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Physical review. B, Condensed matter

Calculation of specific heat and susceptibilities with the use of the Trotter approximation

Richard M. Fye, Richard T. Scalettar

Published Sep 1, 1987• 50 citations
Publisher

Abstract

The Trotter approximation has been used extensively in quantum Monte Carlo simulations. We consider here different ways of calculating the general zero-field susceptibility \ensuremath{\chi} and the specific heat C when a Trotter approximation is used, deriving analytically the error dependence on the imaginary-time increment \ensuremath{\Delta}\ensuremath{\tau} as well as the low-temperature behavior. We find that certain definitions of \ensuremath{\chi} and C exhibit spurious divergences at low temperatures, and suggest the most appropriate ways to extract these quantities. We test our general predictions on two models, and discuss the implications of our results for numerical simulations.

DOI
10.1103/physrevb.36.3833

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The Trotter approximation has been used extensively in quantum Monte Carlo simulations. We consider here different ways of calculating the general zero-field susceptibility \ensuremath{\chi} and the specific heat C when a Trotter approximation is used, deriving analytically the error dependence on the imaginary-time increment \ensuremath{\Delta}\ensuremath{\tau} as well as the low-temperature behavior. We find that certain definitions of \ensuremath{\chi} and C exhibit spurious divergences at low temperatures, and suggest the most appropriate ways to extract these quantities. We test our general predictions on two models, and discuss the implications of our results for numerical simulations.
We explore the errors in the free energy and in operator expectation values when the quantum operator ${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$=${\ensuremath{\prod}}_{l=1}^{L}$${e}^{\mathrm{\ensuremath{-}}(\ensuremath{\Delta}\ensuremath{\tau})H}$ is approximated by ${\ensuremath{\prod}}_{l=1}^{L}$f, where \ensuremath{\Delta}\ensuremath{\tau}=\ensuremath{\beta}/L and f is an approximant to ${e}^{\mathrm{\ensuremath{-}}(\ensuremath{\Delta}\ensuremath{\tau})H}$. We determine analytically the dependence of the resulting errors on \ensuremath{\Delta}\ensuremath{\tau} for \ensuremath{\Delta}\ensuremath{\tau} small, on \ensuremath{\beta} for \ensuremath{\beta} large, and on the size of the system in the limit of a large system. We focus on Trotter approximations, as well as on the expansion of ${e}^{\mathrm{\ensuremath{-}}(\ensuremath{\Delta}\ensuremath{\tau})H}$ in powers of (\ensuremath{\Delta}\ensuremath{\tau})H. Our results are particularly relevant to Monte Carlo studies of quantum systems, and can be used effectively to eliminate the error due to Trotter-like approximations and to provide a guide for choosing a particular type of approximation.
We show that the zero-temperature susceptibility of the one-dimensional, dimerized Hubbard model at quarter-filling can be accurately determined on the basis of exact diagonalization of small clusters. The best procedure is to perform a finite-size scaling of the spin velocity ${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$, and to calculate the susceptibility from the Luttinger-liquid relation \ensuremath{\chi}=2/\ensuremath{\pi}${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$. We show that these results are reliable by comparing them with the analytical results that can be obtained in the weak- and strong-coupling limits. We have also used quantum Monte Carlo simulations to calculate the temperature dependence of the susceptibility for parameters that should be relevant to the Bechgaard salts. This shows that, used together, these numerical techniques are able to give precise estimates of the low-temperature susceptibility of realistic one-dimensional models of correlated electrons.
In the Ising model on a helical lattice with period M (in the limit M →∞ of this lattice, the model corresponds to the two-dimensional Ising model), the exact simultaneous equations to calculate physical quantities are derived. The zero field magnetic susceptibility χ 0 , and also the specific heat C are obtained numerically for the system of M ≤11. Furthermore, in the Ising model with up to the next nearest neighbor interactions on the helical lattice, χ 0 and C are calculated for the system of M ≤9. The results obtained are an excellent approximation to the exact χ 0 and C of the two-dimensional systems.
The temperature dependence of the paramagnetic zero-field susceptibility chi is calculated for the second-neighbour S=1/2 Heisenberg model and for the Baxter-Wu model, from Pade approximants to existing high-temperature series expansions and by Monte Carlo simulations. The data are well represented by a power law chi T=(t')-y using the critical susceptibility exponent gamma over the whole paramagnetic temperature range when a nonlinear scaling variable t' is used; namely t'=1-Tc/T for the second-neighbour models and t'=1-(Tc/T)2 for the Baxter-Wu model.
A Trotter-Suzuki mapping is used to calculate the finite-temperature properties of the one-dimensional supersymmetric t − J model. This approach allows for the exact calculation of various thermodynamical properties by means of the quantum transfer matrix (QTM). The free energy and other interesting quantities are obtained such as the specific heat and compressibility. For the largest eigenvalue of the QTM leading to the free energy a set of just two non-linear integral equations is presented. These equations are studied analytically and numerically for different particle densities and temperatures. The structure of the specific heat is discussed in terms of the elementary charge as well as spin excitations. Special emphasis is placed on the study of the low-temperature behavior confirming scaling predictions by conformal field theory and Luttinger liquid theory. To our knowledge this is the first complete investigation of a strongly correlated electron system on a lattice at finite temperature.
The specific heat and the compressibility for the integrable t – J model are calculated showing Luttinger liquid behavior for low temperatures. A Trotter-Suzuki mapping and the quantum transfer matrix approach are utilized. Using an algebraic Bethe ansatz, this method permits the exact calculation of the free energy and related quantities. A set of just two non-linear integral equations determining these quantities is studied for various particle densities and temperatures. The structure of the specific heat is discussed in terms of the elementary charge as well as spin excitations.