Bilayer XY Model and Critical Phenomena

• Bilayer XY model is a coupled two-layer U(1) system characterized by the competition between in-plane phase stiffness and interlayer locking, which modifies BKT physics and quantum criticality.
• The model appears in various forms—including classical rotors, quantum spins, and hard-core boson mappings—each providing insights into distinct critical behaviors and phase transitions.
• Advanced numerical techniques, such as Monte Carlo simulations and worm algorithm methods, are used to analyze critical exponents, entanglement scaling, and dimensional crossovers in these systems.

The bilayer XY model denotes a family of coupled two-component phase or spin systems defined on two layers and linked by interlayer interactions. In the literature, the term covers both classical bilayer rotor models and quantum spin or electronic bilayers whose low-energy sector has a global $U(1)$ order parameter. Its defining feature is the competition between in-plane phase stiffness and interlayer locking, which reorganizes the spectrum into in-phase and relative sectors and thereby modifies Berezinskii–Kosterlitz–Thouless (BKT) physics, quantum criticality, and entanglement scaling. Depending on the microscopic realization and on the form of the interlayer coupling, the resulting critical behavior can be conventional $O(2)$, composite or paired, $U(1)\times \mathbb{Z}_2$, or fractionalized $XY^\ast$ (Helmes et al., 2014, Masini et al., 2024, Zhang et al., 2023).

1. Definitions and canonical Hamiltonians

A standard classical bilayer XY model consists of two coupled square-lattice layers of planar spins $\theta_{i,m}$, with intralayer coupling $J$ and interlayer coupling $K$, governed by

$$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$

For the bilayer, $M=2$, and it is convenient to denote $\theta_{i,1}\equiv \theta_i$ and $O(2)$0. In the formulation analyzed with the worm algorithm, all couplings are ferromagnetic, $O(2)$1 and $O(2)$2, intralayer bonds lie along $O(2)$3 and $O(2)$4, and for $O(2)$5 periodic boundary conditions are applied only within the layers (Masini et al., 2024).

A standard quantum realization is the spin-$O(2)$6 square-lattice bilayer XY model,

$O(2)$7

with control parameter

$O(2)$8

Here $O(2)$9 is the intralayer exchange and $U(1)\times \mathbb{Z}_2$0 the rung exchange. At small $U(1)\times \mathbb{Z}_2$1, the system has long-range transverse antiferromagnetic order; at large $U(1)\times \mathbb{Z}_2$2, rung-singlet or dimer-like quantum disorder destroys that order (Helmes et al., 2014).

The same quantum model admits a hard-core boson mapping through

$U(1)\times \mathbb{Z}_2$3

so that the bilayer maps to hard-core bosons at half filling with intralayer hopping $U(1)\times \mathbb{Z}_2$4 and interlayer rung hopping $U(1)\times \mathbb{Z}_2$5 (Helmes et al., 2014). This equivalence is central because it makes explicit that the bilayer XY model is not restricted to spin language: it is equally a model of coupled bosonic phases.

2. Quantum critical bilayer XY systems

For the spin-$U(1)\times \mathbb{Z}_2$6 bilayer, the quantum critical point is described by the $U(1)\times \mathbb{Z}_2$7 $U(1)\times \mathbb{Z}_2$8 universality class with dynamical exponent $U(1)\times \mathbb{Z}_2$9, so the $XY^\ast$0-dimensional quantum problem maps to a three-dimensional classical XY theory. Its continuum effective action is the two-component $XY^\ast$1 theory

$XY^\ast$2

with $XY^\ast$3, and the finite-size analysis uses the established $XY^\ast$4 critical exponents

$XY^\ast$5

The transverse staggered structure factor

$XY^\ast$6

obeys the scaling form

$XY^\ast$7

or equivalently

$XY^\ast$8

and quantum Monte Carlo gives

$XY^\ast$9

for the square-lattice bilayer (Helmes et al., 2014).

A closely related anisotropic bilayer spin model interpolates between XY, Heisenberg, and Ising criticality through

$\theta_{i,m}$0

with $\theta_{i,m}$1 and $\theta_{i,m}$2. For $\theta_{i,m}$3, the transition is in the XY universality class, and the paper identifies the underlying microscopic mechanism as the closing of the $\theta_{i,m}$4 triplon pair, so that two soft modes become gapless simultaneously (Devakul et al., 2014). In this sense, the bilayer XY model is a particularly clean realization of an $\theta_{i,m}$5 quantum critical point with a direct microscopic rung-singlet interpretation.

3. Finite-temperature bilayers, helicity modulus, and dimensional crossover

At finite temperature, the classical bilayer remains effectively two-dimensional, so the relevant infrared structure is BKT-like rather than genuinely three-dimensional. In the worm-algorithm formulation, the helicity modulus is computed in the current representation from winding-number fluctuations. For a twist applied to one layer only,

$\theta_{i,m}$6

whereas for an in-phase twist applied to both layers,

$\theta_{i,m}$7

The Nelson–Kosterlitz jump criterion,

$\theta_{i,m}$8

is then used to estimate the paired in-phase transition temperature $\theta_{i,m}$9. For $J$0, finite-size extrapolation yields

$J$1

and the same study reports an intermediate “BKT-paired phase” between the low-temperature algebraic phase and the high-temperature disordered phase (Masini et al., 2024).

A broader layered perspective clarifies what interlayer locking can and cannot do. For the anisotropic stacked XY model with $J$2, the infinite-stack system has a single $J$3 XY critical line for any $J$4, but the bilayer does not. The explicit statement made for $J$5 layers is that there is no true $J$6 long-range order at finite temperature for finite $J$7; instead, a finite $J$8 locks the relative phase and enhances the effective in-plane stiffness of the symmetric mode, so that the bilayer supports a BKT transition of a center-of-mass phase at an elevated temperature relative to a single layer (Kracht et al., 19 Mar 2026). The same analysis introduces a Josephson crossover length

$J$9

which quantifies the scale at which phase locking sets in. For $K$0, observables retain quasi-$K$1 signatures; for $K$2, the layers are locked, but the bilayer remains two-dimensional in universality (Kracht et al., 19 Mar 2026). This suggests that some apparent $K$3-like features in finite samples are crossover effects rather than true symmetry-breaking criticality.

4. Entanglement structure and universal corner terms

At the quantum critical point, the second Rényi entropy

$K$4

obeys an area law supplemented by a universal logarithmic correction from corners. For a subregion with boundary length $K$5,

$K$6

Quantum Monte Carlo on the bilayer XY model finds that strip subregions with smooth boundary exhibit area-law scaling with

$K$7

while square subregions reveal a right-angle corner coefficient

$K$8

The corresponding necklace-lattice values, $K$9 and $$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$0, are very close, supporting universality of the corner coefficient within the $$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$1 class (Helmes et al., 2014).

Series expansions in the easy-plane regime of the bilayer Heisenberg-Ising model sharpen this interpretation. For the XY critical line, the right-angle corner coefficients were reported as

$$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$2

at $$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$3, and

$$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$4

at $$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$5, with agreement within errors along the entire XY line (Devakul et al., 2014). The same work emphasizes a rough proportionality between $$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$6 and the number of simultaneously soft modes at the quantum critical point, comparing Ising ($$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$7), XY ($$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$8), and Heisenberg ($$H = -\sum_{m=1}^{M} \left[ J \sum_{\langle i j\rangle} \cos\!\left(\theta_{i,m}-\theta_{j,m}\right) + K \sum_{i=1}^{N} \cos\!\left(\theta_{i,m}-\theta_{i,m+1}\right) \right].$$9) cases. A plausible implication is that corner entanglement coefficients function as an approximate measure of low-energy degrees of freedom, although the papers frame this as a numerical trend rather than as a theorem.

Methodologically, the bilayer XY entanglement results were obtained with stochastic series expansion quantum Monte Carlo, directed-loop updates, the replica trick in an extended ensemble, and the increment trick for growing subregions efficiently (Helmes et al., 2014). This combination made it possible to separate the non-universal area-law prefactor from the additive universal corner term on finite tori.

5. Paired phases, higher-order couplings, and current controversy

A central contemporary controversy concerns whether the standard bilayer with a simple ferromagnetic interlayer coupling supports an intermediate paired BKT phase. One helicity-modulus study reports a three-region phase diagram consisting of a low-temperature algebraic phase, an intermediate “BKT-paired phase” with algebraic interlayer four-point correlations but exponentially decaying single-layer two-point correlations, and a high-temperature disordered phase (Masini et al., 2024). A later Monte Carlo and analytical reexamination reaches the opposite conclusion: for the two-body interlayer Hamiltonian

$M=2$0

the model does not exhibit a BKT paired phase. Instead, it finds only a superfluid and a disordered normal phase, with coincident single-layer and paired transitions; for $M=2$1, it reports $M=2$2 and explicitly states that there is no singularity at the previously claimed $M=2$3 (Xiao et al., 2 Apr 2025).

The same study proposes that a genuine paired BKT phase requires a different interlayer structure, namely the four-body coupling

$M=2$4

This term stiffens the sum field without locking the relative phase site by site. In that model, the reported phase diagram contains a disordered phase, a paired BKT phase (PSF), and an $M=2$5 phase in which single-layer and paired sectors all have quasi-long-range order. Along the disorder–PSF boundary, the anomalous magnetic dimension $M=2$6 varies continuously, with representative values $M=2$7 at $M=2$8, $M=2$9 at $\theta_{i,1}\equiv \theta_i$0, and $\theta_{i,1}\equiv \theta_i$1 at the paired-XY limit $\theta_{i,1}\equiv \theta_i$2 (Xiao et al., 2 Apr 2025). This continuous variation lies outside the simplest single-layer BKT expectation.

A distinct but related variant replaces ordinary interlayer Josephson locking with second-order Josephson coupling,

$\theta_{i,1}\equiv \theta_i$3

which preserves a residual $\theta_{i,1}\equiv \theta_i$4 symmetry and gives a global $\theta_{i,1}\equiv \theta_i$5 structure. Its dual description is a compact $\theta_{i,1}\equiv \theta_i$6 gauge theory in two dimensions, with the second-order Josephson term generating a confining gauge sector. In that framework, isolated vortex-like dual charges are linearly confined, and the first instability out of the low-temperature phase is argued to be an Ising transition driven by condensation of $\theta_{i,1}\equiv \theta_i$7 domain-wall loops rather than a KT transition of point defects (How et al., 25 Jul 2025). This model therefore marks a sharp boundary between conventional bilayer XY physics and gauge-constrained defect dynamics.

6. Electronic and fractionalized bilayer realizations

The bilayer XY model also appears as an emergent description in electronic bilayers. In zero magnetic field, a bilayer of two-dimensional electron gases can spontaneously break the $\theta_{i,1}\equiv \theta_i$8 layer symmetry and form an interlayer-coherent pseudospin XY ferromagnet. With layer spinor $\theta_{i,1}\equiv \theta_i$9 and pseudospin operators $O(2)$00, the coherence order parameter is

$O(2)$01

or, in uniform form,

$O(2)$02

At zero tunneling, the low-energy phase mode is described by

$O(2)$03

with BKT scale

$O(2)$04

In the balanced case at $O(2)$05, the onset of the XY ferromagnet occurs at $O(2)$06, corresponding in GaAs to a critical density of order $O(2)$07 (Cookmeyer et al., 2023). The bilayer XY model here is therefore not a phenomenological analogy but a phase-only effective theory of spontaneously coherent electron bilayers.

In quantum Hall bilayers at fillings $O(2)$08, the condensed phase is again XY-like at long wavelengths, with interlayer phase

$O(2)$09

and continuum action

$O(2)$10

However, the transition out of this exciton-condensed phase is not ordinary XY: it is controlled by the $O(2)$11 universality class. The critical field $O(2)$12 carries fractional relative-layer charge $O(2)$13, and the physical exciton-condensate order parameter is composite,

$O(2)$14

The thermodynamic exponents remain those of $O(2)$15 XY, with $O(2)$16 and $O(2)$17, but the physical order parameter has a large anomalous dimension $O(2)$18; the estimated critical separation is $O(2)$19 (Zhang et al., 2023). The same fractionalization produces a universal counterflow conductivity with a $O(2)$20 prefactor and an extraordinary-log boundary criticality at the edge. Relative to the ordinary bilayer XY model, this is a case where the long-wavelength ordered phase is XY-like while the critical degrees of freedom are fractional and gauge-structured.

Taken together, these realizations show that the bilayer XY model is best understood as a structural class rather than a single Hamiltonian. Its unifying ingredients are two coupled $O(2)$21 layers, a competition between intralayer stiffness and interlayer locking, and a low-energy decomposition into symmetric and relative sectors. What differs from one realization to another is the operator content of the critical theory: in the standard quantum spin bilayer it is the ordinary $O(2)$22 order parameter; in paired models it is a composite sum field; in the second-order Josephson case it is intertwined with $O(2)$23 domain walls and gauge confinement; and in quantum Hall bilayers it is a composite of a fractional critical boson.