” “atoms”
➡ We have identified several quantitative and qualitative effects:
✓Tied to interactions ✓Tied to the constraint
“dimers”
Implementation of the Hard-Core Constraint • Introduce operators to parameterize on-site Hilbert space (Auerbach, Altman ‘98) † |vaci = |ai, ta,i
a = 0, 1, 2
• They are not independent:
 a
† ta,ita,i
=1
• Representation of Hubbard operators: a†i
p † † = 2t2,it1,i + t1,i t0,i
† † nˆ i = 2t2,it2,i + t1,it1,i
Action of operators
† ai
|2i
† t1,i t0,i
|1i |0i |vaci
Implementation of the Hard-Core Constraint • Hamiltonian: † † † Hpot = µ Â 2t2,i t2,i + t1,i t1,i +U Â t2,i t2,i i
Hkin =
i
i p ⇥† † † † † † † † J Â t1,it0,it0, t + 2(t t t t + t t t t ) + 2t 1, j 1,i 1, j 0,i 2, j j 2,i 0, j 1,i 1, j 2,i t1, j t1,i t2, j hi, ji
• Properties: • Mean field: Gutzwiller energy (classical theory) • interaction: quadratic • Role of interaction and hopping reversed • Strong coupling approach facilitated • hopping: higher order • One phase is redundant: absorb via local gauge transformation t1,i = exp ij0,i |t0,i | ➡ e.g. t_0 can be chosen real
t1,i ! exp ij0,i t1,i ,
t2,i ! exp ij0,i t2,i
Implementation of the Hard-Core Constraint • Resolve the relation between t-operators (zero density) q † † t1,i t0,i = t1,i 1
† t1,i t1,i
† † t2,i t2,i ! t1,i (1
† t1,i t1,i
† t2,i t2,i )
• justification: for projective operators one has from Taylor representation † † X = 1 t1,i t1,i t2,i t2,i • Now we can interpret the remaining operators as standard bosons:
X 2 = X ! f (X) = f (0)(1
Hi = {|ni1i |mi2i }, n, m = 0, 1, 2, ...
• decompose into physical/unphysical space:
Hi = Pi Ui
Pi = {|0i1i |0i2i , |1i1i |0i2i , |0i1i |1i2i } • correct bosonic enhancement factors on physical subspace • the Hamiltonian is an involution on P and U: H = HPP + HUU
• remaining degrees of freedom: “atoms” and “dimers” ➡ similarity to Hubbard-Stratonovich transformation
p
n = 0, 1 “atoms”
• on-site bosonic space
X) + X f (1)
|2i1i |1i1i
|0i1i
|0i2i |1i2i |2i2i “dimers”
Implementation of the Hard-Core Constraint • The partition sum does not mix U and P too: Z = Tr exp bH = TrPP exp bHPP + TrUU exp bHUU
• Need to discriminate contributions from U and P: • Legendre transform of the Free energy
G[c] = W [J] +
Z
T
J c,
• Has functional integral representation: Z
W [J] = log Z[J] dW [J] c⌘ dJ
D dc exp S[c + dc] +
exp G[c] =
S[c = (t1 ,t2 )] =
Work with Effective Action
Z
⇣
Z
Quantum Equation of Motion for J=0
T
J dc,
dG[c] J= c
⌘ † † dt  t1,i ∂tt1,i + t2,i ∂tt2,i + H[t1 ,t2 ] i
• Usually: Effective Action shares all symmetries of S • Here: symmetry principles are supplemented with a constraint principle
Condensation and Thermodynamics • Physical vacuum is continuously connected to the finite density case: Introduce new, expectationless operators by (complex) Euler rotation ~t = (t0 ,t1 ,t2 )T
~b = Rq Rj ~t • Hamiltonian in new coordinates takes form: Quadratic part: Spin waves (Goldstone for n > 0)
H = EGW + HSW + Hint Mean field: Gutzwiller Energy
higher order: interactions
microscopic
thermodynamic
interactions
interactions condensation
long distance
interactions condensation spin waves
The requantized Gutzwiller model • Hamiltonian to cubic order is of Feshbach type: • quadratic part: Hpot = Â(U i
2µ)n2,i
Dimer energy
µn1,i
detuning from atom level
• leading interaction:
i p † ⇥† † † Hkin = J Â t1,it1, j + 2(t2,it1,it1, j + t1,i t1, j t2, j )
two separate atom’s energy
hi, ji
(bilocal) dimer splitting into atoms
• Compare to standard Feshbach models: ⇠ 1/U
detuning
⇠U
here: detuning
➡
Hkin =
we can expect resonant (strong coupling) phenomenology at weak coupling ⇥† J Â t1,i (1 hi, ji
n1,i
n2,i )(1
n1, j
n2, j )t1, j +
p
† 2(t2,i t1,i (1
n1, j
† n2, j )t1, j + t1,i (1
n1,i
† † † n2,i )t1, j t2, j ) + 2t2,i t2, j t1, j t1,i
i
Vacuum Problems
dimer excitation
• The physics at n=0 and n=2 are closely connected: • “vacuum”: no spontaneous symmetry breaking • low lying excitations: • n=0: atoms and dimers on the physical vacuum • n=2: holes and di-holes on the fully packed lattice
n=0 n=2
di-hole excitation
• Two-body problems can be solved exactly
-1
G d(K) =
• Bound state formation:
1 = ˜ | + bn an |U
n=0:
Z
dd q (2⇡)d
˜b + cn /d E
1 P
+
= (1
+
cos qe )
a0 = 1, b0 = 0, c0 = 2
reproduces Schrödinger Equation: benchmark ➡ Square root expansion of constraint fails
Eb /Jz
0
n=0
Eb /Jz
n=2
1
➡
n=2: ➡
a2 = 4, b2 =
˜ b c2 = 4 6 + 3E
d=2
red blue
2
d=3
3
di-hole-bound state formation at finite U in 2D
U/Jz 4
0
1
2
3
4
5
6
U
Beyond Mean Field Phase Diagram • Start here! 0
2
4
6
8 0.0
microscopic
0.5
eM =
1 Ma2
1.0
thermodynamic kF3 n = 2 ,T 3p
1.5
2.0
long distance kld
1/2
n1/3 , T 1/2 , eM
Beyond Mean Field Phase Diagram • Vacuum limit:
• Scattering of a few particles • Requantized Hamiltonian is of Feshbach type 0
• low lying excitations: atoms and dimers on the physical vacuum
• low lying excitations: holes and di-holes on the fully packed lattice
1 2 3 4 5 6 0.0
0.5
1.0
1.5
2.0
➡ n = 0: Schrödinger Equation reproduced: Benchmark ➡ n = 2: finite critical interaction for di-hole bound state in 2D microscopic
eM =
1 Ma2
thermodynamic kF3 n = 2 ,T 3p
long distance kld
1/2
n1/3 , T 1/2 , eM
Beyond Mean Field Phase Diagram • Thermodynamic scales: Nonuniversal shifts on the phase border ˜c = 2⇡ log U
8
p ⇤2
1 0
n
1
˜c = U ˜0 1 + 4 U
n1/4 2
˜c = U
3 + ⇡ log
4
p ⇤2
1
n
3
˜c = U
2(1 + 2
p
n).
4
˜c = U ˜2 1 + U
5
˜0 ⇡ U
4/3, ⇡ 0.42 ˜2 ⇡ 11/3 ⇤ ⇡ 5.50, U
˜2 | 3 |U ˜2 | 2|U
n1/4 ˜c = U
4(1 + 2
6 0.0
0.5
1.0
1.5
2.0
➡ Quantum fluctuations lead to quantitative deviation from MFT ➡ Stronger shifts for small densities: weaker vacuum fluctuations microscopic
eM =
1 Ma2
thermodynamic kF3 n = 2 ,T 3p
long distance kld
1/2
n1/3 , T 1/2 , eM
p
n)
Beyond Mean Field Phase Diagram • Qualitative effects of the constraint and interactions: 0
2
4
6
8 0.0
0.5
1.0
1.5
2.0
➡ Enhancement of symmetry from ➡ Ising quantum critical point near half filling microscopic
eM =
1 Ma2
thermodynamic kF3 n = 2 ,T 3p
long distance kld
1/2
n1/3 , T 1/2 , eM
Symmetry Enhancement in Strong Coupling • Perturbative limit U >> J: expect dimer hardcore model • Interpret EFT as a spin 1/2 model in external field:
• Leading (second) order perturbation theory:
CDW order
v l= =1 2t
➡ Isotropic Heisenberg model (half filling n=1):
• Emergent symmetry: SO(3) rotations vs. SO(2) sim U(1) • Bicritical point with Neel vector order parameter
xy plane: superfluid order
with constraint
• charge density wave and superfluid exactly degenerate • CDW: Translation symmetry breaking • DSF: Phase symmetry breaking
• physically distinct orders can be freely rotated into each other:
without constraint
“continuous supersolid” (similar to fermions: S.C. Zhang ’93)
➡ The symmetry enhancement is unique to the 3-body hardcore constraint
Signatures of “continuous supersolid” • Next (fourth) order perturbation theory: Superfluid preferred
• Proximity to bicritical point governs physics in strong coupling (1) Second collective (pseudo) Goldstone mode CDW(x) DSF(x)
w(q) = tz (leq + 1)(1
0
eq )
KCDW 1/2K DSF
10 (2) Use weak superlattice to rotate Neel order parameter
10
0
1
10
0.8
10
1
1.2
(3) Simulation of 1D experiment in a trap (t-DMRG) 2
2
40 1
1
20 0
20
40
density profile: Onset of CDW
60
0
gap
20
40
60
DSF order in textured regions
0
Second (pseudo) Goldstone mode
Signatures of “continuous supersolid” • Next (fourth) order perturbation theory: Superfluid preferred
• Proximity to bicritical point governs physics in strong coupling (1) Second collective (pseudo) Goldstone mode CDW(x) DSF(x)
w(q) = tz (leq + 1)(1
0
eq )
KCDW 1/2K DSF
10 (2) Use weak superlattice to rotate Neel order parameter
10
0
1
10
0.8
10
1
1.2
(3) Simulation of 1D experiment in a trap (t-DMRG) 2
2
40 1
1
20 0
20
40
density profile: Onset of CDW
60
0
gap
20
40
60
DSF order in textured regions
0
Second (pseudo) Goldstone mode
Infrared Limit: Nature of the Phase Transition • Two near massless modes: Critical atomic field, dimer Goldstone mode • Coleman-Weinberg phenomenon for coupled real fields: Radiatively induced first order PT • Perform the continuum limit and integrate out massive modes: pure Goldstone action
S[J, f] = SI [f] + SG [J] + Sint [J, f] SI [f] =
Z
pure Ising action
∂µ f∂µ f + m2 f2 + lf4
coupling term
Ising field: Real part of atomic field
Sint [J, f] = ik
Z
∂t J f 2
Frey, Balents; Radzihovsky& Ising potential landscape: Z_2 symmetry breaking
➡ Interactions persist to arbitrary long wavelength (cf. decoupling SW) ➡ k 6= 0: Phase transition is driven first order by coupling of Ising and Goldstone mode
Ising Quantum Critical Point around n=1 |k|
• Plot the Ising-Goldstone coupling:
Z
∂t J f
Z
b†2,i ( g2 µ)b2,i
Sint [J, f] = ik G3
~x,t
2
decoupling
0
1
• Symmetry argument: • dimer compressibility must have zero crossing • Ward identies for time-local gauge invariance and atom-dimer phase locking ➡k must have zero crossing: true quantum critical Ising transition
• Estimate correlation length: ➡weakly first order, broad near critical domain
➡ Second order quantum critical behavior is a lattice + constraint effect
2
n
Effective Lattice Hamiltonian
•
Start from our model Hamiltonian, add optical potential:
H=
⇧
x
•
⇤ † ax
⇥ 2m
⇥
µ + V (x) + Vopt (x) ax +
gˆ n2x
⌅
Periodicity of the optical potential suggests expansion of field operators into localized lattice periodic Wannier functions (complete set of orthogonal functions)
ax =
wn (x
xi )bi,n
i,n band index
•
For low enough energies (temperature), we can restrict to lowest band:
4V0 ER , ER = k 2 /(2m) ⇥ n = 0
T, U, J
•
minimum position
Then we obtain the single band (Bose-) Hubbard model
H=
b†i bj
J
i
i,j⇥
J= U =g
dxw0 (x)( dx|w0 (x)|4
2
ˆi in
n ˆi +
µ
/2m⇥
+ 21 U
i
Vopt (x))w0 (x
U
n ˆ i (ˆ ni
1)
i
/2)
n ˆ i = b†i bi
J
Cold Atoms in Optical Lattices Optical Lattices AC-Stark shift: We consider an atom in its electronic ground state exposed to laser light at fixed position ↵x. If the light is far detuned from excited state resonances, the ground state will experience a second-oder AC-Stark shift ⇥Eg (↵x) = (Cold )I(↵x)Atoms with (in) Optical the dynamic polarizability of the atom for frequency , and Lattices I(↵x) the light intensity. Example: for a two-level atom {|g , |e } in the RWA the AC-Starkshift is given Optical Lattices ⇥2 (⇥ by ⇥Eg (↵x) = 4 x) with Rabi frequeny ⇤ and detuning ⇥ = eg (⇤ ⌃ ⇥). Note that nonresonant for red detuning (⇥ < 0) the ground state shifts down ⇥Eg < 0, while AC-Stark shift: We consider an atom in its electronic ground state exposed to laser (⇥ > 0) we have ⇥Eg > 0. for blue detuning laser light at fixed position ↵x. If the light is far detuned from excited state resoSingle Atom in anwill Optical Potential: If we include the motion ofg (↵ the nances, the moving ground state experience a second-oder AC-Stark shift ⇥E x) = Starkdynamic shift Equation atom Schrödinger ( we )I(↵xderive ) with the ( AC ) the polarizability of atom for frequency , and redthe detuned blue detuned I(↵x) the light intensity. ⇥ 2 2 ⌦ (↵x, t) ⇤ (↵x) 2 iExample: for = a two-level ⌦ + atom Vopt (↵x{|g ) , (↵ x,}t)in the(VRWA x)the ⇥ ⇥E x) = ), opt (↵ g (↵ |e AC-Starkshift is given ⌦t 2m 4⇥ ⇥2 (⇥ x) with Rabi frequeny ⇤ and detuning ⇥ = by ⇥Eg (↵x) = 4 eg (⇤ ⌃ ⇥). where appears a (conservative) “optiNote the thatposition for red dependent detuning (⇥AC-Stark < 0) the shift ground state as shifts down ⇥Eg < 0, while calforpotential” for the(⇥ center-of-mass of the atom. blue detuning > 0) we have motion ⇥Eg > 0.
and lattice depth V0 ⇤ |E|2 tunable / controlled by the laser intensity.
Bloch Theorem for Periodic Potentials Review: Bloch’s Theorem for Periodic Potentials pˆ2 2m +V
ˆ = • Consider a Hamiltonian (in 1D) H (ˆ x) with periodic potential V (x) = V (x + a). We are interested in the eigenfunctions H⌃(x) = E⌃(x). (We set = 1). ˆ • We define a translation operator T = e ipa so that T ⌃(x) = ⌃(x + a). - T is unitary, and thus has eigenfunctions T ⇧ (x) = ei ⇧ (x) with ( ⌅, ⌅] real.
=
- Because ⌃ (x + a) = ei ⌃ (x) we can write ⌃ (x) = ei u (x) with periodic Bloch functions u (x) = u (x + a) . • We have [H, T ] = 0, and we can find 20simultaneous eigenfunctions of {H, T } H T with q
q (x) q (x)
= =
E q (x) eiqa q (x)
[ ⇧/a, ⇧/a]. We call q quasimomentum.
• Eigenstates of the Hamiltonian thus have the form iqx (n) ⌃(n) (x) = e uq (x) q
and the Bloch functions
(n) uq (x)
q
[ ⇧/a, ⇧/a]
are eigenstates of
H T
q (x)
= E q (x) iqa q (x) = e q (x)
Bloch Theorem for Periodic Potentials with q
[ ⇧/a, ⇧/a]. We call q quasimomentum.
• Eigenstates of the Hamiltonian thus have the form iqx (n) ⌃(n) (x) = e uq (x) q q
and the Bloch functions ˆ q u(n) (x) ⇥ h q
[ ⇧/a, ⇧/a]
(n) uq (x)
are eigenstates of ⇥ 2 (ˆ p + q) + V (ˆ x) u(n) q (x) = 2m
(n) (n) q uq (x)
Note: the Bloch functions can be expanded in a Fourier series quantum number of the eigenstate (-> Bloch 1 (n) uq (x) = band index)
2⇧
+⇥ ⇤
j= ⇥
(n,q) ikj x
cj
e
(kj = 2⇧j/a) spatial dependence of wave function
(n) u (x) q deep lattices we can ignore tunneling. AsHarmonic approximation (1D): For Discussion:
suming > 0 so that V0 > 0 we have for the lowest states a harmonic oscillator potential 1 2 quasimomentum due to 2 V (x) = V0 sin (kx) ⇧ V0 (kx) ⇧ m⌅ 2 x2 lattice periodicity 2
ˆ q u(n) (x) ⇥ h q
(n) (n) + V (ˆ x) u(n) (x) = E q q uq (x) 2m +⇥ ⇤ (n,q) 1 (n) ikj x u (x) = c e Solution of Schroedinger Equation for 1D optical lattice j = 2⇧j/a) Note: the Bloch qfunctions can be expanded in a (k Fourier series j 2⇧ j= ⇥ +⇥ ⇤ 1 (n,q) (n) uq (x) = cj eikj x (kj = 2⇧j/a) 2⇧ j= ⇥
Solution of the Schrödinger Equation for the Optical Lattice (1D)
Harmonic approximation (1D): For deep lattices we can ignore tunneling. AsSolution of the Schrödinger Equation for the Optical Lattice (1D) suming V0 > 0 we have for the lowest states a harmonic oscillator potential 1 can2 ignore Harmonic approximation (1D): 2For deep lattices tunneling. As2 we V (x) = V0 sin (kx) ⇧ V0 (kx) ⇧ m⌅ x2 suming V0 > 0 we have for the lowest states a harmonic 2 oscillator potential with trapping frequency ⌅ = 4V02 ER / and with 2recoil ER ⇥ 1 frequency 2 2 V (x) = V0 sin (kx) ⇧ V0 (kx) ⇧ m⌅ x (TypciallyER ⌅ kHz, and V0 ⌅ few tens of kHz.) 2
2 2
k /2m.
(Note: the recoil frequency a natural energy scale for the depth of with trapping frequency ⌅ = ER4Vprovides 0 ER / and with recoil frequency / energy ER ⇥ 2 2optical lattice V0 /ER ⌥ 1.) the k /2m. (TypciallyER ⌅ kHz, and V0 ⌅ few tens of kHz.) The ground statefrequency wave function (Note: the recoil ER provides a natural energy scale for the depth of ⇧ the optical lattice V0 /ER ⌥ 1.) 1 x2 /(2a20 ) ⌥n=0 (x) = e The ground state wave function 1/2 ⇧ a0 ⇧ ⌅ 2 xLamb-Dicke /(2a20 ) has size a0 = /m⌅ ⌃ ⌥⇤/2, (x) and=we are1in the limit ⇥ = 2⇧a0 /⇤ ⌃ e n=0 ⇧ 1/2 a0 1.
Band Structure Harmonic approximation (1D): For deep lattices we can ignore tunneling. Assuming V0 > 0 we have for the lowest states a harmonic oscillator potential 2 V01 = 220E V0 = 10E 2 R V (x) = V0 sin (kx) ⇧ V0 (kx) ⇧ m⌅ x2 R
2
with trapping frequency ⌅ = 4V0 ER / and with recoil frequency / energy ER ⇥ 2 2 k /2m. (TypciallyER ⌅ kHz, and V0 ⌅ few tens of kHz.)
(Note: the recoil frequency ER provides a natural energy scale for the depth of the optical lattice V0 /ER ⌥ 1.)
The ground state wave function
⌥n=0 (x) = has size a0 = 1.
⌅
⇧
1 ⇧ 1/2 a0
e
x2 /(2a20 )
/m⌅ ⌃ ⇤/2, and we are in the Lamb-Dicke limit ⇥ = 2⇧a0 /⇤ ⌃
Bloch wave functions and bands (1D): In general the eigen-solutions of the time-independent Schrödinger equation with periodic potential, H⌥nq (x) = nq ⌥nq (x), has the form ⌥nq (x) = eiqx unq (x) with (periodic) Bloch wave functions unq (x) = unq (x + a), q the quasimomentum in the first Brillouin zone ⇧/a < q ⇤ +⇧/a, and n = 0, 1, . . . labelling the Bloch bands. Plots:
(Note: the recoil frequency ER provides a natural energy scale for the depth of the optical lattice Bands V0 /ER ⌥ Lowest Two Bloch for1.) V =5 E 0
The ground state wave function
band separation p !Bloch ⇡ has 4Vsize 0 ER
1.
oscillator level spacing
⌥n=0 (x) = a0 =
⌃
R
⌥
1 ⇧ 1/2 a0
e
x2 /(2a20 )
/m⌅ ⌃ ⇤/2, and we are in the Lamb-Dicke limit ⇥ = 2⇧a0 /⇤ ⌃
Bloch wave functions and bands (1D): In general the eigen-solutions of the bandwidth time-independent Schrödinger equation with periodic potential, H⌥nq (x) = nq ⌥nq (x) iqx has the form ⌥ (x) = e unq (x) with (periodic) Bloch wave functions unq (x) = 4J nq unq (x + a), q the quasimomentum in the first Brillouin zone ⇧/a < q ⇤ +⇧/a, validity: J and nonly = 0, 1, . . . labelling the Bloch bands. lowest bands Compare: Bloch bands in tight binding approximation. Above we introduced the Hamiltonian ⇥ ⇧ ⇧ † † ˆ = J H bi bi+1 + b†i+1 bi = q bq b q q
i
with the tight-binding dispersion relation q
=
2J cos qa
( ⇧/a < q ⇤ ⇧/a).
For well separated bands the Bloch band calculation fits this relation well. This is typically fulfilled for the lowest bands.
For well separated bands the Bloch band calculation fits this relation well. This Wannier functions is typically fulfilled for the lowest bands.
Wannier functions (1D): Instead of Bloch wave functions we can also work with Wannier wave functions ⌥ ⇧ discrete Fourier a ixi q (n) wn (x xi )e , uq (x) = transform 2⇧ x =ia i ⌥ trade ˆ ⇥/a a quasimomentum wn (x xi ) = dq unq (x) e iqxi for site index 2⇧ ⇥/a
which are localized around a particular lattice site xi = ia with i = 0, ±1, . . . .
The Bloch and Wannier wave functions 21are complete set of functions Generalizations:
We can generate 2D, and 3D potentials by adding optical potentials in different spatial directions. For lasers with different frequencies the interference terms average out, and the potentials are additive. Example: in 3D V (x) = 3 2 V sin (kxj ). j=1 0j Laser configurartions / lattice configurations; tricks with light polarization; disorfunction der potential via laser speckles; Wannier back ground harmonic trapping potentials etc. harmonic oscillator function
Rem.: Compare with solid state physics, where periodic potential for electrons are generated by ions oscillating around equilibrium positions (phonons); fast
Laser Control: Kinetic vs. Potential Energy • shallow lattice : weak laser U small
J large
weakly interacting system: J >> U (kinetic energy >> interactions) laser parameters (time dependent)
• deep lattice: intense laser
strongly interacting system: J
