Physical Review Letters

Phases of Two-Dimensional Heisenberg Spin Systems from Strong-Coupling Expansions

Chris J. Hamer, John B. Kogut, Leonard Susskind

Published Nov 13, 1978• 32 citations

Abstract

A quantum-mechanical Hamiltonian formulation of lattice spin systems is used to search for phase transitions in the O(2), O(3), and O(4) Heisenberg models in two dimensions. Strong-coupling expansions which have been calculated through eighth order indicate a phase transition at nonzero coupling for the O(2) model, but the non-Abelian models are predicted to exist only in the strong-coupling phase.

DOI: 10.1103/physrevlett.41.1337

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Physical Review Letters

Phases of Two-Dimensional Heisenberg Spin Systems from Strong-Coupling Expansions

Chris J. Hamer, John B. Kogut, Leonard Susskind

Published Nov 13, 1978• 32 citations

Publisher

Abstract

DOI: 10.1103/physrevlett.41.1337

Related papers

Accurate results from Hamiltonian strong-coupling expansions of the Ising, planar, O(3) , and O(4) spin systems

Chris J. Hamer, John B. Kogut

Strong-coupling expansions of the quantum-Hamiltonian form of the Ising, $O(2)$ , $O(3)$ , and $O(4)$ models in two dimensions are analyzed. The critical point, correlation-length exponent $\ensuremath{\nu}$, and magnetization index $\ensuremath{\beta}$ are found exactly for the Ising model. The susceptibility index $\ensuremath{\gamma}$ and the specific-heat index $\ensuremath{\alpha}$ are computed to within 1% of their known values. The phase transition of the $O(2)$ model is found and the correlation length is predicted to diverge with an essential singularity $\ensuremath{\xi}(T)\ensuremath{\sim}\mathrm{exp}[b{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\sigma}}]$ where $\ensuremath{\sigma}=0.7\ifmmode\pm\else\textpm\fi{}0.1$. The index of the spin-spin correlation function $\ensuremath{\eta}$ is predicted to be 0.26\ifmmode\pm\else\textpm\fi{}0.03 at the critical point in agreement with experiment. The non-Abelian spin systems are argued to exist only in a disordered phase. The asymptotic freedom of these models plays an important role in these arguments by ruling out the possibility of weak-coupling phase transitions.Strong-coupling expansions of the quantum-Hamiltonian form of the Ising, $O(2)$, $O(3)$, and $O(4)$ models in two dimensions are analyzed. The critical point, correlation-length exponent $\ensuremath{\nu}$, and magnetization index $\ensuremath{\beta}$ are found exactly for the Ising model. The susceptibility index $\ensuremath{\gamma}$ and the specific-heat index $\ensuremath{\alpha}$ are computed to within 1% of their known values. The phase transition of the $O(2)$ model is found and the correlation length is predicted to diverge with an essential singularity $\ensuremath{\xi}(T)\ensuremath{\sim}\mathrm{exp}[b{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\sigma}}]$ where $\ensuremath{\sigma}=0.7\ifmmode\pm\else\textpm\fi{}0.1$. The index of the spin-spin correlation function $\ensuremath{\eta}$ is predicted to be 0.26\ifmmode\pm\else\textpm\fi{}0.03 at the critical point in agreement with experiment. The non-Abelian spin systems are argued to exist only in a disordered phase. The asymptotic freedom of these models plays an important role in these arguments by ruling out the possibility of weak-coupling phase transitions.

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This is a study of finite lattice methods applied to quantum spin models on one and two dimensional lattices.The motivation for the use of these techniques is provided by placing them in the context of the wider field of approximation methods in the statistical mechanics of spin models.After reviewing previous finite lattice calculations on quantum spin models, the theory needed to efficiently generate and analyse finite lattice data is given.Finite lattice calculations on three quantum spin models are then presented.The first calculation provides an estimate of the transverse susceptibility of the one dimensional antiferromagnetic XY model, by extrapolation from calculations on rings of up to 10 sites.Good convergence is found for KB T ~ 0•4 and these results are compared with recent c -axis susceptibility measurements on the compound Cs 2 CoC1 4 .Chapters 5 and 6 study the quantum hamiltonian analog of the two dimensional axial next nearest neighbour Ising model, A perturbation analysis using non degenerate Rayleigh-Schrodingerperturbation theory is followed by a mass gap scaling analysis.These analyses provide accurate phase boundaries for, and ellucidate several interesting features of, the phase diagram of this model.In particular, the model is shown to exhibit a phase whose modulation varies continuously with a change in the interaction parameters in the hamiltonian.This "modulated" phase region lies between a normal ordered phase and a paramagnetic phase.Evid~nce is given that the transition from the paramagnetic to modulated phase is exponential in nature.The third model studied is a spin 1 biquadratic Heisenberg model with hamiltonian,<ij> (Jl s .. s. +J2(~ . .~.)2). 1 J 1 J Accurate estimates of the ground state energy in both one and two dimensions are found by calculating this quantity on 4,6,8 and 10 site lattices.Comment is made about the ground state phase diagram, and about the existence of intermediate spin states and states with a complicated magnetic ordering.The thesis concludes by commenting on the performance of finite lattice methods in the three models studied and by outlining certain directions for future advance., ACKNOWLEVGEMENTS I would like to expressThis is a study of finite lattice methods applied to quantum spin models on one and two dimensional lattices.The motivation for the use of these techniques is provided by placing them in the context of the wider field of approximation methods in the statistical mechanics of spin models.After reviewing previous finite lattice calculations on quantum spin models, the theory needed to efficiently generate and analyse finite lattice data is given.Finite lattice calculations on three quantum spin models are then presented.The first calculation provides an estimate of the transverse susceptibility of the one dimensional antiferromagnetic XY model, by extrapolation from calculations on rings of up to 10 sites.Good convergence is found for KB T ~ 0•4 and these results are compared with recent c -axis susceptibility measurements on the compound Cs 2 CoC1 4 .Chapters 5 and 6 study the quantum hamiltonian analog of the two dimensional axial next nearest neighbour Ising model, A perturbation analysis using non degenerate Rayleigh-Schrodingerperturbation theory is followed by a mass gap scaling analysis.These analyses provide accurate phase boundaries for, and ellucidate several interesting features of, the phase diagram of this model.In particular, the model is shown to exhibit a phase whose modulation varies continuously with a change in the interaction parameters in the hamiltonian.This "modulated" phase region lies between a normal ordered phase and a paramagnetic phase.Evid~nce is given that the transition from the paramagnetic to modulated phase is exponential in nature.The third model studied is a spin 1 biquadratic Heisenberg model with hamiltonian,<ij> (Jl s .. s. +J2(~ . .~.)2). 1 J 1 J Accurate estimates of the ground state energy in both one and two dimensions are found by calculating this quantity on 4,6,8 and 10 site lattices.Comment is made about the ground state phase diagram, and about the existence of intermediate spin states and states with a complicated magnetic ordering.The thesis concludes by commenting on the performance of finite lattice methods in the three models studied and by outlining certain directions for future advance., ACKNOWLEVGEMENTS I would like to express

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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.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 GeneralJan 1, 198170 citations

Monte Carlo renormalization-group analysis of the classical Heisenberg model in two dimensions

Stephen H. Shenker, Jan Tobochnik

Monte Carlo renormalization-group methods were applied to the classical three-component Heisenberg model on a two-dimensional lattice. Expectation values of local correlations of spins and various sized block spins were computed using traditional Monte Carlo methods. By matching quantities at different length scales generated by different Monte Carlo Hamiltonians we directly determined the renormalization of the nearest-neighbor coupling. Using these data and the results of high-temperature expansions and low-temperature renormalization-group calculations we have determined that this model does not have a phase transition. We have also obtained the amplitude for the low-temperature divergence of the susceptibility and the correlation length.Monte Carlo renormalization-group methods were applied to the classical three-component Heisenberg model on a two-dimensional lattice. Expectation values of local correlations of spins and various sized block spins were computed using traditional Monte Carlo methods. By matching quantities at different length scales generated by different Monte Carlo Hamiltonians we directly determined the renormalization of the nearest-neighbor coupling. Using these data and the results of high-temperature expansions and low-temperature renormalization-group calculations we have determined that this model does not have a phase transition. We have also obtained the amplitude for the low-temperature divergence of the susceptibility and the correlation length.

Physical review. B, Condensed matterNov 1, 1980259 citations

Strong-coupling expansions of the quantum-Hamiltonian form of the Ising,

O(2)

O(3)

, and

O(4)

models in two dimensions are analyzed. The critical point, correlation-length exponent $\ensuremath{\nu}$, and magnetization index $\ensuremath{\beta}$ are found exactly for the Ising model. The susceptibility index $\ensuremath{\gamma}$ and the specific-heat index $\ensuremath{\alpha}$ are computed to within 1% of their known values. The phase transition of the

O(2)

model is found and the correlation length is predicted to diverge with an essential singularity $\ensuremath{\xi}(T)\ensuremath{\sim}\mathrm{exp}[b{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\sigma}}]$ where $\ensuremath{\sigma}=0.7\ifmmode\pm\else\textpm\fi{}0.1$. The index of the spin-spin correlation function $\ensuremath{\eta}$ is predicted to be 0.26\ifmmode\pm\else\textpm\fi{}0.03 at the critical point in agreement with experiment. The non-Abelian spin systems are argued to exist only in a disordered phase. The asymptotic freedom of these models plays an important role in these arguments by ruling out the possibility of weak-coupling phase transitions.

The Hamiltonian formulation of lattice spin systems is used to study the critical properties of a truncated quantum

O(n)

-spin model in one and

d

spatial dimensions for arbitrary

n

. Strong-coupling expansions for the mass gap, ground-state energy density, and susceptibility have been computed up to 14th order and used to search for phase transitions. For

n=0

we obtain the exponents for the self-avoiding random-walk problem. In the

O(2)

case we find that the correlation length diverges with an essential singularity at the critical point in 1 + 1 dimensions. No phase transition is found for the

O(3)

and

O(4)

models in the same dimension. Exponents for other values of

n

are presented and higher dimensions are considered.

Recent lattice Hamiltonian strong-coupling calculations for the (1+1)-dimensional Ising, O(2), O(3), and O(4) models are extended to

d+1

dimensions and to general

\mathrm{O}(n)

symmetry. This allows a nontrivial numerical check on the rate of convergence of the strong-coupling expansions as the multidimensional models all have standard second-order phase transitions which have been well studied by other methods. Dlog-Pad\'e analysis of the fourth-order mass gap series for the (2+1)-dimensional Ising model show the critical exponent $\ensuremath{\nu}$ to be calculated to about the 10% level of accuracy. The behavior of $\ensuremath{\nu}$ as a function of

d

and

n

is exhibited and compared against known results. Mean-field behavior is recovered exactly in the $d\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, while the Berlin-Kac spherical model is obtained in the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit.