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Physical review. B, Condensed matter

Critical behavior of a one-dimensional frustrated quantumXYmodel

Enzo Granato

Published Feb 1, 1992• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.45.2557

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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.
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 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.
We discuss the critical behavior of several models, all related to the 19-vertex model. These models combine Ising and Gaussian degrees of freedom and have central charge c=3/2 at the critical point. We present arguments why the frustrated XY model, which also has Gaussian and Ising degrees of freedom, might fall into a different universality class. We use the Monte Carlo transfer-matrix method of Bl\"ote and Nightingale to obtain accurate numerical results for the central charge and exponents of the frustrated XY model. In particular, we test the prediction for the central charge c to be equal to 3/2 also in this model. We find c=1.66\ifmmode\pm\else\textpm\fi{}0.04 for the central charge of the frustrated XY model at the critical point. It is not clear at this stage to which universality class this critical behavior belongs.
We study the critical behavior of the fully frustrated XY\mathrm{XY}XY model on a square lattice by means of extensive histogram Monte Carlo simulations. We find a single transition in contrast to the scenario of two distinct transitions. From a finite-size scaling we determine the critical exponents associated with the Ising-like order parameter and find ${T}_{c}(\ensuremath{\infty})=0.4552(2)$, $\ensuremath{\nu}=0.852(2),$ and $\ensuremath{\gamma}=1.531(3)$. These exponents, different from the pure two-dimensional Ising critical exponents, are in agreement with the ones obtained for a mixed XY\mathrm{XY}XY Ising system.
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.