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Physical review. B, Condensed matter

Phase transitions in the quantum Ising and rotor models with a long-range interaction

Amit Dutta, Jayanta K. Bhattacharjee

Published Oct 18, 2001• 161 citations
Publisher

Abstract

We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as ${1/r}^{d+\ensuremath{\sigma}},$ where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when $\ensuremath{\sigma}=2.$ The dynamical exponent varies continuously as the parameter \ensuremath{\sigma} and is unity for $\ensuremath{\sigma}=2.$ The one-dimensional long-range quantum Ising system shows a phase transition at $T=0$ for all values of \ensuremath{\sigma}. The most interesting observation is that the phase diagram for $\ensuremath{\sigma}=d=1$ shows a line of Kosterlitz-Thouless transition at finite temperature even though the $T=0$ transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter \ensuremath{\sigma}. We explore the phase diagram extending the short-range quantum nonlinear \ensuremath{\sigma} model renormalisation group equations to the present case.

DOI
10.1103/physrevb.64.184106

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We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as 1/rd+\ensuremathσ,{1/r}^{d+\ensuremath{\sigma}},1/rd+\ensuremathσ, where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when \ensuremathσ=2.\ensuremath{\sigma}=2.\ensuremathσ=2. The dynamical exponent varies continuously as the parameter \ensuremath{\sigma} and is unity for \ensuremathσ=2.\ensuremath{\sigma}=2.\ensuremathσ=2. The one-dimensional long-range quantum Ising system shows a phase transition at T=0T=0T=0 for all values of \ensuremath{\sigma}. The most interesting observation is that the phase diagram for \ensuremathσ=d=1\ensuremath{\sigma}=d=1\ensuremathσ=d=1 shows a line of Kosterlitz-Thouless transition at finite temperature even though the T=0T=0T=0 transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter \ensuremath{\sigma}. We explore the phase diagram extending the short-range quantum nonlinear \ensuremath{\sigma} model renormalisation group equations to the present case.
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
Several recent experiments in atomic, molecular, and optical systems motivated a huge interest in the study of quantum long-range systems. Our goal in this paper is to present a general description of their critical behavior and phases, devising a treatment valid in ddd dimensions, with an exponent $d+\ensuremath{\sigma}$ for the power-law decay of the couplings in the presence of an O(N)\text{O}(N)O(N) symmetry. By introducing a convenient ansatz for the effective action, we determine the phase diagram for the NNN-component quantum rotor model with long-range interactions, with N=1N=1N=1 corresponding to the Ising model. The phase diagram in the $\ensuremath{\sigma}\text{\ensuremath{-}}d$ plane shows a nontrivial dependence on $\ensuremath{\sigma}$. As a consequence of the fact that the model is quantum, the correlation functions are anisotropic in the spatial and time coordinates for $\ensuremath{\sigma}$ smaller than a critical value, and in this region the isotropy is not restored even at criticality. Results for the correlation length exponent $\ensuremath{\nu}$, the dynamical critical exponent zzz, and a comparison with numerical findings for them are presented.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.
In this work we investigate the quantum phase transitions at zero temperature of the one-dimensional transverse Ising model with an extra term containing four-spin interactions. The competition between the energy couplings of the model leads to an interesting zero-temperature phase diagram. We use a modified Lanczos method to determine the ground state and the first excited state energies of the system, with sizes of up to 20 spins. We apply finite size scaling to the energy gap to obtain the boundary region where ferromagnetic to paramagnetic transition takes place. We also find the critical exponent associated with the correlation length. We find a degenerate ⟨3,1⟩⟨3,1⟩⟨3,1⟩ phase region. The first-order transition boundary between this phase and the paramagnetic phase is determined by analyzing the behavior of the transverse spin susceptibility as the system moves from one region to the other.
Critical exponents for a one-dimensional general continuous Ising model with long-range ferromagnetic interactions decaying as 1/r1+σ are calculated using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between −1 and 1. The critical point behavior is investigated. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents; the singularities in the specific heat and susceptibility depend on σ. For σ=1, there are extremely weak finite size effects: the first-order and the second-order cumulants of the order parameter yield ν=2.42(1). The susceptibility exponent γ=2.259(9). Results for σ=0.7 will be shown and discussed.
By using the Monte Carlo simulation, we investigate the critical behavior of the 1-dimensional spin system with infinite long-range (LR) interaction r^{-(1+σ)}. By changing the value σ, we compare the results with those of the d-dimensional spin system with the nearest-neighbor(NN) interaction only. In the case of XY spin model, we obtain three different type phase transitions, i.e. the mean field type for 0 < σ < 0.5 , the σ-dependent non-trivial one for 0.5 < σ < 1 and 'Berezinskii-Kosterlitz-Thouless (BKT)-like’transition at σ = 1 as in the NN model of d > 4 , 2 < d < 4 and d = 2 , respectively. In the case of q-state clock spin model with σ = 1, we also confirm the BKT-like transition together with the similar q-dependence of the critical behavior to that of the 2-dimensional NN model. These results suggest that the infinite long-range interaction can partly compensate the role of the lattice dimensionality by increasing the effective value d from d = 1 to d = 2.