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Lattice field theories with an energy current

John Cardy, Peter Suranyi

Published Jan 1, 1999• 4 citations
OpenAlex

Abstract

We investigate a lattice scalar field theory in the presence of a bias favouring the establishment of an energy current, as a model for stationary nonequilibrium processes at low temperature in a non-integrable system. There is a transition at a finite value of the bias to a gapless modulated phase which carries a classical current; however, unlike in similar, integrable, models, quantum effects also allow for a non-zero current at arbitrarily small bias. The transition is second order in the magnetically disordered phase, but is pre-empted by a first-order transition in the ferromagnetic case, at least at the mean-field level. 1

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We investigate a lattice scalar field theory in the presence of a bias favouring the establishment of an energy current, as a model for stationary nonequilibrium processes at low temperature in a non-integrable system. There is a transition at a finite value of the bias to a gapless modulated phase which carries a classical current; however, unlike in similar, integrable, models, quantum effects also allow for a non-zero current at arbitrarily small bias. The transition is second order in the magnetically disordered phase, but is pre-empted by a first-order transition in the ferromagnetic case, at least at the mean-field level. 1
The phase diagram of the Heisenberg ferromagnetic model in the presence of a magnetic random field (we have used bimodal distribution) of spin S=1/2 (quantum case) and $S=\infty $ (classical case) on a simple cubic lattice is studied within the framework of the effective-field theory in finite cluster (we have chosen N=2 spins). Integrating out the part of order parameter (equation of state), we obtained an effective Landau expansion for the free energy written in terms of the order parameter Ψ(m)Ψ(m)Ψ(m). Using Maxwell construction we have obtained the phase diagram in the T−HT-HT−H plane for all interval of field. The first-order transition temperature is calculated by the discontinuity of the magnetization at Tc∗(H)T_{c}^{\ast}(H)Tc∗​(H), on the other hand in the continuous transition the magnetization is null at T=Tc(H)T=T_{c}(H)T=Tc​(H). At null temperature (T=0) we have found the \textbf{coexistence} field Hc=3.23JH_{c}=3.23JHc​=3.23J that is independent of spin value. The transition temperature Tc(H)T_{c}(H)Tc​(H) for the classical case ($S=\infty $), in the T−HT-HT−H plane, is larger than the quantum case (S=1/2).
The purpose of this work is to understand the zero temperature phases and the phase transitions of Heisenberg spin systems which can have an extensive, spontaneous magnetic moment; this entails a study of quantum transitions with an order parameter which is also a non-abelian conserved charge. To this end, we introduce and study a new class of lattice models of quantum rotors. We compute their mean-field phase diagrams and present continuum, quantum field-theoretic descriptions of their low energy properties in different regimes. We argue that, in spatial dimensiond=1, the phase transitions in itinerant Fermi systems are in the same universality class as the corresponding transitions in certain rotor models. We discuss implications of our results for itinerant fermions systems in higherdand for other physical systems.
A study is made of an Ising model on the hexagonal closed-packed lattice, with ferromagnetic interactions D between nearest-neighbor spins located in adjacent layers, and antiferromagnetic interactions J between nearest-neighbor spins located in the same layer. The ground states of the model are studied for different values of the parameter κ = —J/D. For κ < 1/2 the ground states are ferromagnetic and for κ > 1/2 the ground state spin configurations consist of stacked identical layers, such that each layer is obtained by stacking rows of alternating spins. At the point (κ — 1/2,t = 0), where t = T/D, there exists a multitude of degenerate ground state spin configurations which are not stable for κ ≠ 1/2. Mean-field theory and low temperature expansions are used to study the phase diagram at low temperatures. Mean-field theory predicts that (κ= 1/2, t = 0) is a multiphase point where an infinite sequence of modulated phases coincide. In the vicinity of the multiphase point, the mean-field phase diagram is found to be similar to the mean-field phase diagram of the three-dimensional ANNNI model near its multiphase point. Low temperature expansions are performed to second order in x, where x = e [sup -2/t], around the phase boundary between the ferromagnetic and the modulated phases. In contrast to standard low temperature expansions, the complete contribution, to order x², is obtained by grouping the contributions from excitations which contribute to arbitrarily high orders in x. The phase boundary between the ferromagnetic and the modulated phases is found to coincide, to order x², with the line onto which Domany mapped a kinetic Ising model on the honeycomb lattice. This strongly suggests that the Domany line is a phase boundary in three-dimensions. Mean-field theory shows that this Ising model contains a continuous minimum-energy surface. A renormalization group method which applies to models which contain continuous minimum-energy surfaces is used to analyze the phase transition between the paramagnetic and the modulated phases. The calculation is performed using a Landau-Ginzburg-Wilson Hamiltonian whose minimum-energy surface consists of a hexagon and which contains fourth-order invariants due to the lattice. The calculation shows that the Hamiltonian does not contain a stable fixed point. This suggests that the paramagnetic-modulated phase transition of this Ising model is a fluctuation-induced first-order transition.
This paper continues our studies of quantum field theories on a lattice. We develop techniques for computing the low-lying spectrum of a lattice Hamiltonian using a variational approach, without recourse either to weak- or strong-coupling expansions. Our variational methods, which are relatively simple and straightforward, are applied to the Ising model in a transverse magnetic field as well as to a free spinless field theory. We demonstrate their accuracy in the vicinity of a phase transition for the Ising model by comparing with known exact solutions.