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Ground-state properties of quantum many-body systems: entangled-plaquette states and variational Monte Carlo

Fabio Mezzacapo, Schuch, N., Boninsegni, M., Cirac, J.

Published Aug 24, 2009• 47 citations
OpenAlex

Abstract

We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any 'sign' instability. We show results for various two-dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatze, is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.

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We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any 'sign' instability. We show results for various two-dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatze, is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.
We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any 'sign' instability. We show results for various two-dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatze, is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.
We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state (GS) properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e. incorporating problem-dependent terms) trial wave functions (WFs). The extrapolation to the thermodynamic limit of our results for lattices comprising up to N =324 spins yields an upper bound of the GS energy per site (in units of the exchange coupling) of − 0.5458(2) [−0.4074(1) for the XX model], while the estimated infinite-lattice order parameter is 0.3178(5) (i.e. ∼64% of the classical value).
We study lattice QED in 2+12+12+1 dimensions coupled to Wilson fermions in the Hamiltonian formulation and develop a sign-problem-free variational Monte Carlo algorithm. Using a continuous basis to describe the gauge degrees of freedom and fermionic Gaussian states for the matter fields, we construct a variational ansatz for the ground state. The parameters of the ansatz are optimized using gradient descent and/or imaginary time evolution based on the time-dependent variational principle. For a lattice with NNN matter sites, the computational complexity for sampling a gauge configuration is O(N2log⁡N)\mathcal{O}(N^2\log N)O(N2logN), and the complexity for computing the energy per gauge configuration is O(N4)\mathcal{O}(N^4)O(N4). Comparing our method to results from exact diagonalization on small lattices, we demonstrate good agreement. In addition, we present preliminary results for the ground state for two fermion flavors, Nf=2N_f=2Nf​=2, in the presence of a non-zero chemical potential, a regime in which the conventional action-based Monte Carlo methods suffer from the sign problem.
We explore a variational Ansatz for lattice quantum systems---named long-range entangled-plaquette state---in which pairs of clusters of adjacent lattice sites are correlated at any distance. The explicit scale-free structure of correlations built in this wave function makes it fit to reproduce critical states with long-range entanglement. The use of complex weights in the Ansatz allows for an efficient optimization of nonpositive definite states in a fully variational fashion, namely without any additional bias (arising, e.g., from pre-imposed sign structures) beyond that imposed by the parametrization of the state coefficients. These two features render the Ansatz particularly appropriate for the study of quantum phase transitions in frustrated systems. Moreover, the Ansatz can be systematically improved by increasing the long-range plaquette size, as well as by the inclusion of even larger adjacent-site plaquettes. We validate our Ansatz and its optimization procedure in the case of the XX and Heisenberg chain, and further apply it to the case of a simple, yet paradigmatic model of frustration, namely the ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Heisenberg chain. For this model we provide clear evidence that our trial wave function faithfully describes both the short-range physics (particularly in terms of ground-state energy) and the long-range one expressed by the Luttinger exponent, and the central charge of the related conformal field theory, which govern the decay of correlations and the scaling of the entanglement entropy, respectively. Finally we successfully reproduce the incommensurate correlations developing in the system at strong frustration, as a result of the flexible representation of sign (phase) structures via complex weights.
We show that the entangled-plaquette variational Ansatz can be adapted to study the two-dimensional t-J model in the presence of two mobile holes. Specifically, we focus on a square lattice comprising up to N=256N=256N=256 sites in the parameter range $0.4\ensuremath{\le}J/t\ensuremath{\le}2.0$. Ground state energies are obtained via the optimization of a wave function in which the weight of a given configuration is expressed in terms of variational coefficients associated with square and linear entangled plaquettes. Our estimates are in excellent agreement with exact results available for the N=16N=16N=16 lattice. By extending our study to considerably larger systems we find, based on the analysis of the long-distance tail of the probability of finding two holes at spatial separation rrr, and on our computed two-hole binding energies, the existence of a two-hole bound state for all the values of J/tJ/tJ/t explored here. It is estimated that ddd-wave binding of the two holes does not occur for $J/t<{J}_{c}/t\ensuremath{\simeq}0.19$.
We propose an efficient numerical method, which combines the advantages of recently developed tensor-network based methods and standard trial wave functions, to study the ground-state properties of quantum many-body systems. In this approach, we apply a projector in the form of a tensor-product operator to an input wave function, such as a Jastrow-type or Hartree-Fock wave function, and optimize the tensor elements via variational Monte Carlo. The entanglement already contained in the input wave function can considerably reduce the bond dimensions compared to the regular tensor-product state representation. In particular, this allows us to also represent states that do not obey the area law of entanglement entropy. In addition, for fermionic systems, the fermion sign structure can be encoded in the input wave function. We show that the optimized states provide good approximations of the ground-state energy and correlation functions in the cases of two-dimensional bosonic and fermonic systems.