Unified renormalization-group approach to the thermodynamic and ground-state properties of quantum lattice systems (original) (raw)
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Physical Review Letters, 2011
A linearized tensor renormalization group (LTRG) algorithm is proposed to calculate the thermodynamic properties of one-dimensional quantum lattice models, that is incorporated with the infinite time-evolving block decimation technique, and allows for treating directly the two-dimensional transfer-matrix tensor network. To illustrate its feasibility, the thermodynamic quantities of the quantum XY spin chain are calculated accurately by the LTRG, and the precision is shown to be comparable with (even better than) the transfer matrix renormalization group (TMRG) method. Unlike the TMRG scheme that can only deal with the infinite chains, the present LTRG algorithm could treat both finite and infinite systems, and may be readily extended to boson and fermion quantum lattice models.
Thermal tensor renormalization group simulations of square-lattice quantum spin models
Physical Review B
In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a thermal correlation length, which justifies the analysis of finite system size for the sake of numerical efficiency. In this paper we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width W = 10. We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc., are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH we obtain at low temperature an energy per site u * g −0.6694(4) and a spontaneous magnetization m * S 0.30(1), which is already consistent with the ground state properties. We extract an exponential divergence vs. T of the structure factor S(M), as well as the correlation length ξ, at the ordering wave vector M = (π, π), which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analysing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature Tc.
Physical Review B, 1979
We apply a recently proposed real-space renormalization-group method to the twodimensional square Ising model in a transverse magnetic field at zero temperature. We do the calculation to firsthand second order in the intercell coupling; in both cases we find a nontrivial fixed point. We compare the obtained critical exponents with accepted values for the threedimensional Ising model from high-temperature series expansions. In first order the agreement is poor, in second order we obtain good agreement with the expected values, In a recent paper', we proposed a real-space renormalization-group (RG) method for quantum spin systems on a lattice at zero temperature. Our method is a systematic perturbation expansion which reduces, to lowest order, to a truncation method discussed by several authors. ' ' In that method, one divides the lattice into cells, diagonalizes exactly the intracell Hamiltonian and takes a truncated basis consisting of the lowest-lying states in each cell. The renormalized Hamiltonian is then the part of the original Hamiltonian spanned by these low-lying states. %e show in Ref. 1 that this procedure can be viewed as a first-order calculation in the intercell coupling, and show how to systematically carry it out to arbitrary order. In Ref. 1 we applied the method to the one-dimensional Ising model in a transverse field to second order and to the two-dimensional triangular Ising model in a transverse field to first order. In the triangular lattice, complications arise if one tries to do a second-order calculation because, due to geometric reasons, one generates a renormalized Hamiltonian that does not have the same threefold symmetry as the original one. In this paper, we apply our method to the two-dimensional square lattice, which is simpler from a geometrical point of view. It is known' that the critical behavior of a d-dimensional Ising model at zero temperature as a function of a transverse field is the same as that of (d+1)dimensional Ising system in zero transverse field as a function of temperature; therefore, we obtain from our calculation the critical exponents for the threedimensional Ising model. %e compare our results with high-temperature expansion results for the critical exponents of a three-dimensional Ising model. %e consider the Hamiltonian where i and j run over nearest neighbors in a square lattice. %e divide the system into square cells of four sites each, as shown in Fig. 1, and diagonalize exactly the intracell Hamiltonian. The cell Hilbert space of 16 states splits into two sets of eight states each, corresponding to states with even and odd number of spins up (or down). The eigenstates and eigenvalues for those two sets can be found analytically and are listed in . Note that the ground state for both limiting cases 5/e 0 and e/5 0 can be formed from the lowest even and odd cell states, i0) and i 1); for 5/~0 the ground state is simply (2a)
Density matrix renormalisation group for a quantum spin chain at non-zero temperature
1996
We apply a recent adaptation of White's density matrix renormalisation group (DMRG) method to a simple quantum spin model, the dimerised XYXYXY chain, in order to assess the applicabilty of the DMRG to quantum systems at non-zero temperature. We find that very reasonable results can be obtained for the thermodynamic functions down to low temperatures using a very small basis set. Low temperature results are found to be most accurate in the case when there is a substantial energy gap.
Exact renormalization-group study of aperiodic Ising quantum chains and directed walks
Physical Review B, 1997
We consider the Ising model and the directed walk on two-dimensional layered lattices and show that the two problems are inherently related: The zero-field thermodynamical properties of the Ising model are contained in the spectrum of the transfer matrix of the directed walk. The critical properties of the two models are connected to the scaling behavior of the eigenvalue spectrum of the transfer matrix which is studied exactly through renormalization for different self-similar distributions of the couplings. The models show very rich bulk and surface critical behaviors with nonuniversal critical exponents, coupling-dependent anisotropic scaling, first-order surface transition, and stretched exponential critical correlations. It is shown that all the nonuniversal critical exponents obtained for the aperiodic Ising models satisfy scaling relations and can be expressed as functions of varying surface magnetic exponents.
Phys Rev B, 1999
A variant of White's density matrix renormalisation group scheme which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-1/2 antiferromagnetic Heisenberg chain and a dimerised XY spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings which are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
Ground-state correlations and finite temperature properties of the transverse Ising model
The European Physical Journal B, 2005
We present a semi-analytic study of Ising spins on a simple square or cubic lattice coupled to a transverse magnetic field of variable strength. The formal analysis employs correlated basis functions (CBF) theory to investigate the properties of the corresponding N-body ground and excited states. For these states we discuss two different ansaetze of correlated trial wave functions and associated longitudinal and transverse excitation modes. The formalism is then generalized to describe the spin system at nonzero temperatures with the help of a suitable functional approximating the Helmholtz free energy. To test the quality of the functional in a first step we perform numerical calculations within the extended formalism but ignore spatial correlations. Numerical results are reported on the energies of the longitudinal and the transverse excitation modes at zero temperature, on critical data at finite temperatures, and on the optimized spontaneous magnetization as a function of temperature and external field strength.