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Ising critical behaviour in the one-dimensional frustrated quantum XY model

Enzo Granato

Published Jun 1, 1993• 0 citations
OpenAlex

Abstract

A generalization of the one-dimensional frustrated quantum XV model is considered in whiell the inter and intraechain coupling constarás of the two mfinite XV (planar rotor) chains have different strengths. The mudei can describe Lhe superconductor-insulator transitron due to chargtng effects in a ladder of Josephson junctions ia a magnetic field with Italf a flux quantum per pliaquet te. From a fluctuation- effective action, this transition is expected to Fie iii the universality class of Lhe two dimensional classical XY-Ising modei. rilhe criticai behaviour is studiral using a Monte Cano transfer erratrix applied to Lhe path-integral representation of the model and a finte sie scaling analysis of data on srnall system sues. II i found that, unlike the previous stirdied case of equal inter and intra chain roupling constai-as, Lhe XV and Ising like excitations of the quantum model decouple for large interchain coupling, giving rise to pule Ising mudei criticai behaviour for lhe chiratity order parameter in good agreement with Lhe results for Lhe XY-Ising modal.

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A generalization of the one-dimensional frustrated quantum XV model is considered in whiell the inter and intraechain coupling constarás of the two mfinite XV (planar rotor) chains have different strengths. The mudei can describe Lhe superconductor-insulator transitron due to chargtng effects in a ladder of Josephson junctions ia a magnetic field with Italf a flux quantum per pliaquet te. From a fluctuation- effective action, this transition is expected to Fie iii the universality class of Lhe two dimensional classical XY-Ising modei. rilhe criticai behaviour is studiral using a Monte Cano transfer erratrix applied to Lhe path-integral representation of the model and a finte sie scaling analysis of data on srnall system sues. II i found that, unlike the previous stirdied case of equal inter and intra chain roupling constai-as, Lhe XV and Ising like excitations of the quantum model decouple for large interchain coupling, giving rise to pule Ising mudei criticai behaviour for lhe chiratity order parameter in good agreement with Lhe results for Lhe XY-Ising modal.
A generalization of the one-dimensional frustrated quantum XY\mathrm{XY}XY model is considered in which the interchain and intrachain coupling constants of the two infinite XY\mathrm{XY}XY (planar rotor) chains have different strengths. The model can describe the superconductor to insulator transition due to charging effects in a ladder of Josephson junctions in a magnetic field with half a flux quantum per plaquette. From a fluctuation-effective action, this transition is expected to be in the universality class of the two-dimensional classical XY\mathrm{XY}XY-Ising model. The critical behavior is studied using a Monte Carlo transfer matrix applied to the path-integral representation of the model and a finite-size-scaling analysis of data on small system sizes. It is found that, unlike the previous studied case of equal interchain and intrachain coupling constants, the XY\mathrm{XY}XY and Ising-like excitations of the quantum model decouple for large interchain coupling, giving rise to pure Ising model critical behavior for the chirality order parameter and a superconductor-insulator transition in the universality class of the two-dimensional classical XY\mathrm{XY}XY model.
A one-dimensional quantum version of the frustrated XY model is introduced which can be physically realized as a ladder of Josephson junctions at half of a flux quantum per plaquette. From a fluctuation effective action, the zero-temperature (superconductor-insulator) transition is predicted to be in the universality class of the two-dimensional classical XY-Ising model. A Monte Carlo transfer matrix is used to calculate critical exponents and central charge. A finite-size-scaling analysis of extensive calculations on small system sizes supports the prediction. The same critical behavior has recently been found for the two-dimensional classical version. Together, the results strongly support an XY-Ising-like critical behavior for these systems.
A one-dimensional quantum version of the frustrated XY (planar rotor) model is considered which can be physically realized as a ladder of Josephson junctions at half a flux quantum per plaquette. This system undergoes a superconductor to insulator transition at zero temperature as a function of charging energy. The critical behavior is studied using a Monte Carlo transfer matrix applied to the path-integral representation of the model and a finite-size-scaling analysis of data on small system sizes. Depending on the ratio between the interchain and intrachain couplings the system can have single or double transitions which is consistent with the prediction that its critical behavior should be described by the two-dimensional classical XY-Ising model.
In this thesis I study the Frustrated XY Model on a square lattice A direct physical realization of such a model can be found in arrays of Josephson junctions In Chapter I examine the Frustrated XY Model on a ladder via a transfer matrix formalism The eigenvalues of the transfer matrix are found numerically giving a determination of the di erent phases of the ladder The spatial periodicity of the ground state exhibits a devil s staircase as a function of the magnetic ux ning factor f If the transverse Josephson coupling is varied a continuous superconducting normal transition in the transverse direction is observed analogous to the breakdown of the Kolomogorov Arnold Moser trajectories in dynamical systems The remainder of the thesis examines the Frustrated XY Model in two dimensions Chapter examines the ground states and low energy excitations of the model In Chapter I compare the critical properties of the model with lling factors f and To obtain a comparison with recent experiments I investigate the e ect of weak quenched bond disorder for f A nite size scaling analysis of extensive Monte Carlo simulations strongly suggests that the critical exponents of the phase transition for f and for f with disorder are those of the pure D Ising model Studying the possible domain walls in the system provides some explanations for these results In order to examine the properties of the model at an incommensurate eld I
We study the phase diagram and critical behaviour of the two-dimensional square-lattice fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau–Ginzburg–Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising XY model. We present a finite-size-scaling analysis of the results of high-precision Monte Carlo simulations on L × L square lattices, up to L = O (10 3 ). In the FFXY model and in the other models, when the transitions are continuous, there are two very close but separate transitions. There is an Ising chiral transition characterized by the onset of chiral long-range order while spins remain paramagnetic. Then, as temperature decreases, the systems undergo a Kosterlitz–Thouless spin transition to a phase with quasi-long-range order. The FFXY model and the other models, in a rather large parameter region, show a crossover behaviour at the chiral and spin transitions that is universal to some extent. We conjecture that this universal behaviour is due to a multicritical point. The numerical data suggest that the relevant multicritical point is a zero-temperature transition. A possible candidate is the O(4) point that controls the low-temperature behaviour of the 4-vector model.
We use extensive Monte Carlo transfer-matrix calculations on infinite strips of widths L up to 30 lattice spacing and a finite-size scaling analysis to obtain critical exponents and conformal anomaly number c for the two-dimensional XY Ising model. This model is expected to describe the critical behavior of a class of systems with simultaneous U(1) and Z2{\mathit{Z}}_{2}Z2​ symmetries of which the fully frustrated XY model is a special case. The effective values obtained for c show a significant decrease with L at different points along the line where the transition to the ordered phase takes place in a single transition. Extrapolations based on power-law corrections give values consistent with c=3/2 although larger values cannot be ruled out. Critical exponents are obtained more accurately and are consistent with previous Monte Carlo simulations suggesting critical behavior and with recent calculations for the frustrated XY model.