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arXiv:2603.25858

Benchmarking the accuracy of superconducting pair-pair correlations within Constrained Path Quantum Monte Carlo

Jodie Roberts, Beau A. Thompson, Rudolf Torsten Clay

Published Mar 26, 2026• 0 citations
arXivHTMLPDF

Abstract

Ground state properties of the Hubbard model are of fundamental importance to understand the mechanism of unconventional superconductivity in the high-T_c cuprates and other materials. One of the most powerful numerical methods for strongly interacting models is quantum Monte Carlo, which however faces a fundamental limitation, the Fermion sign problem. The sign problem can be mitigated using approximate methods such as Constrained Path Monte Carlo, where the sign of the overlap with a trial wavefunction is constrained to remain positive. Beyond the constraint, additional approximations must be made in order to measure observables for operators that do not commute with the Hamiltonian. We examine critically the most commonly used approximation, back propagation, as well as a recently proposed constraint release measurement technique. In comparisons with a variety of systems that can be solved numerically exactly by other methods, we find that back propagation tends to underestimate long-range superconducting pair-pair correlations. Except in special cases such as one dimension, back propagation introduces an uncontrolled approximation even when the exact trial wavefunction is used. The constraint release technique can provide accurate results, with the disadvantages that it much more computationally expensive and reintroduces the sign problem.

DOI
10.48550/arxiv.2603.25858

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Ground state properties of the Hubbard model are of fundamental importance to understand the mechanism of unconventional superconductivity in the high-T_c cuprates and other materials. One of the most powerful numerical methods for strongly interacting models is quantum Monte Carlo, which however faces a fundamental limitation, the Fermion sign problem. The sign problem can be mitigated using approximate methods such as Constrained Path Monte Carlo, where the sign of the overlap with a trial wavefunction is constrained to remain positive. Beyond the constraint, additional approximations must be made in order to measure observables for operators that do not commute with the Hamiltonian. We examine critically the most commonly used approximation, back propagation, as well as a recently proposed constraint release measurement technique. In comparisons with a variety of systems that can be solved numerically exactly by other methods, we find that back propagation tends to underestimate long-range superconducting pair-pair correlations. Except in special cases such as one dimension, back propagation introduces an uncontrolled approximation even when the exact trial wavefunction is used. The constraint release technique can provide accurate results, with the disadvantages that it much more computationally expensive and reintroduces the sign problem.
We introduce an efficient and numerically stable technique to make use of a BCS trial wave function in the computation of correlation functions of strongly correlated quantum fermion systems. The technique is applicable to any projection approach involving paths of independent-fermion propagators, for example, in mean-field or auxiliary-field quantum Monte Carlo (AFQMC) calculations. Within AFQMC, in the absence of the sign problem, the methodology allows the use of a BCS reference state which can greatly reduce the required imaginary time of projection and improves Monte Carlo sampling efficiency and statistical accuracy for systems where pairing correlations are important. When the sign problem is present, the approach provides a powerful generalization of the constrained-path AFQMC technique which usually uses Slater determinant trial wave functions. As a demonstration of the capability of the methodology, we present benchmark results for the attractive Hubbard model, both spin balanced (no sign problem) and with a finite spin polarization (with sign problem).
We describe and discuss a recently proposed quantum Monte Carlo algorithm to compute the ground-state properties of various systems of interacting fermions. In this method, the ground state is projected from an initial wave function by a branching random walk in an overcomplete basis of Slater determinants. By constraining the determinants according to a trial wave function |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉 is exact. We illustrate the method by describing in detail its implementation for the two-dimensional one-band Hubbard model. We show results for lattice sizes up to 16\ifmmode\times\else\texttimes\fi{}16 and for various electron fillings and interaction strengths. With simple single-determinant wave functions as |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, the method yields accurate (often to within a few percent) estimates of the ground-state energy as well as correlation functions, such as those for electron pairing. We conclude by discussing possible extensions of the algorithm.
We present a formulation of the constrained path Monte Carlo method for fermions that uses trial wave functions that include many-body effects. This formulation allows us to implement a whole family of generalized mean-field states as constraints. As an example, we calculated superconducting pairing correlation functions for the two-dimensional repulsive Hubbard model using a BCS trial state as the constraint. We compared the results with the case where a free-electron trial wave function is used. We found that the correlation functions are independent of which state is used as the constraint, which reaffirms the results previously found by Zhang et al. [Phys. Rev. Lett. 78, 4486 (1997)] regarding the suppression of long-range pairing correlations as the system size increases.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.
Understanding strongly correlated electrons is an important long-term goal, not only for uncovering fundamental physics behind, but also for their emergence of lots of novel states which have potential applications in quantum control and quantum computations. Meanwhile, the strongly correlated electrons are usually extremely hard problems, and it is generally impossible to understand them unbiasedly. Quantum Monte Carlo is a typical unbiased numeric method, which does not depend on any perturbation, and it can help us to exactly understand the strongly correlated electrons, so that it is widely used in high energy and condensed matter physics. However, quantum Monte Carlo usually suffers from the notorious sign problem. In this paper, we introduce general ideas to design sign problem free models and discuss the sign bound theory we proposed recently. In the sign bound theory, we build a direct connection between the average sign and the ground state properties of the system. We find usually the average sign has the conventional exponential decay with system size increasing, leading to exponential complexity; but for some cases it can have algebraic decay, so that quantum Monte Carlo simulation still has polynomial complexity. By designing sign problem free or algebraic sign behaved strongly correlated electron models, we can approach to several long outstanding problems, such as the itinerant quantum criticality, the competition between unconventional superconductivity and magnetism, as well as the recently found correlated phases and phase transitions in moiré quantum matter.
We propose a new quantum Monte Carlo algorithm to compute fermion ground-state properties. The ground state is projected from an initial wave function by a branching random walk in an over-complete basis space of Slater determinants. By constraining the determinants according to a trial wave function $|{\ensuremath{\Psi}}_{T}〉$, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if $|{\ensuremath{\Psi}}_{T}〉$ is exact. We report results on the two-dimensional Hubbard model up to size $16\ifmmode\times\else\texttimes\fi{}16$, for various electron fillings and interaction strengths.