Replica Boundary Condition Applied to the ±J Ising Model in Two Dimensions

Real critical systems, such as uniaxial ferromagnets in the 3d Ising universality class, are constrained by boundaries and subject to random couplings. We consider the Wilson-Fisher fixed point in $4-ε$ dimensions subject to a random magnetic field localized on a two-dimensional surface, which becomes co-dimension 1 in the physical $ε\to1$ limit. Using the replica method for the disordered field, we find that the ordinary boundary condition is stable under disorder but also discover a non-trivial ``dirty" boundary condition which can be reached by tuning the disorder strength or the local temperature. We also investigate the logarithmic structure of the defect spectrum and how it emerges via the replica formalism.Real critical systems, such as uniaxial ferromagnets in the 3d Ising universality class, are constrained by boundaries and subject to random couplings. We consider the Wilson-Fisher fixed point in $4-ε$ dimensions subject to a random magnetic field localized on a two-dimensional surface, which becomes co-dimension 1 in the physical $ε\to1$ limit. Using the replica method for the disordered field, we find that the ordinary boundary condition is stable under disorder but also discover a non-trivial ``dirty" boundary condition which can be reached by tuning the disorder strength or the local temperature. We also investigate the logarithmic structure of the defect spectrum and how it emerges via the replica formalism.

arXiv:2605.22628May 21, 20260 citations

Ground State Properties of the ±J Ising Model in Two Dimensions

之康尾関, Yukiyasu Ozeki

Jan 1, 19900 citations

Phase diagram of the ±J Ising model in two dimensions

之康尾関, Yukiyasu Ozeki, 秀稔西森 +1 more

Sep 1, 19870 citations

Tensor renormalization-group study of the surface critical behavior of a frustrated two-layer Ising model

Christophe Chatelain

Abstract Two replicas of a 2D Ising model are coupled by frustrated spin-spin interactions. It is known that this inter-layer coupling is marginal and that the bulk critical behavior belongs to the Ashkin–Teller (AT) universality class, as the J 1 − J 2 Ising model. In this work, the surface critical behavior is studied numerically by tensor renormalization-group calculations. The bond-weight tensor renormalization group algorithm is extended to tackle systems with boundaries. It is observed that the two-fold degeneracy of the surface magnetic scaling dimension of the AT model is lifted in the frustrated two-layer Ising model (F2LIM). The splitting is explained by the breaking of the Z 2 -symmetry under spin reversal of a single Ising replica in the F2LIM. The two distinct surface magnetic scaling dimensions x 1 s and x 2 s of the F2LIM satisfies a simple duality relation x 1 s = 1 / 4 x 2 s .Abstract Two replicas of a 2D Ising model are coupled by frustrated spin-spin interactions. It is known that this inter-layer coupling is marginal and that the bulk critical behavior belongs to the Ashkin–Teller (AT) universality class, as the J 1 − J 2 Ising model. In this work, the surface critical behavior is studied numerically by tensor renormalization-group calculations. The bond-weight tensor renormalization group algorithm is extended to tackle systems with boundaries. It is observed that the two-fold degeneracy of the surface magnetic scaling dimension of the AT model is lifted in the frustrated two-layer Ising model (F2LIM). The splitting is explained by the breaking of the Z 2 -symmetry under spin reversal of a single Ising replica in the F2LIM. The two distinct surface magnetic scaling dimensions x 1 s and x 2 s of the F2LIM satisfies a simple duality relation x 1 s = 1 / 4 x 2 s .

arXiv:2510.21269Journal of Statistical Mechanics Theory and ExperimentFeb 1, 20260 citations

Exploring critical systems under measurements and decoherence via Keldysh field theory

Ruochen Ma

We employ an $n$ -replica Keldysh field theory to investigate the effects of measurements and decoherence on long distance behaviors of quantum critical states. We classify different measurements and decoherence based on their timescales and symmetry properties, and demonstrate that they can be described by $n$ -replica Keldysh field theories with distinct physical and replica symmetries. Low energy effective theories for various scenarios are then derived using the symmetry and fundamental consistency conditions of the Keldysh formalism. We apply this framework to study the critical Ising model in both one and two spatial dimensions. In one dimension, we demonstrate that (1) measurements over a finite period of time along the transverse spin direction do not modify the asymptotic scaling of correlation functions and entanglement entropy, whereas (2) measurements along the longitudinal spin direction lead to an area law entangled phase. We also show that (3) decoherence noises over a finite time can be mapped to specific boundary conditions of a critical Ashkin-Teller model, and the entanglement characteristics of the resulting mixed state can be determined. For measurements and decoherence over an extensive time, we demonstrate that (4) the von Neumann entanglement entropy of a large subsystem can exhibit a (sub-)dominant logarithmic scaling in the stationary state for weak measurement (decoherence) performed in a basis that is symmetric under the Ising symmetry, but (5) reduces to an area law for measurements and decoherence in the longitudinal direction. Our results demonstrate that the Keldysh formalism is a useful tool for systematically studying the effects of measurements and decoherence on long-wavelength physics.We employ an $n$-replica Keldysh field theory to investigate the effects of measurements and decoherence on long distance behaviors of quantum critical states. We classify different measurements and decoherence based on their timescales and symmetry properties, and demonstrate that they can be described by $n$-replica Keldysh field theories with distinct physical and replica symmetries. Low energy effective theories for various scenarios are then derived using the symmetry and fundamental consistency conditions of the Keldysh formalism. We apply this framework to study the critical Ising model in both one and two spatial dimensions. In one dimension, we demonstrate that (1) measurements over a finite period of time along the transverse spin direction do not modify the asymptotic scaling of correlation functions and entanglement entropy, whereas (2) measurements along the longitudinal spin direction lead to an area law entangled phase. We also show that (3) decoherence noises over a finite time can be mapped to specific boundary conditions of a critical Ashkin-Teller model, and the entanglement characteristics of the resulting mixed state can be determined. For measurements and decoherence over an extensive time, we demonstrate that (4) the von Neumann entanglement entropy of a large subsystem can exhibit a (sub-)dominant logarithmic scaling in the stationary state for weak measurement (decoherence) performed in a basis that is symmetric under the Ising symmetry, but (5) reduces to an area law for measurements and decoherence in the longitudinal direction. Our results demonstrate that the Keldysh formalism is a useful tool for systematically studying the effects of measurements and decoherence on long-wavelength physics.

arXiv:2304.08277Apr 17, 20231 citation

Real critical systems, such as uniaxial ferromagnets in the 3d Ising universality class, are constrained by boundaries and subject to random couplings. We consider the Wilson-Fisher fixed point in

4-ε

dimensions subject to a random magnetic field localized on a two-dimensional surface, which becomes co-dimension 1 in the physical

ε\to1

limit. Using the replica method for the disordered field, we find that the ordinary boundary condition is stable under disorder but also discover a non-trivial ``dirty" boundary condition which can be reached by tuning the disorder strength or the local temperature. We also investigate the logarithmic structure of the defect spectrum and how it emerges via the replica formalism.

We employ an

n

-replica Keldysh field theory to investigate the effects of measurements and decoherence on long distance behaviors of quantum critical states. We classify different measurements and decoherence based on their timescales and symmetry properties, and demonstrate that they can be described by

n

-replica Keldysh field theories with distinct physical and replica symmetries. Low energy effective theories for various scenarios are then derived using the symmetry and fundamental consistency conditions of the Keldysh formalism. We apply this framework to study the critical Ising model in both one and two spatial dimensions. In one dimension, we demonstrate that (1) measurements over a finite period of time along the transverse spin direction do not modify the asymptotic scaling of correlation functions and entanglement entropy, whereas (2) measurements along the longitudinal spin direction lead to an area law entangled phase. We also show that (3) decoherence noises over a finite time can be mapped to specific boundary conditions of a critical Ashkin-Teller model, and the entanglement characteristics of the resulting mixed state can be determined. For measurements and decoherence over an extensive time, we demonstrate that (4) the von Neumann entanglement entropy of a large subsystem can exhibit a (sub-)dominant logarithmic scaling in the stationary state for weak measurement (decoherence) performed in a basis that is symmetric under the Ising symmetry, but (5) reduces to an area law for measurements and decoherence in the longitudinal direction. Our results demonstrate that the Keldysh formalism is a useful tool for systematically studying the effects of measurements and decoherence on long-wavelength physics.