arXiv:1803.01864v1 [hep-th] 5 Mar 2018
Mott transition and Hubbard Model in Holography
Yunseok Seo,a,c Geunho Song,a Yong-Hui Qi,a,b and Sang-Jin Sina a
Department of Physics, Hanyang University, Seoul 133-791, Korea Research Institute for Natural Science, Hanyang University, Seoul 133-791, Korea c GIST College, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea b
E-mail:
[email protected],
[email protected],
[email protected],
[email protected] Abstract: We show that the competition between the fermi liquidity and the gap generation, which is the physics of the Hubbard model, can be realized in a canonical holographic model of a fermion with bulk mass and a dipole interaction. The bulk mass near 1/2 tends to protect the fermi liquidity while large dipole coupling attempts to generate the mass gap so that the model is equipped with the basic mechanism of the Mott transition and expected to play the role of Hubbard model in holographic context. In fact, the spectral densities of the holographic model agree with the dynamical mean field theory (DMFT) calculation in two features: the ’three peaks’ and ’the transfer’ of the degree of freedom from the quasi-particle peak to the side peak. The phase diagram of our holographic model contains six phases including bad metals and pseudo-gap as well as the Fermi liquid and Mott insulator, and a line connecting the last two defines a Holographic version of Hubbard model. The experimental data for spectral densities of Vanadium Oxide materials seem to be well fit by our theory. Keywords: Holography, spectral density, Mott transition, Hubbard model
Contents 1 Introduction
1
2 Dirac equation
3
3 The Phases of holographic fermion and the Role of m and p
4
4 Holographic Hubbard Models
7
5 Comparing with experiment
9
6 Discussion
10
A Appendix A.1 Mathematical Detail A.2 Gap generation versus Appearance of semi-metal phase A.3 Role of the bulk mass A.4 Hyperboic Embeddings
11 11 12 13 15
1
Introduction
Metal-Insulator transition (MIT) for strongly interacting system (SIS) [1] is a challenging subject and quantitative understanding of such transition and the origin of the pseudo-gap(PG) is a necessary virtue of any successful theory of strongly correlated system. 1 Applying Holographic method to SIS has been done with the hope of developing a quantitative and efficient theory to account the transport and spectral density data of such system[2, 3]. Since the physics of Mott transition is usually discussed using Hubbard model which capture the competition of hopping and on-site repulsion whose strengths are represented by t and U respectively, it is important to ask whether a holographic method can capture such essence of the Hubbard model. There has been much effort on holographic fermion spectral function starting from [4]. Crucial results for our purpose are of three categories: 2 i) the simple calculational framework and the recovery of marginal non-fermi liquids in holography [7–10] with domain wall geometry, ii) the recovery of fermi liquid [11–13] with bulk mass, iii) holographic gap generation mechanism [14–18] with the dipole term ¯ µν σµν ψ. pψF (1.1) 1
In this paper we consider PG in the original meaning introduced in ref. [1] rather than that of High Tc superconductivity. 2 Discussion of the metal insulator transition in terms of conductivity was also made in ref.’s [5, 6]. Here we discuss it in terms of spectral density.
–1–
With this gap generation mechanism at hand, one might expect that the Mott transition from metal to insulator transition can be discussed immediately. However, even after we fix the non-normalizability of the spectral density of the earlier literatures, which will be mentioned later, it turns out that the MIT by the dipole at a fixed bulk mass is qualitatively different from that of Hubbard model. It is either from bad metal to insulator ( at low mass ) or no insulator at all (for high mass). Furthermore, the detail of the transition has another difference: DMFT calculation [19] of Hubbard model shows that while the quasi-particle peak is narrowed but maintained until the moment of the gap creation as one can see in Figure 1(a), the dipole interaction in the holographic model suppresses the quasi-particle peak from the beginning as one can see in Figure 1(b). The reason of such difference is simple: while holographic theory start from an strongly interacting fermion already at p = 0, the Hubbard model starts from a free fermion at U = 0. We will give a resolution to this problem by utilizing the free fermion behavior near m = 1/2 and the embedding of Hubbard model into holographic one. The result A. Georges et al.: Dynamical mean-field theory of . . . is64 given in Figure 1(c). v, and the substitution into the self-consistency condition implies that G −1 0 ;i v , which is another way of saying that the effective bath in the Anderson model picture has a gap. We know from the theory of an Anderson impurity embedded in an insulating medium that the Kondo effect does not take place. The impurity model ground state is a doubly degenerate local moment. Thus, the superposition of two magnetic HartreeFock solutions is qualitatively a self-consistent ansatz. If this ansatz is placed into Eq. (221), we are led to a closed (approximate) equation for G(i v n ): 4
3
2
2
2
2
20
0 0.7
0
5
2
D G 28D v G 14 ~ 4 v 1D 2U ! G216v 50. (234)
FIG. 30. Local spectral density p D r (v) at T=0, for several values of U, obtained by the iterated perturbation theory approximation. The first four curves (from top to bottom, U/D =1,2,2.5,3) correspond to an increasingly correlated metal, while the bottom one (U/D=4) is an insulator.
(a) Hubbard Model in DMFT
This approximation corresponds to the first-order approximation in the equation of motion decoupling schemes reviewed in Sec. VI.B.4. It is similar in spirit to the Hubbard III approximation Eq. (173) (Hubbard, 1964), which would correspond to pushing this scheme one step further. These approximations are valid for very large U but become quantitatively worse as U is reduced. They would predict a closure the gap at (b) Holography with fixedofmass U c 5D for (234) (U c 5)D for Hubbard III). The failure of these approximations, when continued into the metallic phase, is due to their inability to capture the Kondo effect which builds up the Fermi-liquid quasiparticles. They are qualitatively valid in the Mott insulating phase however. The spectral density of insulating solutions vanish within a gap 2D g /2, v ,1D g /2. Inserting the spectral representation of the local Green’s function into the selfconsistency relation, Eq. (221) implies that S(v+i0 +) must be purely real inside the gap, except for a d-function piece in ImS at v=0, with
0 6
0 -3
0
3
ω
(c) Holography with Embedding
Figure 1. Mott transition in various approaches. (a) Hubbard model in DMFT: The quasi-particle disappears continuously T=0) is at generated. a critical value The figure is from [19]. (b) Holographic model with small fixed mass, peak remains till(atgap U c2 /D.2.92, as explained in more detail in Sec. VII.E. the quasi-particle peak is suppressed and the three peak structure is missing. For large fixed mass, there Insulating phase is2. no MIT. (c) Holographic model with Embeding (m(U ), p(U )). When U/t is large, we begin with a different ansatz based on the observation that in the ‘‘atomic limit’’ t=0 (U/t=`), the spectral function has a gap equal to U. In this limit the exact expression of the Green’s function reads
1
ImS ~ v 1i0 52 pr d ~ vdiagram /2,D holographic /2# ! for v P @ 2D of We will begin with detailed study of !phase fermions as we change (235) and that ReS hasturns the following low-frequency behavior: both bulk 1/2 mass and the dipole term. It out that the phase diagram is much more rich 1/2 1 . (232) G~ iv ! 5 r i v 1U/2 i v 2U/2 1O ~ v ! . (236) ~ v 1i0 ! 2U/25 that we expected. We can realize sixReSphases that v can appear in the holographic model with bulk Since ImG( v 1i0 ) also plays the role of the density of In these expressions, r is given by states of the effective conduction electron bath entering mass and dipole interaction: Fermi liquid, bad metal (without shoulder) and bad metal prime the impurity model, we have to deal with an impurity 1 r~ e ! embedded in an insulator [D(v=0)=0]. It is clear that an 5 E de . with shoulder, pseudo-gap, semi-metal ande gapped insulator(237) phases. We will give the resulting r expansion in powers of the hybridization t does not lead to singularities at low frequency in this case. This is very r can be considered as an order parameter for the insuphase diagram. different from the usual expansion in the hybridization lating phase [the integral in Eq. (237) diverges in the V with a given (flat) density of states that is usually conmetallic phase]. A plot of the spectral function and selfon the phase diagram, we can suggest the physics of Hubbard model or the Mott sideredBased for an Anderson impurity in a metal. Here, t energy in the insulating phase,how obtained within the iteralso enters the conduction bath density of states (via the ated perturbation theory approximation, is also disself-consistency condition) and the gap survives an exin Figs. 30 and 31.model: The accuracy namely, of these results is transition can be realized in the played holographic for a given parameter U/t of the pansion in t/U. An explicit realization of this idea is to more difficult to assess than for the metal, since exact make the following approximation for the local Green’s diagonalization methods are less efficient in this phase. Hubbard model, there is a parameter (p, m) in the holographic model that gives qualitatively function (Rozenberg, Zhang, and Kotliar, 1992): A plot of the gap D vs U estimated by the iterated perturbation theory and exact diagonalization is given in 1/2 1/2 theGsame spectral density. As a result, we can consider p and m as functions of the Hubbard Fig. 32. Within both methods, the insulating solution is 1 , (233) ~ iv !. G ~ i v ! 2U/2 G ~ i v ! 1U/2 found to disappear for U,U (T50), with U model U . Asof two U magchange the (p(U ), m(U . 2.15D (whiletrajectory the iterated perturbation theory method )) in the phase space defines a which can beparameter motivated as the superposition yields U . 2.6D). As discussed below in more detail netic Hartree-Fock solutions or as a resummation of an Embedding ofimplies thethatHubbard model holographic (Sec.into VII.F), the the precise mechanism for themodel. disappear- Any line connecting a point in expansion in D/U. This G(i v );i v for small fermi-liquid phase and a point in the gapped phase would give an holographic model describing Rev. Mod. Phys., Vol. 68, No. 1, January 1996 n at
n
2
g
2
1
n
g
1
2
1`
2
2`
2
2
g
n
21 0
n
21 0
n
c1
IPT c1
–2–
ED c1
a Mott transition. Therefore we may say that one embedding defines one holographic Hubbard model. We will give a few simple embeddings and give the evolution of the model along the increasing U/t. Comparing with dynamical mean field theory (DMFT) results for Hubbard model, two features agree: the appearance of ’three peaks’ and ’transfer’ of the density of state (DOS) to the shoulders. However, the way gap disappears is different from each other after the critical strength of interaction, where both DMFT and holography calculations can not be trusted completely. Comparing with the experiment, the vanadium oxides data seem to fit quite well to out theory.
2
Dirac equation
We start from the fermion action with non-minimal dipole interaction, Z √ SD = d4 x −giψ¯ ΓM DM − m − ip ΓM N FM N ψ + Sbd ,
(2.1)
where the subscript D denotes the Dirac fermion and the covariant derivative is 1 DM = ∂M + ωabM Γab − iqAM . (2.2) 4 For fermions, the equation of motions are first order and we can not fix the values of all the component at the boundary, which make it necessary to introduce ‘Gibbons-Hawking term’ Sbd to guarantee the equation of motion which defined as Z Z √ √ 1 1 d ¯ d x hψψ = ± dd x h(ψ¯− ψ+ + ψ¯+ ψ− ), (2.3) Sbd = ± 2 2 where h = −gg rr , ψ± are the spin-up and down components of the bulk spinors. The sign is to be chosen such that, when we fix the value of ψ+ at the boundary, δSbd cancel the terms including δψ− that comes from the total derivative of δSD . Similar story is true when we fix ψ− . The former defines the standard quantization and the latter does the alternative quantization. The background solution we will use is Reisner-Nordstrom black hole in asymptotic AdS4 spacetime, r2 f (r) 2 L2 r2 2 2 dt + dr + d~x L2 r2 f (r) L2 Q2 M r0 f (r) = 1 + 4 − 3 , A=µ 1− , r r r ds2 = −
(2.4)
where L is AdS radius, r0 is the radius of the black hole and Q = r0 µ, M = r0 (r02 + µ2 ). The temperature of the boundary theory is given by T = f 0 (r0 )/4π and it can be solved for r0 to p give r0 = (2πT + (2πT )2 + 3µ2 )/3. In this paper we treat the fermion as a probe and do not consider its back reaction and further mathematical details of getting the spectral density is described in the appendix. It was pointed out [20] that the high frequency behavior of the spectral function diverges like ω 2m so that the sum of the degree of freedom over frequency is infinite if m is positive. Therefore we need to take the negative bulk mass only in the standard quantization. For the ease of discussion we want to maintain the positivity of the mass which can be done simply by going to the alternative quantization. Summarizing, we work in alternative quantization with positive mass.
–3–
3
The Phases of holographic fermion and the Role of m and p
We study the phases of holographic fermion as function of dipole coupling p and the Dirac bulk mass m. Six phases appear: these are Fermi Liquid (FL), Bad metal, Bad Metal prime(BM’), Pseudo-gap(PG), Gapped(G) and Semi-metal(SM) phases. The phase boundaries are fuzzy since transitions are all smooth. FL, BM, BM’ and SM are metalic phases, G is insulating and PG is between bad-metal and insulator. To illustrate the character of each phase, we give the temperature evolution of typical spectral functions in Figure 2. In this paper, we use the the symmetrized version A(ω) + A(−ω) of the spectral density, to meet the general tendency in the literature. The difference between the bad metal and its primed phase is whether they have shoulder or not. See Figure 2(d) and (b). Similarly, the difference between the Fermi-liquid and semi-metal is also whether or not there is a shoulder. See Figure 2(a) and (c). The difference of Bad metal prime (BM’) and Pseudo gap is whether or not the central peak is higher or lower than the shoulder. See Figure 2(b) and (e). The difference of semi-metal and Gapped phase is whether they have central peak or not. See Figure 2(c) and (f). T 0.1
1.4 T 0.1 T 0.46
T 0.28
1.2
T 0.22
1.2
T 0.82 4
T 0.1
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(c) semi-metal (SM) T 0.1 7
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(e) pseudogap (PG)
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ω
(f) gapped (G)
Figure 2. (a)-(e):Typical Fermion Phases and their Temperature evolutions. a) FL with (p=0.5, m=0.2), (b) BM’ with shoulder (p=2, m=0.1,0.4), (c) SM (p=6, m=0.45), (d) BM without shoulder (p=0.5, m=0.1), (e) PG (p=2.5,m=0.15), (f) G (p=6, m=0.15)
The result of the detailed study of phases are summarized by the phase diagram given in Figure 5(a). The dashed line represents the free fermion wall, the FL phase is located at the upper-left corner and Gapped phase is at the lower-right corner. All other phases are in between the two and can be understood as their proximity and competition effects. Notice also that the phase diagram is divided by the line of m ' 0.35: the lower half region is where gap-generation phenomena is observed as we expected from the presence of the dipole term. However, in the upper region, a new metalic phase appears instead of gapped one. We call it semi-metal phase, because significant fraction of density of state is depleted from the quasi-particle peak near the fermi level and moved to the shoulder region which is called Hubbard side peak. The emergence
–4–
of this new metalic phase in the strongly coupled system was unexpected, which might be as surprising as when DMFT calculation discovered a narrow quasiparticle peak between the two Hubbard bands [21–24]. To understand its appearance in our theory, we first study the effect of the dipole term on the spectral density of near m = 0.5 line, which is drawn in Figure 3. 3
(a) m = 0.5, p = 0
(b) m = 0.5, p = 5
(c) m = 0.47, p = 5
Figure 3. Contour plot of spcectral density for m ∼ 0.5. Red dotted line indicate ω + µ = ±k. (a) At p = 0, the DOS lives only at the IR-light-cone, (b) At p = 5, only a fraction of DOS lives on IR-lightcone. (c) Away from the m = 1/2, the reconnection of spectral curves happen to avoid the ’crossing with interaction’ off the IR-light-cone.
It turns out that the ’free fermion’ line or the IR-light-cone exists as far as m = 0.5 even in the presence of the dipole term although more and more degrees of freedom are depleted from the quasi-particle peak and moved to the shoulder region as we increase the strength of the dipole term. We call the line m = 0.5 ’the free fermion Wall’. When we move away from the line of m = 1/2, the interaction causes the reconnection of spectral curves to avoid the crossing and the IR-light-cone is not sustained. Consequently, the density profile moves from Figure 3(b) to (c). One important effect of the dipole term is the creation of new branch by splitting the the spectral density on the IR lightcone of into two pieces by creating a new spectral density curves away from the IR- light cone of Figure 3(a). As p increases, it push down the new middle band and pushes up the upper middle band so that the net effect is creating and increasing the gap. The third effect is to make the middle band sharper which means it keep transferring the spectral density from the middle peak to the shoulder peak. This is similar to the effect of U in the DMFT calculation of Hubbard model. With the understanding of how new branch of spectral density, we can understand the the role of mass to create the semi-metal phase: as one can see in Figure 4, the role of the mass is to push up the new branch of the the new branch created by the dipole term so that the new branch can cross the fermi level. This effect competes with that of increasing p which lower the new conduction branch, but the effect of mass is order of magnitude stronger. For m > 0.35 the new branch always crosses and new conduction band is available accordingly. This is the mechanism of the SM phase and once we understand the presence of the SM phase, the bad metal regime can be understood as the transition zone from gapless to the SM phases. We draw 3
Previously, the free fermion phase near the m = 0.5 was noticed by Leiden group [11] at p = 0 and here we study it in the presence of the gap generating dipole term.
–5–
the appearance of the new conduction band as we increase the mass for the given p in Figure 4. Notice that the new band touch the fermi level at m = 0.35 and for all p. Such tangential behavior generate the caustic singularity in phase diagram in the phase boundary of (SM)-G phases. Apparent band crossing in the figure (c) is due to the symmetrization A(ω) + A(−ω).
(a) m = 0.25
(b) m = 0.35
(c) m = 0.45
Figure 4. m-evolution at p = 6 shows the origin of the the semi-metalic phase: mass pushes up(down) the lower (upper) conduction band.
The appearance of PG, BM and BM’ phases can be best understood as the proximity effects of two extreme zones: one is the fermi liquid at the upper left corner of the phase diagram and and the other is gapped phases at the lower right corner. For small mass, the dipole interaction leads to metal-insulator transition as p increases: For example, at p = m = 0, the dispersion curve is a straight lines which we call ’IR light-cone. As p increases, the linear dispersion curve splits into two semi-parabolas. As p grows further, the upper one goes up further and lower one goes down, which is why gap is generated for large p. For p > 4.8 and m = 0, gap is dynamically generated as it was first shown by Phillips et.al [14]. For larger mass, the gap generation requests slightly higher values of dipole strength. The pseudo-gap is nothing but the intermediate zone of this smooth transition, namely 0.8 < p < 4.8. In the phase diagram Figure 5, there is a large territory of PG regime and the color grading qualitatively indicates the change of its character from one boundary to the other. Similarly, for larger mass, the dipole term drives BM’ -SM transition. When m ' 0.35, the new band crosses the fermi level. This is why strong dipole interaction leads to new metal rather then a Mott insulator. As p increases the new band is narrowed and sharpened but it never disappears even for very large dipole strength. For the details of evolution of phases along the constant m-line or p-line, See the figures in appendix. Finally, the effect of the temperature is two folds: first it smooths out sharpness of the quasi-particle peak as one can see in In FL and BM phase. See FIgure 2 (a)(d). Secondly it diminishes the interaction effect like ’transfer’ so that in BM’,PS and SM phases, the transfer of DOS from quasi-particle peak to side peak is not enough so that the the quasi-particle peak at higher temperature is higher than that of the lower temperature. See FIgure 2 (b)(c)(e). Such character gives us a prediction that some, if not all, transition metal oxides have temperature intervals where conductivity increases, contrary to the intuition.
–6–
m
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(a) Phase diagram
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(b) Embedding of Hubbard models
Figure 5. (a) Phase diagram in (p, m) space. Six phases appear and all transitions are smooth crossover except SM-G transition. (b)An embedding defines a holographic Hubbard model: α = 1/50 (1/20) for yellow (red) line in straight line embedding. β =8/9 (4/3) for blue (green) curve in hyperbolic embedding.
4
Holographic Hubbard Models
Traditionally, the physics of Mott transition is usually discussed using Hubbard model which describe the transition as the competition of hopping and onsite coulomb repulsion. So one of the test of the usefulness of holographic method is to see if it can encode the physics of Hubbard model at least qualitatively. It is given by the Hamiltonian X † X X H = −t( ciσ cjσ + H.c.) + U niσ ni-σ + V ni nj + V2 + V3 + (4.1)
where and σ is for spin index, t is the hopping rate, U represents the on-site Coulomb repulsion and V is for the nearest off-site repulsion and Vn is next to nearest repulsion. These can be set to be zero in leading order. The purpose of this section is to discuss how to realize the physics of the Hubbard model in terms of the simplest holographic model and to compare the result with DMFT calculations and experiment. Here, the question is, for any Hubbard model with given values of t, U, V, Vi , · · · , can we find m and p that produce the same or at least qualitatively the same spectral density. The answer seems to be yes, because we already have seen that our phase diagram contains all six phases that are available for the Hubbard model, therefore if we consider a line in the phase diagram connecting two points, one in Fermi liquid phase and the other in Gapped phase, then it will give a mapping of an Hubbard model into a holographic model and each such a path in (p, m) space defines a ”holographic Hubbard model”. Here we consider the simplest embedding for U >> V and Vi = 0 for i ≥ 2. Since both p and U are gap generation mechanism in each side, it is natural to identify these: p = U/t. To identify the m in terms of the Hubbard model parameter, we need to observe following two: i) m = 1/2 line has singularly strong effect of restoring the free fermion nature in the sense that there is a metalic spectrum around fermi sea with dispersion relation ω + µ = ±k, whatever large value p may have, as we can see in Figure 3. Therefore we expect that t → ∞ when
–7–
m → 1/2 − 0+ ; ii) m, by which we mean mL with L being the radius of AdS, is related to the long range coulomb repulsion V . Then it is reasonable to identify V /t = (1/2 − m) and V = αU with α 1/2, the second term(E+ ) dominate but it can be cancelled by counter terms [30], which do not contribute any finite terms to the effective action. Second, in the standard quantization where we fix ψ+ at the boundary, A’s are the source terms. While in the alternative quantizaton where we fix ψ− at the boundary, Di is taken to be the source. Therefore, the retarded Green’s function in standard quantization, is given by D1 D2 R G = diag i , −i ≡ diag(GR m > 0, (A.9) + , G− ), A1 A2 while that in alternative quantization is given by A1 A2 ˜ ˜R ˜R G = diag i , −i ≡ diag(G + , G− ) D1 D2 R = −diag(1/GR + , 1/G− ),
m > 0.
(A.10)
˜ R , the Green function for Since GR for m < 0 case, can be also obtained by GR → −1/GR , G the alternative quantization for m > 0, is the same as that for −m in the standard quantization: ˜ R (ω, k; m) = −1/GR (ω, k; m) = GR (ω, k; −m). G ± ± ∓ Introducing the ξ± by ξ+ = i
y− , and z+
ξ− = −i
z− , y+
(A.11)
(A.12)
the equations of motion Eqs.(A.2) can be recast into two independent equations for ξ± : r gxx 0 √ √ √ 2 . (A.13) ξ± = −2m gxx ξ± + [u(r) + p gxx A0t (r) ∓ k] + [u(r) − p gxx A0t (r) ± k]ξ± grr and the Green functions for m < 1/2 can be written as 2m GR ξ± (r, ω, k). ± (ω, k) = lim r r→∞
A.2
(A.14)
Gap generation versus Appearance of semi-metal phase
We first study the lower half of the phase diagram by calculating its evolution along the line m = 0.1 with increasing the dipole strength. The result is is drawn in Figures 8 (a)-(h), where three different phases appear: 1. Fig. 8 (a,e) p = 1, Bad metal phase with broadened peak with low height at fermi level, 2. Fig. 8 (b,f) p = 2, Psuedo-gap phase with incomplete depletion of DOS at fermi level. 3. Fig. 8 (c,g) p = 6, Gapped phase; Fig. 8 (d,h) p = 8, Gapped phase with increased gap size.
– 12 –
(b) p = 2
1.5
1.0
(d) p = 8
10
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(a) p = 1
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(e) p = 1
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ω
ω
(f) p = 2
(g) p = 6
2
4
0 -4
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0
2
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(h) p = 8
Figure 8. (a)-(d) on p-evolution of the DOS along the line m = 0.1 show appearance of and pseudo-gap and gapped phase; (e)-(h) spectral function along vertical red line in each figure (a)-(d).
The overall feature of the evolution is from metalic to the insulating phase with pseudo gap phase in the middle and it agrees with our expectation. We now show the evolution along the line of m = 0.45 with increasing p to demonstrate the changes of phase in the upper half part of the phase diagram. The ω-plots in Figures 9(e)-(h) are is along the red vertical redlines in Figure 9(a)-(d) respectively. We can see three different phases: 1. Gapless metalic phase with linear dispersion: it is a fermi liquid (FL) regime. 2. The bad metal phase due to development of incomplete generation of conduction band. 3. New metalic phase which we call semi-metal due to the development of the conduction band. Semi-metal because half of the DOS at the fermi sea is depleted and moved to shoulder region. A.3
Role of the bulk mass
We now study the role of mass more systematically by calculating the evolution of the DOS at two nonzero fixed values of p, that is along two vertical lines p = 2.5 and p = 6.0 in phase dragram. In the Figure 10(a)-(h), the m-evolution along p = 0.2 line is drawn, where a few physically interesting phases appear. From the figures 10(i)-(p), we can see that the bulk mass sharpens the peak at the fermi surface consistently regardless of the value of p. One can see that increasing m pushes up the lower middle band so that gap is reduced. When the middle band crosses the fermi level, central peak appears signaling the creation of the semi-metalic phase. For both cases the final stage is SM phase.
– 13 –
(a) p = 1
(b) p = 2
A(ω)
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(c) p = 3
1
(e) p = 1
2
3
ω
-3
-2
-1
0
1
2
3
ω
-3
(f) p = 2
-2
-1
(d) p = 6
A(ω)
A(ω)
1.5
1.5
1.0
1.0
0.5
0.5
0
1
(g) p = 3
2
3
ω
-3
-2
-1
0
1
2
3
ω
(h) p = 6
Figure 9. (a)-(d):p-evolution of the DOS at m = 0.45 shows appearance of and bad metal prime and semi-metal; (e)-(h) spectral function along vertical red line in each figure (a)-(d).
The figure 11(a) shows that in the absence of the bulk mass, the DOS peak in ω-plot goes away very rapidly as soon as we turn on temperature. That is, quasi-particles are fragile at finite temperature, which is the character of non-fermi liquids. On the other hand, if we turn on the mass, the fermi surface peaks becomes sharper as we can see in the Figure 11(b). The same phenomena can be seen in the k-plot: in Figure 11(c) and (d), we contrasted the temperature evolution of the fermi surface without and with bulk mass. which is contrasted with the nonzero mass case where the fermi surface is remained at substantially high temperature. See the Figure 11(c). That is, the fermi liquid behavior is enhanced by the presence of the bulk mass. Here again, we can see the role of mass is the stabilizer of the quasi-particle nature in holographic matter. As m → 1/2, such ’quasi-particle stabilizing tendency’ increases infinitely so that the system is a fermi liquid whatever strength is the dipole term. In fact, the spectral function shows that the dispersion curve is straight line just as if we study a free fermions. Therefore it is natural to think the bulk fermions with mass near 1/2 describe a weakly interacting system, i.e, quasiparticles. For applications to the realistic material, having such a dial to make the system fermi liquid in a limit is very useful and in fact essential because in the real experiments, one can tune the coupling by applying pressure or doping rate. In fact, for p = 0, it can be understood if we notice that ψ is the dual of the operator with dimension ∆ = d/2 − m which is dimension of free fermion when m = 1/2 so that the fermion with m = 1/2 in alternate quantization is dual to the free fermion [8, 11]. In the presence of the dipole coupling whose role is to introduce a gap which break the conformal symmetry dynamically, there is no guarantee that such trend continue to hold. Our observation is that, nevertheless, such free fermion nature at m = 1/2 persists even in the presence of the dipole interaction regardless of its strength. We call it Free
– 14 –
(a) m = 0 p = 2.5
(b) m = 0.15 p = 2.5
A(ω)
(c) m = 0.3 p = 2.5
A(ω)
(d) m = 0.45 p = 2.5 A(ω)
A(ω)
1.4
1.4
1.2 1.0
1.2
1.2
1.0 0.8
1.0
1.0
0.8 0.8
0.8
0.6 0.6
0.6
0.6 0.4
0.4
0.4 0.2
0.2
-4
2
-2
4
ω
-1.5
-1.0
0.5
-0.5
0.4
0.2
1.0
1.5
ω
-1.5
-1.0
0.2
0.5
-0.5
1.0
1.5
ω
-2
1
-1
2
(e) m = 0 p = 2.5
(f) m = 0.15 p = 2.5
(g) m = 0.3 p = 2.5
(h) m = 0.45 p = 2.5
(i) m = 0 p = 6
(j) m = 0.15 p = 6
(k) m = 0.3 p = 6
(l) m = 0.45 p = 6
A(ω)
A(ω)
ω
A(ω)
A(ω)
12
2.0
25
20
10
20
8
1.5
15 15
6
1.0
10 10
4
-4
-2
0.5
5
2
2
4
(m) m = 0 p = 6
ω
-4
-2
5
2
4
ω -4
(n) m = 0.15 p = 6
-2
2
4
(o) m = 0.3 p = 6
ω
-4
-2
2
4
ω
(p) m = 0.45 p = 6
Figure 10. (a)-(h): m- evolution of the spectral density with increasing m with p = 2.5. (i)-(p):Evolution of the spectral density at p = 6.0. For both cases the final stage is SM phase.
fermion Wall in m-p phase diagram. After careful examination of full range of phase diagram, we found that at mass region m > 0.35, some metalic phases exist always. A.4
Hyperboic Embeddings
Following embedding is interesting since it illuminate many hidden aspects of the theory including the physical origin of the bulk mass. We set V 6= 0 but fixed instead of being related to U. p = (U + V )/t, and 2m = V /U ≤ 1. (A.15)
– 15 –
60
m0
T 0
m 0.224
T 0.1 50 T 0.4
m 0.448
15
A(k)
A(k)
ω=0
ω=0
40
30
10
20
5
10
0
0 -2
-1
0
1
-2
2
-1
0
1
2
k
k
(a) Fast decay of FS by Temperature
(b) Mass as FS stabilizer 10
T 0.1
3.0
T 0.1
T 0.2 2.5
T 0.2 8
T 0.3
6
A(k)
A(k)
ω=0
ω=0
2.0
T 0.3
1.5
4 1.0
2 0.5
0
0.0 -2
-1
0
1
2
-2
-1
0
1
2
k
k
(c) T-evolution of FS at m = 0
(d) T-evolution of FS at m = 0.3
Figure 11. Role of mass in stabilizing the fermi surface(FS): (a)T = 0, 0.1, 0.4 at m = p = 0. (b)m = 0, 0.224, 0.448 at T = 0.1 and p = 0. (c) with zero mass, (c) with non-zero mass m = 0.3.
The constraint V /U ≤ 1 comes since the off-site repulsion is weaker than the on-site one. The corresponding holographic Hubbard model is a hyperbola p = β(
1 V + 1), with β = = f ixed. 2m t
(A.16)
This embedding has following consistency properties. • In large U limit, both t-U -V model and its gravity dual m-p model are in Gapped phase and even in the smallest U limit, it does not pass the obvious free fermion point (m, p) = (1/2, 0) because U ≥ V . • The first embedding with β = 4/3 (Blue curve) passes SM → BM 0 → P G → G and the second one with β = 8/9 (Green curve) passes F L → BM 0 → P G → G phases as U increases, The latter is also qualitatively consistent with evolution of Hubbard model calculated with multi-site DMFT. The graph is given below. Although we know the reason for the free fermion nature of m = 1/2 by counting the conformal weight, it is still somewhat mysterious from the interaction point of view: How such freeness is achieved in a theory of strong interaction? The bulk mass enter into dynamics only in a combination mL with L being the AdS radius which encodes the interaction strength of the boundary theory through L ∼ λ1/4 . In fact, 1/2 is the maximal value of mL within the unitarity bound. The most natural way to identify the mL’s gauge coupling dependance is the off-site coulomb interaction V , and following argument shows a plausible way to understand the free
– 16 –
0.5 70 2.5 0.4
0.4
0.3
0.3
60
1.0
0.2
50
A(ω)
1.5
A(ω)
A(ω)
A(ω)
2.0
0.2
40 30 20
0.5
0.1
0.1
0.0
0.0
10
0.0 -3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
0 -3
-2
-1
ω
ω
(a)
0
1
2
3
-3
(b)
0
-1
1
2
3
1
2
3
ω
(c)
(d)
0.6
2.0 0.7
0.5
0.6
0.4
0.4
1.5
0.5
A(ω)
A(ω)
0.6
0.3
A(ω)
0.8
A(ω)
-2
ω
0.4
0.2
0.2
0.1
-3
-2
-1
0
ω
(e)
1
2
3
0.5
0.1
0.0
0.0
1.0
0.3
0.2
0.0
0.0 -3
-2
-1
0
1
2
3
-3
ω
(f)
-2
-1
0
1
2
3
-3
-2
0
-1
ω
ω
(g)
(h)
Figure 12. U -evolutions at kc for V 6= 0 cases (Hyperbolic Embedding) . (a)-(d) for upper (Blue) curve, (e)-(h) for lower (Green) curve, Spectral functions at kc for V 6= 0 (Green Curve).
fermion nature. And this is indeed the basis of identification m = V /2U together with U > V in hyperbolic embedding. In [31], it is argued that off-site long range Coulomb interaction V increases the effective hopping rate t by tef f ≈ t/(1 − V /U ). The corresponding effective t-U model has effective coupling U/tef f ∼ U (1 − V /U )/t, which can be significantly reduced in the limit U → V . Therefore the system should be in a fermi liquid phase. In fact U can never reach V since onsite repulsion should be always larger than off-site one. In this sense, m = 1/2 is a singular limit which can not be actually reached by interacting fermions.
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