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Physical review. B, Condensed matter

Unified renormalization-group approach to the thermodynamic and ground-state properties of quantum lattice systems

Attilio L. Stella, Carlo Vanderzande, Raf Dekeyser

Published Feb 1, 1983• 33 citations
Publisher

Abstract

It is shown that some recently proposed iterative approaches to the ground state of quantum systems can be obtained as zero-temperature limits of free-energy-preserving renormalization transformations, if the perturbative expansion is handled in the appropriate way. Besides giving new insight into the ground-state approaches and their mutual relationships, these results allow us to perform consistent approximate calculations of the free energy at all temperatures and to get global descriptions of the critical properties. Free-energy and specific-heat calculations are reported for $\mathrm{XY}$ and Heisenberg spin-\textonehalf{} chains and for the triangular $\mathrm{XY}$ model. It is also demonstrated how this extension into finite temperatures allows us to compute in a consistent way the $z$ exponent, and to obtain substantial improvement in the numerical values for the ground-state energy density of the transverse Ising model.

DOI
10.1103/physrevb.27.1812

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It is shown that some recently proposed iterative approaches to the ground state of quantum systems can be obtained as zero-temperature limits of free-energy-preserving renormalization transformations, if the perturbative expansion is handled in the appropriate way. Besides giving new insight into the ground-state approaches and their mutual relationships, these results allow us to perform consistent approximate calculations of the free energy at all temperatures and to get global descriptions of the critical properties. Free-energy and specific-heat calculations are reported for XY\mathrm{XY}XY and Heisenberg spin-\textonehalf{} chains and for the triangular XY\mathrm{XY}XY model. It is also demonstrated how this extension into finite temperatures allows us to compute in a consistent way the zzz exponent, and to obtain substantial improvement in the numerical values for the ground-state energy density of the transverse Ising model.
The results of a new hybrid method for calculating ground-state energies for lattice Hamiltonians are presented. This method combines the ttt expansion with the real-space renormalization-group approach, with the hope of extracting infinite-volume physics at $t\ensuremath{\rightarrow}\ensuremath{\infty}$ from calculations of only a few powers of ttt. We calculate the ground-state energy for the (1+1)-dimensional anisotropic Heisenberg model and the (1+1)-dimensional Ising model. Using a blocking that treats the sites and links symmetrically, we were able to determine the exact critical point of the Ising system. In both of these systems we see that the operator renormalization-group method substantially improves upon the results of either the ttt expansion or real-space renormalization-group methods used separately.
Ground-state fidelity (GSF) and quantum renormalization group (QRG) theory have proven to be useful tools in the study of quantum critical systems. Here we lay out a general, unified formalism of GSF and QRG; specifically, we propose a method for calculating GSF through QRG, obviating the need for calculating or approximating ground states. This method thus enhances the characterization of quantum criticality as well as scaling analysis of relevant properties with system size. We illustrate the formalism in the one-dimensional Ising model in a transverse field (ITF) and the anisotropic spin-1/2 Heisenberg ( XXZ ) model. Explicitly, we find the scaling behavior of the GSF for the ITF model in both small- and large-size limits, the corresponding critical exponents, the exact value of the GSF in the thermodynamic limit and a closed form for the GSF for arbitrary size and system parameters. In the case of the XXZ model, we also present an analytic expression for the GSF, which captures well the criticality of the model, hence excluding doubts that GSF might be an insufficient tool for signaling criticality in this model.
The linear renormalization-group transformation is proposed to study critical temperatures and temperature dependence of the thermodynamic values (free energy, specific heat) of the weakly interacting spin chains. The method is examined in two classical systems: the standard Ising model on a rectangular lattice and the Ising spin chains coupled by pair-pair interactions. In the latter case the transformation does not exhibit any nontrivial fixed point. This result agrees with the exactly determined free energy of the system under consideration. The method is also applied to the weakly interacting quantum XY spin chains. The critical temperature as a function of the interchain interactions and the temperature dependence of the specific heat are found.
A real-space renormalization-group theory is proposed for the anisotropic Heisenberg chain which yields physically reasonable results for all values of the temperature and of the anisotropy. The method is based on a four-spin cluster and a transformation which preserves the sublattice structure of the antiferromagnet. At low temperatures the transformation maps the four lowest energy levels of the four-spin cluster onto the four levels of two spins, giving ground-state energies in excellent agreement with exact results. The specific-heat exponent for the isotropic antiferromagnet is given correctly, but the exponent for larger transverse coupling is somewhat too small. A scaling form is proposed for the free energy near the isotropic antiferromagnetic fixed point, and the crossover to Ising-like behavior is discussed.
A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.