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Lin Yang, Hao Weichang, Guo Huaiming. Quantum Monte Carlo study of hard-core bosons in Creutz ladder with zero flux . Chinese Physics B, 2018, 27(1): 010204  
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Quantum Monte Carlo study of hard-core bosons in Creutz ladder with zero flux
Lin Yang1, Hao Weichang2, Guo Huaiming1, †
Department of Physics, Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), Beihang University, Beijing 100191, China
Department of Physics, Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), Beihang University, Beijing 100191, China

 

† Corresponding author. E-mail: hmguo@buaa.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11274032, 11774019, 51472016, and 51672018).

Abstract

The quantum phase of hard-core bosons in Creutz ladder with zero flux is studied. For a specific regime of the parameters (tx = tp, ty < 0), the exact ground-state is found analytically, which is a dimerized insulator with one electron bound in each rung of the ladder. For the case tx,ty,tp > 0, the system is exactly studied using quantum Monte Carlo (QMC) method without a sign problem. It is found that the system is a Mott insulator for small tp and a quantum phase transition to a superfluid phase is driven by increasing tp. The critical is determined precisely by a scaling analysis. Since it is possible that the Creutz ladder is realized experimentally, the theoretical results are interesting to the cold-atom experiments.

PACS: 02.70.Ss;05.30.Rt
Keyword:quantum Monte Carlo method;Creutz ladder;Mott insulator
1. Introduction

Creutz model is a cross-linked ladder threaded by a flux, in which Creutz shows that edge states emerge at open boundaries.[1,2] While Creutz’s study is in the context of lattice gauge theory, the model can be thought as a one-dimensional (1D) topological insulator, which belongs to AIII class and is characterized by an integer.[3] The Creutz ladder is closely related to two-dimensional Bernevig–Hughes–Zhang model describing two-dimensional topological insulator HgTe quantum well and it can be obtained through a dimensional reduction from the latter.[4–8] Based on the Creutz ladder, many studies have been carried out, such as the effects of interaction and disorder in the topological phase. For specific parameters, Creutz ladder has flat bands, in which fermions and bosons exhibit exotic quantum phases.[11–13] With induced s-wave superconductivity in the Creutz ladder, a transition from fractionally charged solitons to Majorana bound states is found.[14] The Creutz ladder may be realized in cold atoms and their concrete implementations have been proposed.[15,16]

Besides the Creutz ladder, its variational versions have also attracted increasing interests, one of which is the ladder without next-nearest-neighbor interchain hopping.[17,18] It is found that free-fermions and hard-core bosons in such a ladder exhibit different physics. One of them is that the charge gap exists as long as the transverse hopping ty ≠ 0 for hard-core bosons, while a large enough ty is needed to open a charge gap for fermions. It is also shown that the Meissner currents and a vortex phase emerge in the presence of a uniform magnetic field.[19,20]

Fermions and bosons on a ladder subjected to magnetic fields have been the topic of current theoretical and experimental works. It is noted that the studies of Creutz ladder mainly focus on fermions and the behaviors of bosons in Creutz ladder are largely unexplored. In this work, the quantum phase of hard-core bosons in Creutz ladder with zero flux is studied. For a specific regime of the parameters, we analytically obtain the ground-state, which is a dimerized insulator with one electron bound in each rung of the ladder. Then we apply quantum Monte Carlo (QMC) method to study a parameters’ regime without a sign problem.[21,22,25–29] We find that the system is a Mott insulator for small next-nearest-neighbor interchain hopping tp and a quantum phase transition to a superfluid phase is driven by increasing tp. By performing a scaling analysis, the critical is precisely determined. Since it is possible that the Creutz ladder is realized experimentally, the theoretical results are interesting to the cold-atom experiments.

2. Results
2.1. Creutz ladder

The Creutz ladder is described by the Hamiltonian where is the creation (annihilation) operator of fermion on the i-th site with l = 1,2 labeling the legs of the ladder; tx, ty, and tp are the longitudinal, transverse, and next-nearest-neighbor interchain hopping amplitudes respectively, and tx is set to 1 as the energy scale; 2ϕ is a magnetic flux threading each plaquette; is the number operator; and μ is the chemical potential. In the momentum space, the Hamiltonian is given byℋ(k) = −(h0I + hxσx + hzσz), where h0 = 2tx cosϕ cos k, hx = ty + 2tp cos k, hz = 2tx sinϕ sin k; σx and σz are Pauli matrices; and I is the identity matrix. Its energy spectrum is obtained directly as . The system is gapped except the cases when sin ϕ = 0 and |ty| ≤ 2 or |ty| = ± 2. The topological property of the insulators can be characterized by the Berry phase, which is defined as[23,24] with φ(k) the eigenfunction of the occupied states. For |ty| < 2, the Berry phase γ = π and the insulator is topological nontrivial, while the insulators with |ty| > 2 are trivial with γ = 0.

Then we load hard-core bosons into the Creutz ladder. The Hamiltonian of such system has the same form as the one in Eq. (1) except that the creation (annihilation) operator of fermion is replaced by the corresponding ones of hard-core bosons . The hard-core boson operators obey commutation relationship for i ≠ j, but anti-commutation one on the sole site. This makes the hard-core bosons exhibit different physical properties from the fermions. We first compare the ground-state energies of bosons and fermions with the same form of the Hamiltonian. As shown in Fig. 1, hard-core bosons usually have a lower ground-state energy than fermions. However if the hopping is frustrated due to the presence of the magnetic flux, hard-core bosons can have a higher ground-state energy. The momentum distributions of hard-core bosons and fermions are also different. For noninteracting fermions, each momentum eigenstate below the Fermi energy are filled and nf(k) = 1 for k < kF, whereas hard-core bosons tend to condense near the bottom of the band and the momentum distribution exhibits a peak near the corresponding momentum.

Fig. 1. (color online) The ground-state energies of fermions and hard-core bosons in Creutz ladder as a function of the flux ϕ.

In the remainder of the paper, we study the quantum phases of hard-core bosons in the Creutz ladder. We focus on the case with zero flux, i.e., ϕ = 0, for which the quantum phases can be exactly studied using QMC without a sign problem.

2.2. Quantum phases of hard-core bosons in Creutz ladder with zero flux
2.2.1. Exact ground state for the case tx = tp, ty < 0

The Hamiltonian has a fully dimerized eigenstate |ψ〉 = |0〉1 ⊗ |0〉2 … ⊗|0〉L at half filling, with where i = 1,…,L labels the rung of the ladder and |n1,n2〉 is product state with n1(n2) = 0,1 the hard-core boson numbers on the sites of the rung. To show that |ψ〉 is an eigenstate, we define two local operators on each rung It is noted that . In terms of the rung operators, the Hamiltonian can be written as If tx = tp, the last term vanishes and |ψ〉 is the eigenstate of with the eigenenergy E = Lty. So when ty is negative, the above eigenstate may be the ground state with the energy per rung −|ty|. We also show the exact diagonalization (ED) result in Fig. 2. From a critical ty, the energy from ED is exactly −|ty|, thus the dimerized eigenstate described in Eq. (3) is the ground state.

Fig. 2. (color online) The energy per rung of the ground state as a function of ty. The hopping amplitudes tx = tp = 1.
2.2.2. QMC study of the case ty > 0

When the parameters tx, ty, and tp in Eq. (1) are all positive, the Hamiltonian can be exactly studied using QMC with SSE algorithm. The SSE method calculates the mean density of particles by where n labels the total particle number of the system and L is the system size. The corresponding superfluid density is thus given by[18] where β is the inverse temperature, W is the winding number of the particle world lines, and d is the dimensionality.

We first study the case without next-nearest-neighbor interchain hopping tp to confirm our methodology. The results are demonstrated in Fig. 3. A plateau with vanishing superfluid density in the ρ–μ curve is clearly visible at half-filling for ty = 2, implying the existence of a charge gap and that the system is in a Mott insulating state. However for ty = 1 the ρ–μ curve becomes almost continuous and it is hard to determine whether the plateau exists. In fact the plateau shows its presence through the sharp decrease of the superfluid density. A further scaling analysis shows that the plateau exists down to ty = 0. These results are consistent with the previous work, thus our QMC simulations are reliable.

Fig. 3. (color online) QMC results for the particle density ρ and the superfluid density ρSF as a function of the chemical potential μ without next-nearest-neighbor inter-chain hoppings. The size of the ladder is 2 × 64.

In the following we explore whether the Mott insulator still exists after the next-nearest-neighbor interchain hopping tp are included. The average density ρ at different tp are calculated, shown in Fig. 4. It has been known that at tp = 0 the system is a Mott insulator with a finite charge gap. As tp is included and increased, the plateau representing the size of the gap gets narrower. At around tp = 0.6, the plateau is not visible in the ρ–μ curve. A key issue is whether a critical value exists, beyond which the system becomes gapless superfluid phase. To address this question, we perform scaling analysis of the superfluid density. A superfluid phase is characterized by the persistence of nonzero ρSF as the lattice size increases.

Fig. 4. (color online) The density ρ as a function of μ obtained on 2 × 64 Creutz ladder with zero flux for ty = 2 and tp = 0, 0.2, 0.6. The plateaux get narrower when tp is increased.

To perform the scaling analysis, the superfluid densities are calculated on lattices with different sizes. The length of the chain L is taken from 16 up to 256. The inverse of the temperature β = 4L is low enough to ensure the ground state. For a Mott insulating state, the superfluid density as a function of the lattice size L is expected as ρSF ~ e−L/ξ. As shown in Fig. 5, the above formula fits the data well. A further check of the wellness is to collapse the data, i.e., those in Fig. 5 are analyzed by rescaling the x-axis L → L/ξ (see Fig. 6). Then the correlation length ξ can be extracted. If a critical point exists, the correlation length should be divergent at the critical point.

Fig. 5. (color online) The superfluid density versus the length of the ladder L for various values of tp with ty = 2. tp varies from 0.6 to 2.0 with the interval 0.1.
Fig. 6. (color online) Collapse the data in Fig. 5 by rescaling the x-axis L → L/ξ.

We plot ξ curve versus tp in Fig. 7. It is obvious that ξ increases exponentially with tp. The function form is used to fit the data. The fitting process gives . Thus the correlation length diverges here and the superfluid density is finite in the thermodynamic limit. The Mott insulating state is driven to a gapless superfluid phase by a finite next-nearest-neighbor interchain hopping. We also analyze the cases with other ty using the same method. For ty = 1, the scaling analysis yields the critical value .

Fig. 7. (color online) The correlation length ξ versus next-nearest-neighbor interchain hopping tp. The function form is used to fit the points.
3. Conclusion

The quantum phase of hard-core bosons in Creutz ladder with zero flux is studied. For a specific regime of the parameters (tx = tp, ty < 0), the exact ground-state is found analytically, which is a dimerized insulator with one electron bound in each rung of the ladder. For the case tx, ty, tp > 0, the system is exactly studied using QMC method without a sign problem. It is found that the system is a Mott insulator for small tp and a quantum phase transition to a superfluid phase is driven by increasing tp. The critical is determined precisely by a scaling analysis. Since it is possible that the Creutz ladder is realized experimentally, the theoretical results are interesting to the cold-atom experiments.

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