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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

New method for the Hamiltonian formulation for lattice spin systems

Hamzeh H. Roomany, H. W. Wyld, L. Holloway

Published Mar 15, 1980• 72 citations
Publisher

Abstract

The Lanczos scheme for finding low-lying eigenvalues of a sparse matrix of large dimension is applied to solving the Hamiltonian formulation for $Z(2)$ and $Z(3)$ Ising spin systems in 1+1 dimensions. The location of the fixed points and the values of the critical indices are consistent with known results.

DOI
10.1103/physrevd.21.1557

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Finite-lattice approach to the O(2) and O(3) models in 1 + 1 dimensions and the (2+1)-dimensional Ising model

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We describe an extension of an approximation scheme proposed earlier for calculating the critical properties of spin-(1/2) Ising systems in terms of the duo-diagonal, sparse matrices of Kramers and Wannier. The computational framework is refined and extended to accommodate a variety of two-, three-, and higher-dimensional lattices as well as interactions other than nearest neighbor. The relation between the underlying symmetries of the lattice and the structure of its associated phase diagram is clarified. Making use of a probabilistic interpretation for the components of the eigenvectors associated with the maximum eigenvalue of the duo-diagonal matrices, it is shown that this makes practical the computation of various thermodynamic quantities---as well as correlation functions---in terms of the expectation values of appropriate operators taken between these eigenvectors. When combined with the techniques of finite-size scaling the present scheme is confirmed to be effective for extracting the critical-point properties of the infinite lattice.

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An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds $n$ lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.

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Finite-lattice methods in quantum Hamiltonian field theory. I. O(2) and O(3) Heisenberg models

Chris J. Hamer, Michael N. Barber

For pt.I see ibid., vol.14, p.241-57 (1981). Two efficient methods for finding the low-lying states of Hamiltonians on finite lattices are described. The first involves constructing a finite representation of the Hamiltonian using strong-coupling eigenstates, while the second is based on the Lanczos recursion method. The methods are used to determine the mass gap of the O(2) and O(3) Heisenberg Hamiltonians in (1+1) dimensions for a sequence of finite chains. The critical behaviour of the infinite chain is then analysed by extrapolating the finite-lattice estimates using finite-size scaling. A remarkably sensitive test is developed for the presence of a phase transition. For the O(2) model data this test yields strong evidence for a phase transition with the weak-coupling phase massless, while, in the O(3) case the test supports, although more weakly, the absence of any transition.

Journal of Physics A Mathematical and General•Jan 1, 1981•70 citations
The Lanczos scheme for finding low-lying eigenvalues of a sparse matrix of large dimension is applied to solving the Hamiltonian formulation for Z(2)Z(2)Z(2) and Z(3)Z(3)Z(3) Ising spin systems in 1+1 dimensions. The location of the fixed points and the values of the critical indices are consistent with known results.
We describe an extension of an approximation scheme proposed earlier for calculating the critical properties of spin-(1/2) Ising systems in terms of the duo-diagonal, sparse matrices of Kramers and Wannier. The computational framework is refined and extended to accommodate a variety of two-, three-, and higher-dimensional lattices as well as interactions other than nearest neighbor. The relation between the underlying symmetries of the lattice and the structure of its associated phase diagram is clarified. Making use of a probabilistic interpretation for the components of the eigenvectors associated with the maximum eigenvalue of the duo-diagonal matrices, it is shown that this makes practical the computation of various thermodynamic quantities---as well as correlation functions---in terms of the expectation values of appropriate operators taken between these eigenvectors. When combined with the techniques of finite-size scaling the present scheme is confirmed to be effective for extracting the critical-point properties of the infinite lattice.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds nnn lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
The Lanczos method in operator form is applied to a general lattice Hamiltonian and expressions for the first few Lanczos matrices in terms of the connected Hamiltonian moments 〈Hn{\mathit{H}}^{\mathit{n}}Hn〉c{\mathrm{〉}}_{\mathit{c}}〉c​ and the number of plaquettes, Np{\mathit{N}}_{\mathit{p}}Np​, are obtained. Expansions in 1/Np{\mathit{N}}_{\mathit{p}}Np​ suggest a very simple general form for the first few terms in the 1/Np{\mathit{N}}_{\mathit{p}}Np​ expansions for all ${\mathrm{\ensuremath{\alpha}}}_{\mathit{n}}$ and ${\mathrm{\ensuremath{\beta}}}_{\mathit{n}}$. For the one-dimensional Heisenberg spin chain it is demonstrated that the ground-state eigenvalue of the tridiagonal Lanczos matrix derived from this plaquette expansion approaches the true infinite lattice limit as the number of terms in the plaquette expansion is increased.
A variant of White's density-matrix renormalization-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-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain and a dimerized XY\mathrm{XY}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 that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
For pt.I see ibid., vol.14, p.241-57 (1981). Two efficient methods for finding the low-lying states of Hamiltonians on finite lattices are described. The first involves constructing a finite representation of the Hamiltonian using strong-coupling eigenstates, while the second is based on the Lanczos recursion method. The methods are used to determine the mass gap of the O(2) and O(3) Heisenberg Hamiltonians in (1+1) dimensions for a sequence of finite chains. The critical behaviour of the infinite chain is then analysed by extrapolating the finite-lattice estimates using finite-size scaling. A remarkably sensitive test is developed for the presence of a phase transition. For the O(2) model data this test yields strong evidence for a phase transition with the weak-coupling phase massless, while, in the O(3) case the test supports, although more weakly, the absence of any transition.