Cumulant t -Expansion for Strongly Correlated Electrons on a Lattice
- PDF / 485,253 Bytes
- 8 Pages / 612 x 792 pts (letter) Page_size
- 1 Downloads / 250 Views
ONIC PROPERTIES OF SOLID
Cumulant t-Expansion for Strongly Correlated Electrons on a Lattice A. K. Zhuravleva,* a
Mikheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620108 Russia *e-mail: [email protected] Received March 17, 2020; revised April 13, 2020; accepted April 14, 2020
Abstract—A systematic nonperturbative scheme for calculating the ground state energy is adapted for studying systems of strongly correlated electrons on a lattice. It includes a method for calculating the cumulants of the Hamiltonian and a method, using the t-expansion, for constructing successive approximations to the ground state energy by these cumulants. The scheme is applied to spinless fermion and Hubbard models, and a method is proposed to overcome the problems found in previous attempts to use this scheme to study the Hubbard model.
DOI: 10.1134/S1063776120090113
1. INTRODUCTION The problem of studying the properties of quantum systems with strong interaction between electrons is one of the most difficult problems in theoretical condensed matter physics. As a rule, analytical research methods are not sufficient here, and one has to resort to numerical methods. However, serious difficulties arise even here. Direct exact diagonalization faces the problem of exponential increase in the dimension of the Hilbert space with increasing size of the system and is therefore limited to small clusters even when using the Lanczos algorithm [1]. The quantum Monte Carlo method [2] can be used for larger systems; however, for fermions at low temperatures its accuracy is low due to the so-called sign problem [3]. A sophisticated diagonalization technique with the truncation of high-energy states—the density matrix renormalization group (DMRG) method [4]—gives excellent results for the ground state energy of one-dimensional Fermi systems, but it encounters difficulties when applied to two- and three-dimensional cases [5]. Against this background, there still exists a certain interest in constructing regular expansions [6], an attractive feature of which is the relative simplicity of calculating the expansion terms. Unfortunately, a series expansion in powers of the coupling constant usually diverges [7]. However, there are other regular methods: for example, this is the high-temperature expansion in statistical physics [8]. Less known is the so-called t-expansion [9], which we briefly outline here. Suppose given a Hamiltonian Hˆ and an initial
state |φ0 normalized to unity. Introduce an auxiliary function
ˆ −Htˆ |φ0 φ0 |He (1) . E (t ) = ˆ −Ht φ0 |e |φ0 Then, if the state |φ0 has a nonzero overlap with the ground state |ψ0, then the ground state energy E0 satisfies
E0 = lim E (t ).
(2)
t →∞
Define the moments
μm = φ0 |Hˆ m |φ0 (m = 0, 1, 2, …) and the cumulants [10] I m +1 = μm +1 −
m −1
m
p I p =0
p +1μ m − p
(3)
(4)
(to avoid misunderstanding, we note that the quantities Im in [6, 11, 12] were called connected moments). Then function (1) can be expressed [9] as a po
Data Loading...
