Realizing the Haldane Phase with Bosons in Optical Lattices

Realizing the Haldane phase with bosons in optical lattices Junjun Xu,1, 2 Qiang Gu,1, ∗ and Erich J. Mueller2, †

arXiv:1703.05842v1 [cond-mat.quant-gas] 16 Mar 2017

2

1 Department of Physics, University of Science and Technology Beijing, Beijing 100083, China Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA (Dated: March 20, 2017)

We analyze an experimentally realizable model of bosons in a zig-zag optical lattice, showing that by rapidly modulating the magnetic field one can tune interaction parameters and realize an analog of the Haldane phase. We explain how quantum gas microscopy can be used to detect this phase’s non-local string order and its topological edge states. We model the detection process. We also find that this model supports a supersolid phase, but argue that it only occurs at parameter values which would be challenging to realize in an experiment.

In the past 30 years, one of the dominant themes in condensed matter theory has been the search for models where the collective excitations behave like particles whose properties are unlike any known fundamental particle. While many such fractionalized and topologically ordered models have been found [1], very few of them have been experimentally realized. Here we show how to build on a setup proposed by the NIST cold atom experimental group [2] to explore one of the iconic fractionalized phases, the Haldane phase of a spin-1 chain [3]. In 1983, Haldane showed that the properties of integer and half-integer spin chains can be profoundly different [3, 4]. Over the following decade, several researchers explored the rich properties of the integer spin chain, finding half integer spin edge modes [5–7], and non-local string order [8–10]. More recently, Dalla Torre, Berg, and Altman noted that similar physics should occur for spinless bosons hopping on a one-dimensional lattice: the occupation numbers on each site plays the role of the different spin states [11, 12]. Subsequently, analogs of the Haldane phase have been predicted for a number of one-dimensional Bose-Hubbard models with off-site interactions [13–16]. One enlarges the parameter range over which the Haldane phase is stable if there is a constraint on the maximum number of particles per site. By combining a number of experimental techniques, we show how to realize a model which would be expected to support the Haldane phase. We use Density Matrix Renormalization Group (DMRG) techniques to calculate the properties of this model [17, 18], and explain how to detect the exotic signatures of the Haldane phase. In a system of 1D lattice bosons, the Haldane (HI) phase lies at the intersection of the density wave (DW) phase, where double occupied sites (doublons) alternate with empty sites (holons), the Mott insulator (MI) phase, where each site is occupied by a single atom, and the superfluid (SF) phase, where the quasiparticles (doublons and holons) are free to move around. In the Haldane phase the quasiparticles are fluid but ordered: their spacing varies, but as one moves from left to right the next quasiparticle after a doublon is a holon, and vice-versa.

This ground state is four-fold degenerate in a large but finite system with hard-wall boundary conditions – corresponding to the flavors of the leftmost and rightmost quasiparticles – which are bound to the edges of the system. Anisimovas et al. showed that by using a onedimensional spin dependent optical lattice and Raman induced hopping, one could produce the zig-zag lattice illustrated in Fig. 1(a), described by the tight-binding model [2] X † H = tD (c1,j c−1,j + c†1,j−1 c−1,j + H.c.) j

−t

X

(c†s,j+1 cs,j + c†s,j cs,j+1 ) +

j,s

+ U2

X

[n1,j + n1,j−1 ]n−1,j .

U1 X ns,j (ns,j − 1) 2 j,s (1)

j

Here s = ±1 labels the spin state of the atoms and j is the position of the atom along the lattices. tD and t characterize the hopping between and within these two spin states. U1 and U2 describe the on-site and nearest interspecies interaction. Following Anisimovas et al, in the top figure we interpret the spin as a “synthetic dimension.” Physically, the Wannier states of the two spin states lie in a line, alternating one spin-state with another, as illustrated in the bottom figure. We will generally use this latter interpretation, and introduce operators b2j = c−1,j and b2j+1 = c+1,j , in which case tD , t are interpreted as nearest and next-nearest neighbor hopping parameters, while U1 , U2 are on-site P and nearest neighbor interaction parameters, i.e. H = tD i (b†i+1 bi +b†i bi+1 )− P P P t i (b†i+2 bi + b†i bi+2 ) + U21 i ni (ni − 1) + U2 i ni+1 ni . Aside from the longer range hopping, Eq. (1) maps onto the model introduced by Dalla Torre et al. in considering polar molecules in optical lattices. Unfortunately, based on their analysis, one expects that the Haldane phase is not stable when U1 /U2 is large – which is the physically relevant regime considered in [2]. (Our numerics confirm this expectation.) Here we argue that by using a Feshbach resonance, one can reduce U1 , driving the system into the Haldane phase. The lossy nature of bosonic

2

5

MI SF

U1 /tD

4 3

O

6

0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ■■■■ ◆ ■■■ ●● ◆◆ ●◆ ■■ ●◆ ●◆ ●◆ ■■ ●◆ ●◆ ■■ ●◆ ●◆ ●◆ ■■ ● ◆ ● ■ ◆ ● ■ ◆ ● ■ ◆ ● ■ ■ ◆ ● ■ ■ ■ ◆ ● ■ ■■ ◆ ● ■■ ■■ ◆ ■■■ ● ■■■■ ◆ ● ■■■■■■■■■■■■■■ ◆ ●◆ ●● ◆◆ ●●●●●●●●●●●●●●●●●●●●●●●●●●● ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆

HI

t /tD

2 U1 as

B B2

U0 Ueff 0

B1

B2

1

DW

0 0.0

0.2

0.4

0.6

0.8

1.0

t /tD B

B1

time

FIG. 1. (a) Schematic of the experimental system, which can be interpreted as a zig-zag ladder or a one-dimensional lattice with next-nearest neighbor hopping. The green and orange colors label two different spin states s = ±1. (b) On-site interaction U1 in F = 1, mF = 0 (green dashed) and mF = 1 (solid orange) hyperfine states of 87 Rb. The background interaction strength U0 corresponds to the value of U1 away from the Feshbach resonances near the zero-crossings at B1 and B2 . Rapidly switching the magnetic field between B1 and B2 , as illustrated in (c), yields an effective time-averaged on-site interaction Ueff = U0 /2 in both channels.

Feshbach resonances aids us, as the quantum Zeno effect converts the resulting large 3-body recombination rate into a suppression of the probability of having more than two particles on any given site – further stabilizing the Haldane phase. More concretely, we consider the F = 1, mF = 1, 0 states of 87 Rb. The coefficient U1 is proportional to the scattering length associated with two atoms in the same magnetic sublevel. As illustrated in Fig. 1(b), this scattering length can be manipulated by applying a magnetic field. Near B1 ∼ 661.43G there is a zero-crossing where the interactions between two mF = 0 atoms vanish, while near B2 ∼ 685.43G there is a similar zero crossing for mF = 1 [19]. We envision rapidly switching the magnetic field between these two fields, as illustrated in Fig. 1(c). As long as the switching time is short compared to the other scales in the problem (h/U, h/t ∼ h/ER ≈ 0.27ms for laser wavelength λ = 789nm) the effective interaction in each spin channel will be given Rby time-averaging t the instantaneous Hamiltonian Ueff = 0 U (τ )dτ [20–22]. Even though at any given time the interactions in the two channels will be different, this time averaged interaction is the same for each spin species, and U1 will be the same on all sites. This technique effectively halves the strength of the on-site interaction as in Fig. 1(b). The coefficient U2 is largely unaffected. We find that one can achieve a ratio of on-site to nearest neighbor interaction of U1 /U2 ≈ 1.6, for a lattice depth of V0 = 2ER . By

FIG. 2. Phase diagram of the model in Eq. (1), showing regions where experiments observables exceed selected thresholds. Here U2 /tD = 2.5, and the maximum occupation of any site is 2. The solid and dashed boundaries correspond to O = 0.05/0.1. Inset shows these order parameters as for U1 /tD = 1 with the solid blue/dotted green/dashed orange lines corresponding to the string/parity/density wave order parameters.

appropriately tuning the transverse confinement, one can take U2 /tD = 2.5, yielding U1 /tD = 4. Figure 2 shows the phase diagram for this model, revealing that these parameters place the system within the Haldane phase regime. The zero crossings are very close to Feshbach resonances, and hence induce a large 3-body loss rate K3 . In the present circumstance this is advantageous. Following the logic in [23], when K3 is large, there is a strong suppression of the process in which a third particle hops onto a site containing two other particles. This suppression can be modelled by a complex on-site three-body repulsion of strength U3b ∼ −ihK3 n2 /12. We estimate that one can get |U3b | ∼ 10ER for a typical on-site particle density n ∼ 2.1 × 1015 cm−3 and typical on-resonance three-body rate K3 ∼ 10−25 cm6 /s. Since |U3b | is larger than the other scales in the problem, it can be replaced by a constraint that no more than two particles can occupy any site. With this constraint we use the DMRG to calculate the properties of the model in Eq. (1). This technique is typically understood as systematically optimizing a variational wavefunction in the form of a matrix product state [18]. The degree of approximation is controlled by the bond dimension d. We considered experimentally realizable L = 60 sites with one particle per site, and typically take d = 35. We systematically varied the bond dimension, and found no significant difference in our results if we further increased d. The algorithm is more efficient if we alter the boundary conditions to break the potential four-fold degeneracy of the Haldane-phase groundstate, and the potential two-fold degeneracy of the density wave groundstate. These boundary conditions pin a vacancy

3

HI

1 0 2 0 1 1 1 1 2 0 2 0 2 1 1 1 0 2 0 1 1 1 0 2 0 1 2 0 2 1 1 0 1 2 0 2 0 2 0 1

SF

1 1 1 0 1 2 1 1 2 0 2 0 2 0 1 1 2 1 0 0 0 2 1 2 1 2 0 1 2 2 0 1 0 1 2 1 0 1 1 2

DW

0 2 0 2 0 2 0 2 0 2 0 2 0 1 2 0 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 1 0 2 0 2 0 2 0 2

MI

1 1 1 1 1 1 0 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 2 1

SS

0 2 1 1 0 2 0 1 1 3 0 2 0 1 0 1 0 3 0 2 0 1 1 2 0 2 0 2 0 1 2 1 0 0 3 1 0 0 2 2

FIG. 3. Typical configurations of occupation numbers extracted from the central 20 sites of our DMRG wavefunctions, modelling single-shot quantum gas microscope images. Configurations correspond to different phases: HI(U1 /tD = 4, U2 /tD = 2.5 for t/tD = 0), SF(U1 /tD = 4, U2 /tD = 2.5 for t/tD = 1), DW(U1 /tD = 1, U2 /tD = 2.5 for t/tD = 0), MI(U1 /tD = 6, U2 /tD = 2.5 for t/tD = 0), and SS(U1 /tD = 0, U2 /tD = 2.5 for t/tD = 0.8 for maximum occupation number 3). Each circle resembles a single site, and the number in the circle tells how many atoms are on this site. We show two independent realizations for each phase.

at the left-most site, and doublon at the right-most site, but do not alter the physics in the bulk. The order in the Haldane, Mott insulator, and density wave phases are encoded in string, parity and density wave correlation functions [8, 12] D E P str Cij = δni eiπ i