A simple method is proposed and tested for obtaining accelerated convergence of quantum systems on small lattices with N sites. The main idea is to perform exact diagonalizations with some added irrelevant parameter, and use this parameter to accelerate the convergence to the infinite-lattice limit. In this paper different boundary conditions are used to improve the convergence for the Heisenberg model. In particular, we find that the application of the method to the d=1 antiferromagnetic Heisenberg model changes the rate of convergence of the ground-state energy per site, \ensuremath{\Vert} ${\mathit{E}}_{0}$ (N)- ${\mathit{E}}_{0}$ (\ensuremath{\infty})\ensuremath{\Vert}\ensuremath{\sim}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathit{x}}$, to x\ensuremath{\approxeq}4 from the value x=2, which is found using only periodic boundary conditions.A simple method is proposed and tested for obtaining accelerated convergence of quantum systems on small lattices with N sites. The main idea is to perform exact diagonalizations with some added irrelevant parameter, and use this parameter to accelerate the convergence to the infinite-lattice limit. In this paper different boundary conditions are used to improve the convergence for the Heisenberg model. In particular, we find that the application of the method to the d=1 antiferromagnetic Heisenberg model changes the rate of convergence of the ground-state energy per site, \ensuremath{\Vert}${\mathit{E}}_{0}$(N)-${\mathit{E}}_{0}$(\ensuremath{\infty})\ensuremath{\Vert}\ensuremath{\sim}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathit{x}}$, to x\ensuremath{\approxeq}4 from the value x=2, which is found using only periodic boundary conditions.

Physical review. B, Condensed matterSep 1, 19930 citations

Convergence of model Heisenberg dynamics

Е. В. Гусев

Theoretical and Mathematical PhysicsOct 1, 19880 citations

Efficient Classical Simulation of Weakly Interacting Fermion Dynamics

Chu Zhao, Iman Marvian, Yu Tong

We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable $A$ at time $t$ . By analyzing the convergence of the perturbative expansion in the interaction strength $λ$ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime $λ|t|^{2D+1}=\mathcal{O}(1)$ , when the Hamiltonian is geometrically local on a $D$ -dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to $λ|t|=\mathcal{O}(1)$ with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to $λ|t|=\mathcal{O}(1)$ , modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable $A$ at time $t$. By analyzing the convergence of the perturbative expansion in the interaction strength $λ$ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime $λ|t|^{2D+1}=\mathcal{O}(1)$, when the Hamiltonian is geometrically local on a $D$-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to $λ|t|=\mathcal{O}(1)$ with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to $λ|t|=\mathcal{O}(1)$, modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.

quant-pharXiv:2608.19448Aug 21, 20260 citations

Renormalization group treatment of quantum Heisenberg model

I. Golhirsch, Evgeny Levich, Victor Yakhot

Jan 1, 19770 citations

Examining Convergence of Cluster Perturbation Theory and Cluster Coupled Cluster for J1-J2 Heisenberg Spin Lattices

Jeffrey Keyes, Carlos A. Jiménez-Hoyos

In previous work, it was shown that using the cluster mean field(cMF) wavefunction as the zeroth order wavefunction for perturbation theory and coupled cluster produces a reasonable approximation for the J 1-J 2 Heisenberg model in the thermodynamic limit(TDL). However, this work was limited to cPT4 and cCCSD. The central claim of cMF is that it takes strongly correlated systems and makes them weakly correlated so that PT weakly correlated methods can be used. To confirm that this claim is true, it is important to show that the PT and CC series converge. In this paper, we extend the previous calculations to cPT7 and cCCSDTQ5 where we show that the energy is converging. We also discuss how to perform cluster based calculations efficiently in the thermodynamic limit, which is crucial for these more expensive calculations.In previous work, it was shown that using the cluster mean field(cMF) wavefunction as the zeroth order wavefunction for perturbation theory and coupled cluster produces a reasonable approximation for the J 1-J 2 Heisenberg model in the thermodynamic limit(TDL). However, this work was limited to cPT4 and cCCSD. The central claim of cMF is that it takes strongly correlated systems and makes them weakly correlated so that PT weakly correlated methods can be used. To confirm that this claim is true, it is important to show that the PT and CC series converge. In this paper, we extend the previous calculations to cPT7 and cCCSDTQ5 where we show that the energy is converging. We also discuss how to perform cluster based calculations efficiently in the thermodynamic limit, which is crucial for these more expensive calculations.

arXiv:2609.08768Sep 8, 20260 citations

Finite quantum Heisenberg spin models and their approach to the classical limit

Larry Engelhardt, Marshall Luban, Christian Schröder

We determine the temperature range over which classical Heisenberg spin models closely reproduce the zero-field susceptibility of the corresponding quantum Heisenberg models for a finite number $N$ of interacting quantum spins $s$ . Using mostly quantum and classical Monte Carlo methods, as well as analytical methods where applicable, we have explored a variety of geometries, including polygons, open chains, and all Platonic and several Archimedean polytopes. These systems range in size from $N=2$ to 120, and we have considered values of $s$ from $1∕2$ to 50 for both antiferromagnetic and ferromagnetic exchange. Particular attention is devoted to quantifying the slow convergence of the large $s$ quantum data to the limiting classical data. This is motivated by the desire to define conditions where classical Monte Carlo methods can provide useful predictions for finite quantum Heisenberg spin systems.We determine the temperature range over which classical Heisenberg spin models closely reproduce the zero-field susceptibility of the corresponding quantum Heisenberg models for a finite number $N$ of interacting quantum spins $s$. Using mostly quantum and classical Monte Carlo methods, as well as analytical methods where applicable, we have explored a variety of geometries, including polygons, open chains, and all Platonic and several Archimedean polytopes. These systems range in size from $N=2$ to 120, and we have considered values of $s$ from $1∕2$ to 50 for both antiferromagnetic and ferromagnetic exchange. Particular attention is devoted to quantifying the slow convergence of the large $s$ quantum data to the limiting classical data. This is motivated by the desire to define conditions where classical Monte Carlo methods can provide useful predictions for finite quantum Heisenberg spin systems.

Physical Review BAug 11, 200621 citations

{\mathit{E}}_{0}

(N)-

{\mathit{E}}_{0}

(\ensuremath{\infty})\ensuremath{\Vert}\ensuremath{\sim}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathit{x}}$, to x\ensuremath{\approxeq}4 from the value x=2, which is found using only periodic boundary conditions.

We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable

A

at time

t

. By analyzing the convergence of the perturbative expansion in the interaction strength

λ

for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime

λ|t|^{2D+1}=\mathcal{O}(1)

, when the Hamiltonian is geometrically local on a

D

-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to

λ|t|=\mathcal{O}(1)

with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to

λ|t|=\mathcal{O}(1)

, modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.

We determine the temperature range over which classical Heisenberg spin models closely reproduce the zero-field susceptibility of the corresponding quantum Heisenberg models for a finite number

N

of interacting quantum spins

s

. Using mostly quantum and classical Monte Carlo methods, as well as analytical methods where applicable, we have explored a variety of geometries, including polygons, open chains, and all Platonic and several Archimedean polytopes. These systems range in size from

N=2

to 120, and we have considered values of

s

from

1∕2

to 50 for both antiferromagnetic and ferromagnetic exchange. Particular attention is devoted to quantifying the slow convergence of the large

s

quantum data to the limiting classical data. This is motivated by the desire to define conditions where classical Monte Carlo methods can provide useful predictions for finite quantum Heisenberg spin systems.