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Physical Review Letters

Simulations of Discrete Quantum Systems in Continuous Euclidean Time

Bernard B. Beard, U.‐J. Wiese

Published Dec 16, 1996• 329 citations
Publisher

Abstract

We propose a new method to study path integrals for discrete quantum systems in which we work directly in the Euclidean time continuum. The method is of general interest. Here it is applied to the Heisenberg quantum antiferromagnet using a continuous-time version of a loop cluster algorithm. This algorithm is exploited to confirm the predictions of chiral perturbation theory in the extreme low temperature regime, down to $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.01J$. A fit of the low-energy parameters of chiral perturbation theory gives excellent agreement with previous results and with experiments.

DOI
10.1103/physrevlett.77.5130

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We propose a new method to study path integrals for discrete quantum systems in which we work directly in the Euclidean time continuum. The method is of general interest. Here it is applied to the Heisenberg quantum antiferromagnet using a continuous-time version of a loop cluster algorithm. This algorithm is exploited to confirm the predictions of chiral perturbation theory in the extreme low temperature regime, down to T=0.01JT\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.01JT=0.01J. A fit of the low-energy parameters of chiral perturbation theory gives excellent agreement with previous results and with experiments.
The correlation length of the square-lattice spin-1/2 Heisenberg antiferromagnet is studied in the low-temperature (asymptotic-scaling) regime. Our novel approach combines a very efficient loop cluster algorithm---operating directly in the Euclidean time continuum---with finite-size scaling. This enables us to probe correlation lengths up to $\ensuremath{\xi}\ensuremath{\approx}350000$ lattice spacings, more than 3 orders of magnitude larger than in any previous study. We resolve a conundrum concerning the applicability of asymptotic-scaling formulas to experimentally and numerically determined correlation lengths. Our results have direct implications for the zero-temperature behavior of spin-1/2 ladders.
Motivated by the numerical simulation of systems which display quantum phase transitions, we present a novel application of the meron-cluster algorithm to simulate the quantum antiferromagnetic Heisenberg model coupled to an external uniform magnetic field both in one and in two dimensions.In the infinite volume limit and at zero temperature we found numerical evidence that supports a quantum phase transition very close to the critical values B c = 2 and B c = 4 for the system in one and two dimensions, respectively.For the one dimensional system, we have compared the numerical data obtained with analytical predictions for the magnetization density as a function of the external field obtained by scaling-behaviour analysis and Bethe Ansatz techniques.Since there is no analytical solution for the two dimensional case, we have compared our results with the magnetization density obtained by scaling relations for small lattice sizes and with the approximated thermodynamical limit at zero temperature guessed by scaling relations.Moreover, we have compared the numerical data with other numerical simulations performed by using different algorithms in one and two dimensions, like the directed loop method.The numerical data obtained are in perfect agreement with all these previous results, which confirms that the meron-algorithm is reliable for quantum Monte Carlo simulations and applicable both in one and two dimensions.Finally, we have computed the integrated autocorrelation time to measure the efficiency of the meron algorithm in one dimension.
This thesis is divided into three chapters. In the first chapter we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization group (NRG) methods. At zero temperature for a wide range of U , the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems The second chapter outlines the density matrix embedding methodology in the context of electronic structure applications. We formulate analytical gradients for energies obtained from DMET focusing on two scenarios: RHF-in-RHF embedding and FCI-in-RHF embedding. The former involves solving the small embedded system at Restricted Hartree-Fock (RHF) level of theory. This serves to check the validity of the formulas by reproducing the RHF results on the full system for energies and gradients. The latter scenario employs full configuration interaction (FCI) as a high level solver for the small embedded system. Our results show that only Hellmann-Feynman terms, which involve derivatives of one- and two-electron terms in the atomic orbital basis, are required to calculate energy gradients in both cases. We applied our methodology to the problem of H 10 ring dissociation where the analytical gradients matched those obtained numerically. The gradient formulation is applicable to geometry optimization of strongly correlated molecules and solids. It can also be used in ab initio molecular dynamics where forces on nuclei are obtained from DMET energy gradients. In the final chapter we focus on the study of finite-temperature equilibrium properties of quantum systems in continuous space. We formulate an expansion of the partition function in continuous-time and use Monte Carlo to sample terms in the resulting infinite series. Such a strategy has been highly successful in quantum lattice models but has found scant application in off-lattice systems. The Monte Carlo estimate of the average energy of quantum particles in continuous space subject to simple model potentials is found to converge with low statistical error to the exact solutions even when very high perturbation order is required. We outline two ways in which the algorithm can be applied to more complex problems. First, by drawing an analogy between the formulation in continuous-time with the discretized, Trotter factorized version of standard path integral Monte Carlo (PIMC). This allows one to use the suite of standard PIMC moves to carry out the position sampling required to obtain the weight of each time configuration. Finally, we propose an alternate route by fitting the many-particle potential with multidimensional Gaussians which provides an analytical form for the position integrals.
We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The "algorithmic discontinuities" between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic S=1/2 Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the XY model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For LxL lattices with L up to 64, we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: chi( perpendicular )=0.0659+/-0.0002.
We present a general strategy to extend quantum cluster algorithms for S=12S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}S=21​ spin systems, such as the loop algorithm, to those with an arbitrary size of spins. The partition function of a high- SSS spin system is generally represented by the path integral of a S=12S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}S=21​ model with special boundary conditions in the imaginary-time direction. We introduce additional graphs for the boundary part and give the labeling probability explicitly, which completes the algorithm together with an existing S=12S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}S=21​ algorithm. As a demonstration, we simulate the integer-spin antiferromagnetic Heisenberg chains. The magnitude of the first excitation gap is estimated to be 0.41048(6), 0.08917(4), and 0.01002(3) for S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1, 2, and 3, respectively.
Abstract We provide a new paradigm for quantum simulation based on path integration that allows quantum speedups to be observed for problems that are more naturally expressed using the path integral formalism rather than the conventional Hamiltonian formalism. We provide two novel quantum algorithms based on Hamiltonian versions of the path integral formulation and another for Lagrangians of the form m 2 x ˙ 2 − V ( x ) , while avoiding the sign problems that preclude efficient Monte-Carlo algorithms for such problems. This Lagrangian path integral algorithm is based on a new rigorous derivation of a discrete version of the Lagrangian path integral. Our first Hamiltonian path integral method breaks up the paths into short timesteps. It is efficient under appropriate sparsity assumptions and requires a number of queries to oracles that give the eigenvalues and overlaps between the eigenvectors of the Hamiltonian terms that scales as t o ( 1 ) / ϵ o ( 1 ) for simulation time t and error ϵ . The second approach uses long-time path integrals for near-adiabatic systems and has query complexity that scales as O ( 1 / ϵ ) if the energy eigenvalue gaps and simulation time is sufficiently long. Finally, we show that our Lagrangian simulation algorithm requires a number of queries to an oracle that computes the discrete Lagrangian that scales for a system with η particles in D + 1 dimensions as O ~ ( η D t 2 / ϵ ) if V ( x ) is bounded and finite and the wavefunction obeys appropriate position and momentum cutoffs. This shows that Lagrangian dynamics can be efficiently simulated on quantum computers and opens up the possibility for quantum field theories for which the Hamiltonian is unknown to be efficiently simulated on quantum computers.