Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1902.09901 · Journal of Physics A Mathematical and Theoretical

Critical points of coupled vector-Ising systems. Exact results

Gesualdo Delfino A, Noel Lamsen

Published Jul 9, 2019• 15 citations
arXivHTMLPDF

Abstract

Abstract We show that scale-invariant scattering theory allows to exactly determine the critical points of two-dimensional systems with coupled and Ising order parameters. The results are obtained for N continuous and include criticality of the loop gas type. In particular, for N = 1 we exhibit three critical lines intersecting at the Berezinskii–Kosterlitz–Thouless transition point of the Gaussian model and related to the Z 4 symmetry of the isotropic Ashkin–Teller model. For N = 2 we classify the critical points that can arise in the XY-Ising model and provide exact answers about the critical exponents of the fully frustrated XY model.

DOI
10.1088/1751-8121/ab3055

Related papers

Criticality in the disordered N -color Ashkin–Teller model

Youssef Makoudi, Gesualdo Delfino A

Abstract The N -color Ashkin–Teller model corresponds to N Ising models coupled by four-spin interactions. We consider the two-dimensional case in presence of quenched disorder and use scale invariant scattering theory to determine all the solutions of the exact renormalization group fixed points equations. The weak disorder sector is characterized by a solution that, for any fixed N larger than 1, is a line of fixed points with Ising thermal exponents and continuously varying magnetic exponents. The number of fixed point solutions allowed by the symmetries of the model increases at strong disorder illustrating the growing dependence on the distributions of the two random couplings. The presence of some critical exponents which do not depend on the symmetry parameter N confirms this type of superuniversality as a peculiar feature of random criticality.

arXiv:2605.29810•Journal of Statistical Mechanics Theory and Experiment•Aug 1, 2026•0 citations

Critical behavior of two-dimensional fully frustrated XY systems

Martin Hasenbusch, Andrea Pelissetto, Ettore Vicari

We study the phase diagram of the two-dimensional 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 Monte Carlo simulations on square lattices L × L , L ≲ 10 3 . We show that the low-temperature phase of these models is controlled by the same line of Gaussian fixed points as in the standard XY model. We find that, if a model undergoes a unique transition by varying temperature, then the transition is of first order. In the opposite case we observe two very close transitions: a transition associated with the spin degrees of freedom and, as temperature increases, a transition where chiral modes become critical. If they are continuous, they belong to the Kosterlitz-Thouless and to the Ising universality class, respectively. Ising and Kosterlitz-Thouless behavior is observed only after a preasymptotic regime, which is universal to some extent. In the chiral case, the approach is nonmonotonic for most observables, and there is a wide region in which finite-size scaling is controlled by an effective exponent ν eff ≈ 0.8. This explains the result ν ≈ 0.8 of many previous studies using smaller lattices.

arXiv:cond-mat/0509511•Journal of Physics Conference Series•Jun 1, 2006•4 citations

Nonuniversal critical behavior and first-order transitions in a coupled XY -Ising model

Jooyoung Lee, Enzo Granato, J. Michael Kosterlitz

We study, by a Migdal-Kadanoff approximation and Monte Carlo simulations, the phase diagram of a two-dimensional coupled XY-Ising model. This model can describe phase transitions in different systems with underlying continuous and ${\mathit{Z}}_{2}$ symmetries. Depending on the parameters, we find separate XY, Ising and first-order transitions. Also, a line of continuous transitions is found with simultaneous loss of XY and Ising order and varying critical exponents. The fully frustrated XY and Josephson-junction systems can be considered to lie along different paths in the model which can result in nonuniversal behavior if the transition is a single one.

Physical review. B, Condensed matter•Sep 1, 1991•71 citations

Multicritical behaviour in the fully frustratedXYmodel and related systems

Martin Hasenbusch B, Andrea Pelissetto C, Ettore Vicari

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.

arXiv:cond-mat/0509682•Journal of Statistical Mechanics Theory and Experiment•Dec 5, 2005•89 citations

Critical points in coupled Potts models and correlated percolation

Noel Lamsen, Youness Diouane, Gesualdo Delfino A

Abstract We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a q-state Potts model coupled to an r-state Potts model in two dimensions. For integer values of q and r the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin–Teller case ( q = r = 2 ). Since our results extend to continuous values of the number of states, we can access the limit r → 1 corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot in general be obtained from those of Fortuin–Kasteleyn clusters by analytical continuation.

arXiv:2208.14844•Journal of Statistical Mechanics Theory and Experiment•Jan 1, 2023•5 citations

Conformal anomaly and critical exponents of theXYIsing model

M. P. Nightingale, Enzo Granato, J. Michael Kosterlitz

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 ${\mathit{Z}}_{2}$ 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.

arXiv:cond-mat/9505048•Physical review. B, Condensed matter•Sep 1, 1995•55 citations
Abstract We show that scale-invariant scattering theory allows to exactly determine the critical points of two-dimensional systems with coupled and Ising order parameters. The results are obtained for N continuous and include criticality of the loop gas type. In particular, for N = 1 we exhibit three critical lines intersecting at the Berezinskii–Kosterlitz–Thouless transition point of the Gaussian model and related to the Z 4 symmetry of the isotropic Ashkin–Teller model. For N = 2 we classify the critical points that can arise in the XY-Ising model and provide exact answers about the critical exponents of the fully frustrated XY model.
Abstract The N -color Ashkin–Teller model corresponds to N Ising models coupled by four-spin interactions. We consider the two-dimensional case in presence of quenched disorder and use scale invariant scattering theory to determine all the solutions of the exact renormalization group fixed points equations. The weak disorder sector is characterized by a solution that, for any fixed N larger than 1, is a line of fixed points with Ising thermal exponents and continuously varying magnetic exponents. The number of fixed point solutions allowed by the symmetries of the model increases at strong disorder illustrating the growing dependence on the distributions of the two random couplings. The presence of some critical exponents which do not depend on the symmetry parameter N confirms this type of superuniversality as a peculiar feature of random criticality.
We study the phase diagram of the two-dimensional 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 Monte Carlo simulations on square lattices L × L , L ≲ 10 3 . We show that the low-temperature phase of these models is controlled by the same line of Gaussian fixed points as in the standard XY model. We find that, if a model undergoes a unique transition by varying temperature, then the transition is of first order. In the opposite case we observe two very close transitions: a transition associated with the spin degrees of freedom and, as temperature increases, a transition where chiral modes become critical. If they are continuous, they belong to the Kosterlitz-Thouless and to the Ising universality class, respectively. Ising and Kosterlitz-Thouless behavior is observed only after a preasymptotic regime, which is universal to some extent. In the chiral case, the approach is nonmonotonic for most observables, and there is a wide region in which finite-size scaling is controlled by an effective exponent ν eff ≈ 0.8. This explains the result ν ≈ 0.8 of many previous studies using smaller lattices.
We study, by a Migdal-Kadanoff approximation and Monte Carlo simulations, the phase diagram of a two-dimensional coupled XY-Ising model. This model can describe phase transitions in different systems with underlying continuous and Z2{\mathit{Z}}_{2}Z2​ symmetries. Depending on the parameters, we find separate XY, Ising and first-order transitions. Also, a line of continuous transitions is found with simultaneous loss of XY and Ising order and varying critical exponents. The fully frustrated XY and Josephson-junction systems can be considered to lie along different paths in the model which can result in nonuniversal behavior if the transition is a single one.
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.
Abstract We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a q-state Potts model coupled to an r-state Potts model in two dimensions. For integer values of q and r the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin–Teller case ( q = r = 2 ). Since our results extend to continuous values of the number of states, we can access the limit r → 1 corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot in general be obtained from those of Fortuin–Kasteleyn clusters by analytical continuation.
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.