Expansion Quantum Monte Carlo Method Research Articles

Classical model emerges in quantum entanglement: Quantum Monte Carlo study for an Ising-Heisenberg bilayer

By developing a cluster sampling of stochastic series expansion quantum Monte Carlo method, we investigate a spin-$1/2$ model on a bilayer square lattice with intra-layer ferromagnetic (FM) Ising coupling and inter-layer antiferromagnetic Heisenberg interaction. The continuous quantum phase transition which occurs at $g_c=3.045(2)$ between the FM Ising phase and the dimerized phase is studied via large scale simulations. From the analyzes of critical exponents we show that this phase transition belongs to the (2+1)-dimensional Ising universality class. Besides, the quantum entanglement is strong between the two layers, especially in dimerized phase. The effective Hamiltonian of single layer seems like a transverse field Ising model. However, we found the quantum entanglement Hamiltonian is a pure classical Ising model without any quantum fluctuations. Furthermore, we give a more general explanation about how a classical entanglement Hamiltonian emerges.

Bulk and edge dynamics of a two-dimensional Affleck-Kennedy-Lieb-Tasaki model

We study the dynamical properties of both bulk and edge spins of a two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model mainly by using the stochastic series expansion quantum Monte Carlo method with stochastic analytic continuation. In the deep AKLT phase, we obtain a spin spectrum with flat band, which is a strong evidence for a localized state. Through the spectrum analysis, we see a clear continuous phase transition from the AKLT phase to the N\'eel phase in the model, and the energy gap becomes closed at the corresponding momentum point. In comparison with linear spin-wave theory, the differences show that there are strong interactions among magnons at high energies. With open boundary condition, the gap of edge spins in the AKLT phase closes at both the $\Gamma$ point and the $\pi$ point interestingly to emerge into a flat-band-like Luttinger liquid phase, which can be explained by symmetry and perturbation approximation. This paper helps us to better understand the completely different dynamical behaviors of bulk and edge spins in the symmetry protected topological phase.

Pair hopping in systems of strongly interacting hard-core bosons

We have used the Stochastic Series Expansion quantum Monte Carlo method to study interacting hard-core bosons on the square lattice, with pair-hopping processes supplementing the standard single-particle hopping. Such pair hopping arises in effective models for frustrated quantum magnets. Our goal is to investigate the effects of the pair hopping process on the commonly observed superfluid, insulating (Mott), and super-solid ground-state phases in the standard hard-core boson model with various interaction terms. The model is specifically motivated by the observation of finite dispersion of 2-magnon bound states in neutron diffraction experiments SrCu$_2$(BO$_3$)$_2$. Our results show that the pair hopping has different effects on Mott phases at different filling fractions, "melting" them at different critical pair-hopping amplitudes. Thus, it appears that pair hopping may have an important role in determining which out of a potentially large number of Mott phases (stabilized by details of the charge-diagonal interaction terms) actually survive the totality of quantum fluctuations present.

Criticality and scaling corrections for two-dimensional Heisenberg models in plaquette patterns with strong and weak couplings

We use the stochastic series expansion quantum Monte Carlo method to study the Heisenberg models on the square lattice with strong and weak couplings in the form of three different plaquette arrangements known as checkerboard models C$2\times2$, C$2\times4$ and C$4\times4$. The $a\times b$ here stands for the shape of plaquette consisting with spins connected by strong couplings. Through detailed analysis of finite-size scaling study, the critical point of C$2\times2$ model is improved as $g_{c}=0.548524(3)$ compared with previous studies with $g$ to be the ratio of weak and strong couplings in the models. For C$2\times4$ and C$4\times4$ we give $g_{c}=0.456978(2)$ and $0.314451(3)$. We also study the critical exponents $\nu$, $\eta$, and the universal property of Binder ratio to give further evidence that all quantum phase transitions in these three models are in the three-dimensional O(3) universality class. Furthermore, our fitting results show the importance of effective corrections in the scaling study of these models.

Bond disorder in even-leg Heisenberg ladders

Random bond disorder in antiferromagnetic spin-1/2 Heisenberg ladders is investigated using the Stochastic Series Expansion Quantum Monte Carlo method. We find that the effects of individual bond impurities vary strongly, depending on their position on legs and rungs. We initially focus on how the distribution of local bond energies depends on the impurity concentration. Then we study how the phase diagram of even-leg ladders is affected by random bond doping. We observe Bose glass phases in two regimes ($h' < h \lesssim h_{c1}$ and $h" < h <h_{c2}$) and a Bose-Einstein Condensate in-between. Their presence are discussed in the relation to the local bond energies.

Dimensional crossover in spin-1 Heisenberg antiferromagnets: a quantum Monte Carlo study

We present results of large scale simulations of the spin-1 Heisenberg antiferromagnet on a tetragonal lattice. The stochastic series expansion quantum Monte Carlo method is used to calculate equilibrium thermodynamic variables in the presence of an external magnetic field. In particular, the low temperature magnetization curve is investigated in the quasi-one-dimensional (Q1D), quasi-two-dimensional (Q2D), and three-dimensional (3D) limits. Starting from the 3D limit, the Q1D (Q2D) limit is achieved by reducing the in-plane (out-of-plane) spin coupling strength towards zero. In the Q1D limit, a Haldane gap appears in the magnetization curve at low magnetic field. Additionally, near the saturation field the slope of the magnetization curve increases substantially, approaching the infinite-slope behavior of a one-dimensional spin-1 chain. A similar (though less pronounced) effect is seen in the Q2D limit. We also study the effect of uniaxial single-ion anisotropy on the magnetization curve for Q1D and Q2D systems. Our results will be important in understanding the field-induced behavior of a class of low-dimensional Ni-based quantum magnets.

Phase diagram of the 3D quantum anisotropic XY model—A quantum Monte Carlo calculation

In this work we apply the stochastic series expansion quantum Monte Carlo method to study the quantum phase transition of the spin 1 three-dimensional XY model with easy-plane anisotropy D. We simulate this model in cubic lattices (L×L×L) with L∈(4,24) and periodic boundary condition. Using finite size scaling we obtained the phase diagram for the model, the critical exponent zν=0.501(5) and the quantum critical point Dc=9.7950(3)J. Using a low temperature expansion for the magnetic susceptibility we obtained zν=0.59(1).

Simulation of two-dimensional many-particle hardcore bosons by using the quantum Monte Carlo method

In this paper, the stochastic series expansion quantum Monte Carlo method is employed to investigate the thermodynamic properties of hardcore Bose-Hubbard model in two-dimensional space. The two-dimensional hardcore Bose-Hubbard model can be mapped into the two-dimensional antiferromagnetic quasi-Heisenberg model under transform of bosonic operators. There is an additional term which is proportional to the total number of sites compared with real Heisenberg model and it is difficult for simulation. Using a nonlocal operator-loop update, it allows one to simulate thousands of sites. Our simulation results show that, first, energy decreases with the increase of density of particles in a range from 0 to 0.5, and finally approaches to a fixed value. Moreover, with the size of square lattice increasing, energy also increases. Second, when we fix the system size, energy and magnetization increase with temperature, but not with of chemical potential. When we increase the system size, energy increases, while, the magnetization decreases. Third, specific heat is independent of chemical potential, but it dramatically increases with temperature and approaches to a peak, then decreases slowly. According to Landau theory of superfluidity, the tends of curve for energy and specific heat fit the research of He II in the Landau two-fluid model. Fourth, different square lattice linear system sizes have a little influence on tiny differences to the reciprocal of uniform susceptibility. There are small fluctuations in a range from 0 to 0.5(J/kB), where J is the coupling energy, kB is the Boltzmann constant, but the reciprocal of uniform susceptibility increases with temperature increasing in a range from 0.5 to 2(J/kB). The tends of curve are similar to those of Kondo effect.

A new method to study phase transition in 3D Heisenberg model

It is shown that partial entropy, which is the classical analog of von Neumann entropy in quantum theory, is an effective tool to study the thermodynamic phase transitions in the physical systems. This method captures the intrinsic characters of critical fluctuations and does not need the pre-assumed order parameter. As an example, the finite temperature phase transition in the quantum three-dimensional spin-1/2 Heisenberg model is studied, where the stochastic series expansion quantum Monte Carlo method with operator-loop update is used. It is found that close to the critical temperature, the derivative of partial entropy displays a maximum value and shows finite size scaling behaviors, from which the critical temperature and critical exponents are determined.

Quantum Monte Carlo study of 2D spin Peierls state with multimode distortions

2D spin-Peierls states are investigated by using stochastic series expansion quantum Monte Carlo method. The obtained numerical data support the previous result derived through the exact diagonalization (ED) calculations for a small cluster, indicating that the lowest energy state of the square-lattice 2D spin-Peierls system is the so-called multimode spin-Peierls (MMSP) state with lattice distortions involving Fourier components having not only the nesting wave vector but also other wave vectors parallel to it, as long as the spin–lattice coupling constant exceeds a certain critical value, and that the transition between the antiferromagnetic SDW state and the MMSP state, due to the change of the spin–lattice coupling strength, is of the first order. Since the Monte Carlo method allows us to treat larger system sizes compared to the ED method, it is possible to discuss the size effect of the transition and the degeneracy among different distortion patterns.

Application of the perturbation series expansion quantum Monte Carlo method to multiorbital systems having Hund's coupling

We propose a new quantum Monte Carlo method especially intended to couple with the dynamical mean-field theory. The algorithm is not only much more efficient than the conventional Hirsch–Fye algorithm, but is applicable to multiorbital systems having an SU(2)-symmetric Hund's coupling as well.

Stochastic series expansion algorithm for theS=1∕2XYmodel with four-site ring exchange

We describe a stochastic series expansion quantum Monte Carlo method for a two-dimensional S = 1/2 XY model (or, equivalently, hard-core bosons at half filling) which in addition to the standard pair interaction J includes a four-site term K that flips spins on a square plaquette. The model has three ordered ground state phases; for K/J approximately < or = 8 it has long-range xy spin order (superfluid bosons), for K/J approximately > or = 15 it has staggered spin order in the z direction (charge-density wave), and between these phases it is in a state with columnar order in the bond and plaquette energy densities. We discuss an implementation of directed-loop updates for the SSE simulations of this model and also introduce a "multibranch" cluster update which significantly reduces the autocorrelation times for large K/J. In addition to the pure J-K model, which in the z basis has only off-diagonal terms, we also discuss modifications of the algorithm needed when various diagonal interactions are included.

Magnetic ordering in a doped frustrated spin-Peierls system

Based on a model of a quasi-one dimensional spin-Peierls system doped with nonmagnetic impurities, an effective two-dimensional Hamiltonian of randomly distributed $S=1∕2$ spins interacting via long-range pairwise interaction is studied using a stochastic series expansion quantum Monte Carlo method. The susceptibility shows Curie-like behavior at the lowest temperatures reached although the staggered magnetisation is found to be finite for $T\ensuremath{\rightarrow}0$. The doping dependance of the corresponding three-dimensional N\'eel temperature is also computed.

Quantum antiferromagnetism in quasicrystals.

The antiferromagnetic Heisenberg model is studied on a two-dimensional bipartite quasiperiodic lattice. Using the stochastic series expansion quantum Monte Carlo method, the distribution of local staggered magnetic moments is determined on finite square approximants with up to 1393 sites, and a nontrivial inhomogeneous ground state is found. A hierarchical structure in the values of the moments is observed which arises from the self-similarity of the quasiperiodic lattice. The computed spin structure factor shows antiferromagnetic modulations that can be measured in neutron scattering and nuclear magnetic resonance experiments. This generic model is a first step towards understanding magnetic quasicrystals such as the recently discovered Zn-Mg-Ho icosahedral structure.

Quantum Monte Carlo simulation of thin magnetic films

The stochastic series expansion quantum Monte Carlo method is used to study thin ferromagnetic films, described by a Heisenberg model including local anisotropies. The magnetization curve is calculated, and the results compared to Schwinger boson and many-body Green's function calculations. A transverse field is introduced in order to study the reorientation effect, in which the magnetization changes from out-of-plane to in-plane. Since the approximate theoretical approaches above differ significantly from each other, and the Monte Carlo method is free of systematic errors, the calculation provides an unbiased check of the approximate treatments. By studying quantum spin models with local anisotropies, varying spin size, and a transverse field, we also demonstrate the general applicability of the recent cluster-loop formulation of the stochastic series expansion quantum Monte Carlo method.

Bond-order-wave phase and quantum phase transitions in the one-dimensional extended Hubbard model

We use a stochastic series expansion quantum Monte Carlo method to study the phase diagram of the one-dimensional extended Hubbard model at half filling for small to intermediate values of the on-site (U) and nearest-neighbor (V) repulsions. We confirm the existence of a novel, long-range-ordered bond-order-wave (BOW) phase recently predicted by Nakamura (J. Phys. Soc. Jpn. 68, 3123 (1999)) in a small region of the parameter space between the familiar charge-density-wave (CDW) state for V > U/2 and the state with dominant spin-density-wave (SDW) fluctuations for V < U/2. We discuss the nature of the transitions among these states and evaluate some of the critical exponents. Further, we determine accurately the position of the multi-critical point, (U_m,V_m)= (4.7 +/- 0.1, 2.51 +/- 0.04) (in energy units where the hopping integral is normalized to unity), above which the two continuous SDW-BOW-CDW transitions are replaced by one discontinuous (first-order) direct SDW-CDW transition. We also discuss the evolution of the CDW and BOW states upon hole doping. We find that in both cases the ground state is a Luther-Emery liquid, i.e., the spin gap remains but the charge gap existing at half-filling is immediately closed upon doping. The charge and bond-order correlations decay with distance r as r^{-K_rho}, where K_rho is approximately 0.5 for the parameters we have considered. We also discuss advantages of using parallel tempering (or exchange Monte Carlo) -- an extended ensemble method that we here combine with quantum Monte Carlo -- in studies of quantum phase transitions.

Finite-size scaling of the ground-state parameters of the two-dimensional Heisenberg model

The ground-state parameters of the two-dimensional $S=1/2$ antiferromagnetic Heisenberg model are calculated using the stochastic series expansion quantum Monte Carlo method for $L\ifmmode\times\else\texttimes\fi{}L$ lattices with $L$ up to $16$. The finite-size results for the energy $E$, the sublattice magnetization $M$, the long-wavelength susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}(q=2\ensuremath{\pi}/L)$, and the spin stiffness ${\ensuremath{\rho}}_{s},$ are extrapolated to the thermodynamic limit using fits to polynomials in $1/L$, constrained by scaling forms previously obtained from renormalization-group calculations for the nonlinear $\ensuremath{\sigma}$ model and chiral perturbation theory. The results are fully consistent with the predicted leading finite-size corrections, and are of sufficient accuracy for extracting also subleading terms. The subleading energy correction $(\ensuremath{\sim}{1/L}^{4})$ agrees with chiral perturbation theory to within a statistical error of a few percent, thus providing numerical confirmation of the finite-size scaling forms to this order. The extrapolated ground- state energy per spin is $E=\ensuremath{-}0.669437(5)$. The result from previous Green's function Monte Carlo (GFMC) calculations is slightly higher than this value, most likely due to a small systematic error originating from ``population control'' bias in GFMC. The other extrapolated parameters are $M=0.3070(3)$, ${\ensuremath{\rho}}_{s}=0.175(2)$, ${\ensuremath{\chi}}_{\ensuremath{\perp}}=0.0625(9)$, and the spin-wave velocity $c=1.673(7)$. The statistical errors are comparable with those of previous estimates obtained by fitting loop algorithm quantum Monte Carlo data to finite-temperature scaling forms. Both $M$ and ${\ensuremath{\rho}}_{s}$ obtained from the finite-$T$ data are, however, a few error bars higher than the present estimates. It is argued that the $T=0$ extrapolations performed here are less sensitive to effects of neglected higher-order corrections, and therefore should be more reliable.