Introduction

Quantum information and condensed matter physics have been increasingly cross-fertilizing each other in recent years1,2. Within this trend, quantum entanglement was proposed to detect the field-theoretical and topological properties of quantum many-body systems3,4,5,6. For example, it usually offers a direct connection to the conformal field theory (CFT) and categorical description7,8,9,10,11,12,13,14,15,16,17. Beyond the entanglement entropy (EE), Li and Haldane proposed that the entanglement spectrum (ES) is a more fundamental entanglement characteristic to quantify the intrinsic information of many-body systems2,18,19,20,21. They also suggested a profound correspondence between the ES of an entangled region with the energy spectra on the related virtual edges, which is the so-called Li-Haldane conjecture. Thereafter, low-lying ES has been widely used as a fingerprint to investigate CFT and topology in highly entangled quantum systems22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45.

However, most of the ES studies so far have focused on (quasi) 1D systems numerically, due to the exponential growth of computation complexity and memory cost. The existing numerical methods such as the exact diagonalization (ED) and the density matrix renormalization group (DMRG) have significant limitations for approaching large entangling region. We find it noteworthy that the quantum Monte Carlo (QMC) serves as a powerful tool for studying large-size and higher dimensional open quantum many-body systems46,47. QMC has been refined to extract the entanglement spectral function combined with numerical analytic continuation36,37,38,43,44,45,48,49,50,51,52,53,54,55,56. Though the method overcomes the exponential wall problem and can obtain large-scale entanglement spectral functions, it suffers from limited precision and fails to distinguish the fine levels, while the level structure always carries important information about CFT and topology. Moreover, the spectral function is also different from the spectrum, which we will further compare to demonstrate that their characteristics sometimes may be totally different. The other way to extract ES through QMC is to reconstruct it according to the knowledge of many integer Rényi entropies because integer Rényi entropies can be calculated by QMC30. The difficulty of this scheme is that the calculation of n-th Rényi entropy itself is not an easy task especially for higher n, while the higher n-th entropy contains more information of low-lying levels.

All in all, the reduced density matrix (RDM) as a generator of almost all the entanglement quantities, takes a key role in quantum many-body physics but is limited seriously by the huge degree of freedom. From the RDM, almost all physical quantities in the reduced degree of freedom can be extracted. Once the RDM can be simulated in large size, the game can be changed. Therefore, a practical scheme to extract fine RDM with reduced computation complexity is urgent to be proposed. In this paper, we develop a protocol that can efficiently compute the RDM via QMC simulation for the entangling region with long boundaries and in higher dimensions. The large-scale ES was selected to verify the precision of the method for two reasons: On the one hand, unlike EE, ES needs to deal with a large number of degrees of freedom; on the other hand, the requirement of accuracy is extremely high because ES is the logarithm of RDM. To demonstrate the strength of our method, we carefully investigated a Heisenberg ladder, selecting one chain as the entangling region. Compared with previous work studying the same system20,32,57, we can not only address a much larger size but also clarify some misunderstandings in past work20,48 through our large-scale calculations. Furthermore, we study a 2D Heisenberg model to reveal the nontrivial continuous symmetry-breaking phase using the structure of the tower of states (TOSs) in the ES. Except for obtaining the ES numerically, we put forward an efficient way to restore the entanglement Hamiltonian (EH) in operator form from the RDM data. It is important to emphasize that our method is not limited to these particular systems, as it can be widely applied to any models that can be simulated via QMC.

Results

Method

In a quantum many-body system, the ES of a subsystem (entangling region) A with the rest of the system B (environment) is constructed via the RDM. The RDM is defined as the partial trace of the total density matrix ρ over a complete basis of B, i.e., \({\rho }_{A}={{{\rm{Tr}}}}_{B}(\rho )\). The RDM ρA can be interpreted as an effective thermal density matrix \({e}^{-{{{\mathcal{H}}}}_{E}}\) at temperature T = 1 through an EH \({{{\mathcal{H}}}}_{E}\), namely \({{{\mathcal{H}}}}_{E}=-{{\rm{\ln }}}({\rho }_{A})\). The spectrum of the EH is usually denoted as the ES.

Due to the exponential growth of the computation complexity and the memory cost, it is intractable for the existing numerical methods, such as the ED and the DMRG, to approach entangling regions with long boundaries and higher dimensions. Because of the finite computer memory, these numerical methods are limited to short boundaries only. Another way is to extract the low-lying ES through the QMC simulation combined with stochastic analytic continuation (SAC), both in bosonic43,48,58 and fermionic systems36,37,38. Especially in bosonic systems, the ES of spin ladders can be obtained even under a long length48. However, the ES obtained in this way can not show the fine levels but only the dispersion and weight of the spectral function. Nevertheless, the refined structure of ES, e.g., degeneracy, is important for identifying the CFT and topology.

The solution comes from QMC + ED: Tracing the environment via QMC and obtaining the exact low-lying levels through ED. The RDM element \(| {C}_{A}\left.\right\rangle \left\langle \right.{C}_{A}^{{\prime} }|\) can be written in the path integral language as

$${{\rho }_{A}}_{{C}_{A},{C}_{A}^{{\prime} }}\propto \mathop{\sum}\limits_{\{{C}_{B}\}}\left\langle {C}_{A},{C}_{B}| {e}^{-\beta H}| {C}_{A}^{{\prime} },{C}_{B}\right\rangle,\,\beta \to \infty,$$
(1)

where \({C}_{A},\,{C}_{A}^{{\prime} }\) are the configurations of the RDM, and CB is the configurations of environment B.

Commonly, we aim to simulate a partition function \({\sum }_{\{C\}} \left\langle \right.C| {e}^{-\beta H}| C\left.\right\rangle\), where {C} represents a complete, orthogonal basis set in Hilbert space. The configurations C in the bra \(\left\langle \right....|\) and ket \(| ...\left.\right\rangle\) are same, which means that the boundary condition of the imaginary time in the path integral is periodic. If we want to calculate matrix elements \(\left\langle \right.C| {e}^{-\beta H}| {C}^{{\prime} }\left.\right\rangle\) in the basis set {C}, we can achieve this by setting an open boundary condition for the imaginary time in the path integral, allowing C and \({C}^{{\prime} }\) to differ.

Following the above understanding, Eq. (1) can be treated as a general partition function under a special boundary condition along the imaginary time axis, as Fig. 1a shows, which is much more convenient to simulate by QMC. We deal with this special partition function in Eq. (1) in the framework of stochastic series expansion (SSE) method59,60,61,62,63. Of course, it can be simulated by other path integral QMC64,65,66,67,68,69. The only difference compared with the normal SSE is that we open the boundary of the imaginary time in region A and keep the periodic boundary condition (PBC) to that of environment B.

Fig. 1: The schematic diagram of sampling reduced density matrix and the characters of Heisenberg ladder.
Fig. 1: The schematic diagram of sampling reduced density matrix and the characters of Heisenberg ladder.
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a A path integral configuration of a reduced density matrix. The length of imaginary time is β, and the time boundary of B/A is periodic/open. b Lattice of the Spin-1/2 Heisenberg ladder from an overhead view, with spins in A and B depicted by red and blue circles, respectively. c Its phase diagram by denoting \({J}_{{{\rm{leg}}}}=\cos \theta\) and \({J}_{{{\rm{rung}}}}=\sin \theta\).

The value of each element in the RDM can be approached with the frequency of the samplings. It means that the \({{\rho }_{A}}_{{C}_{A},{C}_{A}^{{\prime} }}\) is proportional to \({N}_{{C}_{A},{C}_{A}^{{\prime} }}/{N}_{{{\rm{total}}}}\), where the \({N}_{{C}_{A},{C}_{A}^{{\prime} }}\) is the number of samplings that the upper/below imaginary time boundary configuration is \({C}_{A}/{C}_{A}^{{\prime} }\) and Ntotal is the total amount of samplings. It’s worth noting that the sign should be considered if the weight is not positive.

Example 1: antiferromagnetic (AFM) Heisenberg ladder

To demonstrate the power of our method, we compute the ES of two-leg Heisenberg ladder where the entangled boundary splits the ladder into two chains, as shown in Fig. 1b. The ES of this model has been carefully studied via ED for small sizes in previous work20 and it has a well-known phase diagram (See Fig. 1c). The spins on the ladder are coupled through the nearest neighbor interactions, where Jleg denotes the intra-chain strength along the leg and Jrung denotes the inter-chain strength on the rung. We first simulate with Jleg = 1 and Jrung = 1.732 (θ = π/3) at β = 100. The reason for this choice of parameters is that β = 100 is a temperature low enough for a gapped rung singlet (I) phase, which has been studied with ED in ref. 20, therefore it is convenient for us to compare the results.

Firstly, we present a comparison of the spectrum obtained through QMC+ED and directly through ED with system size L = 8, as illustrated in Fig. 2a. It is evident that the spectrum obtained via QMC+ED approaches that of ED as the number of samplings increases, and higher levels require a greater number of sampling iterations. It is easy to be understood because the QMC of course samples the lower energy configurations with heavier weight preferentially, thus the low-lying levels converge first.

Fig. 2: Comparison of entanglement spectra using different methods.
Fig. 2: Comparison of entanglement spectra using different methods.
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a The comparison of low-lying entanglement excitation spectra obtained by pure ED and QMC+ED with different Monte Carlo steps. The red lines are the results obtained by pure ED, and the black dots are the results obtained by QMC+ED. Here we choose L = 8. b Entanglement excitation spectra versus total momenta k in the chain direction for θ = π/3.

After demonstrating the feasibility of this method, we want to answer some physical questions. For example, there is a discrepancy for the ES in the rung singlet (I) phase of the Heisenberg ladder in previous works20,48. The ES here is expected to bear the low-energy CFT structure, i.e., the ground level of ES, ξ0, will scale as ξ0/L = e0 + d1/L2 + O(1/L3) where the d1 = πcv/6 according to the CFT predication with the central charge c = 1 and v is the velocity of ES near the gapless point20, i.e., the Cloiseaux-Pearson spectrum of the quantum spin chain, \(v| \sin k|\)70. The fitting for the fine levels of the ES by ED at L = 14 (its largest size) gave v ~ 2.36, while the fitting for the rough spectral function of entanglement by QMC combined with SAC at L = 100 pointed to a double value v ~ 4.5848. Therefore, the later paper attributed the different results to the finite size effect. Since our method avoids the limitation of the environment size, we can calculate larger sizes to see how the v scales with L. If the ES has an obvious finite size effect, the v certainly will increase following the size.

The spectra of the system with various, but larger system sizes obtained by QMC+ED are shown in Fig. 2b, and the results are consistent with the ED results shown in ref. 20. It demonstrates the efficiency of our method for larger systems so that we further inspect the finite size effect here. In Fig. 3a, we draw a fitting line for the velocity v of different size L. In the inset of Fig. 3a, we fit the velocities of different sizes by the first two data points. All the lines are close to each other, representing that their velocities are similar. The fitting line in Fig. 3a turns out that the velocity is almost unchanged with size L and convergent to  ~2.41, which is inconsistent with the result of spectral function by QMC+SAC where v ~ 4.5848.

Fig. 3: Extrapolation and spectral analysis of entanglement spectrum in the spin ladder system.
Fig. 3: Extrapolation and spectral analysis of entanglement spectrum in the spin ladder system.
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a Extrapolation of velocity v as a function of 1/L. The inset shows the fitting of velocity vL based on the numerical results for various size L. b Spectral function of the system with size L = 24 and its corresponding velocity v fitting. c Entanglement spectrum versus total momenta k in the chain direction for θ = 2π/3 with the system size L = 20. The red line is the quadratic fitting curve in the entanglement spectrum, given by ω = 0.73k2. d The entanglement spectral function of FM Heisenberg ladder with L = 20. The white line is the quadratic fitting curve near the gapless point.

In order to show the spectral property of the system and avoid any puzzling inconsistency between the ES and entanglement spectral function, we also examine the spectral function of entanglement. For, a physical observable denoted by \({{\mathcal{O}}}\), the spectral function S(ω), in general, can be written as

$$S(\omega )=\frac{1}{\pi }\mathop{\sum}\limits_{m,n}{e}^{-\beta {E}_{n}}| \left\langle \right.n| {{\mathcal{O}}}| m\left.\right\rangle {| }^{2}\delta (\omega -\left[{E}_{m}-{E}_{n}\right])$$
(2)

where \(| n(m)\left.\right\rangle\) and En(m) are the eigenstates and eigenvalues of the given Hamiltonian, respectively. This definition applies to both the original Hamiltonian and the EH. In this work, we focus on the spectral function of the EH. In the following, we choose \({{\mathcal{O}}}\) as the spin operator \({S}_{k}^{z}\), and fix \(| n\left.\right\rangle\) as the ground state, i.e., En ≡ E0 is the ground state energy (it is exact when β → ∞). The choice is same as the ref. 48, which is convenient for comparisons. It is worth noting that the Hamiltonian and energy levels in Eq. (2) should be replaced as EH and ES levels when we talk about the ES. Figure 3b shows the spectral function of the system with size L = 24, and we also give the velocity v ~ 2.29. It reveals that the discrepancy between the ES and entanglement spectral function is not because of the difference in the definition. We find the reason may be the settings of constant missing a factor “2” in the ref. 48.

Example 2: ferromagnetic (FM) Heisenberg ladder

Furthermore, we simulate the case with ferromagnetic Jleg = −1 and antiferromagnetic Jrung = 1.732 at β = 100 on the two-leg Heisenberg ladder for L = 8, 12, 16, 20 and compare the results with those in ref. 20, in which case the ladder is in the gapped rung singlet (II) phase.

According to refs. 48,71,72, the lowest-energy excitations at momentum k = 2π/L correspond to the one-magnon state and behave as \(E=2{J}_{{{\rm{eff}}}}\,{\sin }^{2}(k/2)\), where Jeff is an effective chain coupling. The results in ref. 20 did not show the quadratic behavior due to the small system size. Other studies explained this non-quadratic dispersion via long-range interactions2,32,57. Meanwhile, the QMC+SAC simulation at L = 100 in the ref. 48 shows the spectral function of entanglement is indeed a quadratic dispersion, which further implies the finite size effect of ES here.

In order to check whether the loss of quadratic dispersion is because of the finite size effect, we further try to calculate the ES and its spectral function for larger sizes. Figure 3c shows the results with L = 20, in which there is no obvious quadratic dispersion near k = 0. Actually, we have not seen an obvious change of the dispersion from the linear to the quadratic when increasing the size.

In fact, the answer to the discrepancy comes from the difference between the ES and entanglement spectral function instead of the finite size effect. The results of the FM Heisenberg ladder indicate that the energy spectrum and ordinary spectral function exhibit distinct behaviors at low-energy levels. As shown in Fig. 3d, the ES in L = 20 does not show a quadratic dispersion, while the spectral function does. It is worth noting that we measure the excitation of \({S}_{k}^{z}\) in the spectral function [Eq.(2)], which is actually a one-magnon dispersion. It means the excitation of one magnon is indeed quadratic here, even in small size. We try to draw the curve of the spectral function in the ES levels as the red line shown in Fig. 3c, which further demonstrates that the spectral function goes up quickly beyond the lowest levels of ES. We find that some previous studies in quantum magnetism show, that a pure 1D FM Heisenberg chain, its one-magnon excitation is also higher than the two-magnon bound state73,74. It supports our conclusion that the one-magnon is not the lowest excitation in the FM chain. Thus, the spectral function behaves differently from the low-lying ES in the ferromagnetic case.

Example 3: 2D Heisenberg model

For a d ≥ 2 dimensional quantum O(N) model, its ground state spontaneously beaks the continuous symmetry and is labeled by Néel order. Meanwhile, its energy spectrum has the characteristic of TOS structure. The energies of the TOS in finite size are75,76

$${E}_{S}(L)-{E}_{0}(L)=\frac{S(S+N-2)}{2{\chi }_{\perp }{L}^{d}}$$
(3)

where S is the total spin momentum of the system, χ⊥ is the transverse susceptibility in the thermodynamic limit. Except the energy spectrum77, the ES is also expected having TOS when the ground state is continuous symmetry breaking78.

We then calculate a PBC 20 × 20 square lattice AFM Heisenberg model with two different geometries of A. If we choose the A as a chain as the yellow dots (A region) in Fig. 4a displays, it presents a linear dispersion ES reflecting the Néel order of this AFM model, as shown in Fig. 4b. Because the Heisenberg model belongs to quantum O(3) models, thus the ES levels of TOS is proportional to S(S + 1), where the lowest edge of the ES contains the TOS levels2,78,79.

Fig. 4: The entanglement spectrum of entanglement Hamiltonian in 2D system.
Fig. 4: The entanglement spectrum of entanglement Hamiltonian in 2D system.
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a The AFM Heisenberg model on the square lattice. The dashed lines are used to illustrate the bipartition into two subsystems. The yellow dashed lines illustrate the cutting method that A is a ring, and the red dashed lines illustrate the method that \({A}^{{\prime} }\) is a block. For each cutting method, the other part is denoted as B. We consider the square lattice model with size 20 × 20. We show the entanglement spectrum of EH corresponding to (b) A with length L = 20 and (c) \({A}^{{\prime} }\) with size 4 × 4. The TOS levels are connected by a red line. All the data are calculated in the total Sz = 0 sector.

Meanwhile, if we set A as a 4 × 4 region as the red dots (\({A}^{{\prime} }\) region) in Fig. 4a, the ES also shows a TOS character, as shown in Fig. 4c. Usually, the TOS holds in cornerless cutting cases, this result reveals that the cornered cutting affects TOS little, at least, in the Heisenberg model. For this L = 4 square region, we obtain that the velocity of the TOS fitting line is about 0.38. According to the Eq. (3), we gain the χ⊥ ~ 0.08. The χ⊥ here is basically consistent with the QMC result (~0.0776) of the square lattice AFM Heisenberg model in thermodynamic limit obtained through size extrapolation. (Although the ES should not be exactly the same as the energy spectrum, they are always similar according to the Li-Haldane-Poilblanc conjecture.)

Restore EH

There is no general method for restoring the EH in operator-form directly from the original Hamiltonian. In this work, we propose a two-step strategy to determine the EH, spanned by a specific set of orthogonal operators \(\{{{{\mathcal{O}}}}_{n}\}\), without any approximations or a priori assumptions. For a uniform lattice with L sites, the operator \({{{\mathcal{O}}}}_{n}\) is defined as a product of orthogonal operators \({{{\mathcal{Q}}}}_{i}^{p}\) at each lattice site, i.e., \({{{\mathcal{O}}}}_{n}={\otimes }_{i=1}^{L}{{{\mathcal{Q}}}}_{i}^{{k}_{i}^{n}}\), where \({k}_{i}^{n}\) indexes the operator at site i. For the Hilbert space at each lattice site, with dimension of \({{\mathcal{N}}}\), it is natural to set the operator \({{{\mathcal{Q}}}}_{i}^{p}\) as one of generators of SU(\({{\mathcal{N}}}\)) group, including the identity operator as an adjoint. Thus, p ranges from 1 to \({{{\mathcal{N}}}}^{2}\), and the operator \({{{\mathcal{O}}}}_{n}\) has \({{{\mathcal{N}}}}^{2L}\) possible configurations. As an example, in the AFM Heisenberg ladder, where \({{\mathcal{N}}}=2\), we identify the operators as follows: \({{{\mathcal{Q}}}}_{i}^{0}={\mathbb{1}}\), \({{{\mathcal{Q}}}}_{i}^{1}=2{S}_{i}^{z}\), \({{{\mathcal{Q}}}}_{i}^{2}=\sqrt{2}{S}_{i}^{+}\), and \({{{\mathcal{Q}}}}_{i}^{3}=\sqrt{2}{S}_{i}^{-}\).

In the two-step strategy, the first step is to extract the numerical RDM simulated by QMC. The EH can then be calculated from the RDM using the relation \({{{\mathcal{H}}}}_{E}=-{{\rm{\ln }}}({\rho }_{A})\), which can be expressed in operator-form as \({{{\mathcal{H}}}}_{E}={\sum }_{n}{J}_{n}{{{\mathcal{O}}}}_{n}\). Here the Jn means the coefficient of the operator \({{{\mathcal{O}}}}_{n}\) in the EH. The next step involves determining the coefficient Jn. The numerical EH matrix can be projected into the space of orthogonal operators via the equation \({J}_{n}={{\rm{Tr}}}({{{\mathcal{H}}}}_{E}{{{\mathcal{O}}}}_{n})/{{{\mathcal{N}}}}^{L}\), due to \({{\rm{Tr}}}({{{\mathcal{Q}}}}_{i}^{p}{{{\mathcal{Q}}}}_{i}^{q})={\delta }_{p,q}{{\mathcal{N}}}\). In summary, our scheme is both exact and straightforward, allowing the EH to be restored in operator form.

To demonstrate the power of our method, we restore the EH from the RDM of Heisenberg ladder (our example 1). After consideration of the symmetry, the EH of this system should be a Heisenberg chain with possible long-range interactions44. According to the 2nd order perturbation calculation80, the EH of this ladder in the limit Jleg ≪ Jrung is

$${{{\mathcal{H}}}}_{E}\approx {J}_{1}\mathop{\sum}\limits_{i}{S}_{i}\cdot {S}_{i+1}+{J}_{2}\mathop{\sum}\limits_{i}{S}_{i}\cdot {S}_{i+2}$$
(4)

where the couplings are \({J}_{1}=2\frac{{J}_{{{\rm{leg}}}}}{{J}_{{{\rm{rung}}}}}\) and \({J}_{2}=-\frac{1}{2}{\left(\frac{{J}_{{{\rm{leg}}}}}{{J}_{{{\rm{rung}}}}}\right)}^{2}\).

We simulate the AFM Heisenberg ladder at Jrung = 50, 2, 0.5 with fixed Jleg = 1 to obtain the RDM and then restore the related couplings according to the above scheme. The estimates for J1 and J2 are displayed in Table 1. The case with a large Jrung = 50 fits well with the perturbation, while Jrung = 2 also provides a good match. However, when Jrung = 0.5 < Jleg = 1, the perturbation theory no longer holds. In such a case, we find that ∣J2∣ goes much closer to ∣J1∣, indicating the emergence of a long-range interaction in the EH, as shown in refs. 58,81. At the same time, the coupling decays much more slowly with distance.

Table 1 The couplings restored from the sampled RDM for the system with various Jrung for the Heisenberg ladder
Full size table

Discussion

The advantage and disadvantage

We have to emphasize that the computational complexity of this method is exponential/polynomial for the number of degrees of freedom in A/B. Since we trace all the degrees of freedom of B and sample them, the complexity of the B part is the same as the normal QMC, which increases in the power law. At the same time, we sample all the degrees of freedom of A without any trace operation, thus its complexity is proportional to the dimension of the RDM, which is exponentially increasing. Therefore, the advantage of this method is that it is basically not limited to the size of the environment, and it enables the extraction of both the RDM and fine levels of the ES.

However, since the important information is always contained in the low-lying ES, and the important sampling of QMC just presents the low-lying levels first, actually we can set a cutoff for the sampling to avoid the additional cost for high levels. For example, focusing on the first excited gap of the ES (the Schmidt gap82,83) could significantly reduce the required sampling. As shown in Fig. 2a, the first gap converges very quickly during the Monte Carlo sampling.

To compare with ED, this method can obtain ρA for much larger system sizes. Compared with the QMC+SAC scheme, it is not restricted to the cut geometry between A and B, and it can extract the fine levels of low-lying ES. (In the QMC+SAC method, because the spectral function is gained from the imaginary time correlation function via numerical analytic continuation, it requires that A should have PBCs to guarantee a well-defined momentum k.) In other words, the analytic continuation depends closely on the cut form of the entanglement. However, our method can study any geometry for the A. Similarly, although the Bisognano-Wichmann theorem gives the detailed form of EH, it holds under some strict conditions (e.g., the entanglement region has no corner, the original Hamiltonian satisfies translation symmetry) and may lose the effectiveness in many lattice models84,85,86,87.

Conclusions and outlooks

We propose a practical scheme to extract the high-precision RDM (displayed in fine levels of ES) from the method of QMC simulation combined with ED. It can extract the fine information of low-lying ES for large-scale quantum many-body systems whose environment has huge degrees of freedom. Using this method, we explain the previous discrepancy for the ES in a Heisenberg ladder system. Furthermore, we calculate the ES of a 2D AFM Heisenberg model within two different cut geometries. Both the cornered and cornerless cases show the TOS character, which reflects the continuous symmetry breaking here. Moreover, we have put forward an efficient and simple way to restore the EH from the data of sampled RDM.

In summary, the RDM can be obtained in a larger size via QMC within this simple frame. Therefore, other observables such as von Neumann entropy, entanglement negativity, and off-diagonal measurements can also be extracted in this way88. This low-technical-barrier algorithm can be widely used in QMC simulations and is easy to parallelize on a computer.

Methods

We use a QMC algorithm combined with the ED technique in this work. Details have been explained in the main text.