Defect magnetic critical behaviour of the two-dimensional Ashkin–Teller model with an asymmetric line defect

When a line defect (modified bonds along a single row of one Ising sublattice) is introduced into the critical two-dimensional Ashkin–Teller model, the local magnetization exponents of the two spin species are not agreed upon. A field-theoretical calculation predicts that the defect exponent of the directly perturbed σ spins equals the exactly known defective-Ising value — a function of the defect strength only, independent of the four-spin coupling ε — while the untouched τ spins keep the universal Ising value 1/8. Density-matrix renormalization group and Monte Carlo computations instead find that both exponents vary with ε and the defect strength, with enhanced defects flowing to an extraordinary-interface fixed point (a defect line that stays locally ordered at criticality) and complementary local ordering for ε<0. A 2022 Monte Carlo study by the original field-theory group attributes the discrepancy to an inappropriate gauge-invariant regularization and claims reconciliation through a more general regularization with ε-dependent counterterms, but this derivation has not been independently verified and no accepted exponent formula or defect renormalization-group phase diagram exists. The problem is to determine the defect critical behaviour of this model and thereby settle the field-theory-versus-numerics contradiction.

Background

The Ashkin–Teller (AT) model (Ashkin and Teller, 1943) is a four-state spin model on the square lattice that can be written as two superposed Ising models, with spin variables $\sigma_i=\pm1$ and $\tau_i=\pm1$ on every site, coupled by a four-spin interaction $K_4\,\sigma_i\sigma_j\tau_i\tau_j$ between nearest-neighbour pairs in addition to the usual Ising couplings $K$; its ratio $\epsilon=K_4/K$ measures the coupling strength between the two Ising copies. The model is equivalent, by duality, to the eight-vertex model solved by Baxter, and is the canonical example of nonuniversal critical behaviour in two dimensions: along its critical line separating the ferromagnetic and paramagnetic phases (given, in the classical parametrization, by $e^{-2K_4}=\sinh 2K$; the ferromagnetic segment runs from the four-state Potts point at $\epsilon=1$ through the decoupled-Ising point $\epsilon=0$ down to $\epsilon=-1/\sqrt2$, below which the self-dual line enters a critical fan, a finite critical region rather than a single transition), most critical exponents — e.g. those of the energy-density correlations, with anomalous dimension $x_e=\pi/[2\arccos(-\epsilon)]$ — vary continuously with $\epsilon$, whereas the bulk magnetization scaling dimension of both spin species stays fixed at the Ising value $x_m=1/8$.

A line defect is obtained by altering the bonds along a single line of the lattice. In the exactly solved two-dimensional Ising model with such a defect, the magnetization exponent on the defect line is a continuous function of the defect strength (computed by Bariev and by McCoy and Perk, 1980), while the energy-density exponent there is unchanged — the textbook example of local nonuniversality, reviewed by Iglói, Peschel and Turban (1993). Line defects come in two dual variants (chain and ladder defects, related by duality with the corresponding strengths linked by $\tilde{h}\leftrightarrow 1/J$), and in the AT model one may make the defect asymmetric by modifying only the bonds of the σ Ising copy and leaving the τ copy untouched. The defect strength is parametrized as a ladder coupling $J$ ($J=1$ for the pure system) or a chain-defect transverse field $\tilde{h}$ in the DMRG work, as an added line coupling $K_l$ (effective line coupling $K+K_l$) in the Monte Carlo work, and as a continuum strength $\mu$ in the field theory.

For this asymmetric defect, Naón (2009) performed a field-theoretical calculation: in the continuum limit the AT model becomes two Majorana fermion species coupled through their energy densities, and the squared spin–spin correlator on the defect line factorizes into the defective-Ising (Bariev) correlator times a factor that becomes independent of the distance in the thermodynamic limit. The prediction is that the σ-spin defect exponent equals the Bariev value $\Delta_\sigma=\frac{1}{8}(1+4\mu)^2$ (where $\mu$, $|\mu|<1/4$, is the continuum defect strength: $\mu=-1/4$ corresponds to a perfectly ordered line, $\mu=1/4$ to a cut line) — independent of $\epsilon$ — while the τ-spin defect exponent keeps the universal value $\Delta_\tau=1/8$; the four-spin coupling is declared an irrelevant perturbation at the defect, an 'explicit confirmation of universality'.

Numerics contradict this. Lajkó and Iglói (2011) studied the Hamiltonian (quantum-chain) version of the model by the density matrix renormalization group (DMRG, a variational method for one-dimensional quantum systems giving near-exact ground states of long chains), extracting the defect magnetization exponents $x^\sigma_m$, $x^\tau_m$ by finite-size scaling of the local magnetization $m_0(L)\sim L^{-x_m}$. For $\epsilon>0$ they found the two exponents to be nearly equal and to depend on both $\epsilon$ and the defect strength, contradicting the field-theory prediction in both respects. They interpreted their data in terms of interface critical phenomena: for enhanced defect couplings the exponents extrapolate to zero, i.e. the defect is locally ordered at criticality — behaviour of an extraordinary-interface (EI) fixed point, the analogue of an ordered free surface — while for reduced couplings the flow is to an ordinary-interface (OI) fixed point with exponents close to those of a free surface. For negative four-spin coupling $\epsilon<0$ (in the Hamiltonian formulation, a regime whose classical counterpart carries negative Boltzmann weights) the behaviour is complementary: for an enhanced defect the perturbed σ subsystem becomes locally ordered while the τ subsystem stays locally disordered, with the roles interchanged for a reduced defect. They proposed a renormalization-group phase diagram with these OI/EI flows, explicitly opposing the field-theoretical picture of an irrelevant $\epsilon$. Monte Carlo simulations of the classical model (Duchowney, Naón and Iucci, preprint 2016; published 2022 in Physica A) confirmed the DMRG findings. The same 2016/2022 authors — the group that produced the original field-theory result — then diagnosed the contradiction: the path-integral computation of Naón (2009) evaluates a fermionic determinant with a gauge-invariant regularization, which forces all $\epsilon$-dependent contributions out of the Jacobian, although the four-spin coupling itself breaks the underlying gauge symmetry. They argue that a more general regularization, including $\epsilon$-dependent counterterms, restores the $\epsilon$-dependence of the defect exponents and reconciles the analytical and numerical results. As of 2026 this reconciliation exists only as this group's proposal: it has not been independently checked, the DMRG side has not published a response, and no closed-form defect-exponent formula or accepted defect phase diagram is available, so the magnetic critical behaviour of the asymmetrically defect AT model remains unsettled.

Problem Statement

Determine the magnetic critical behaviour at an asymmetric line defect (a single line of modified σ-spin bonds, τ bonds untouched) in the two-dimensional Ashkin–Teller model on its critical line. Concretely, all three of the following parts are jointly required: (i) the defect magnetization scaling dimensions $x^\sigma_m$ and $x^\tau_m$ of the two spin species as functions of the four-spin coupling ε and the defect strength, for both signs of ε — in particular whether they coincide and how they depend on ε, versus the field-theoretical prediction $\Delta_\sigma=\frac{1}{8}(1+4\mu)^2$ (with $\mu$ the continuum defect strength; independent of $\epsilon$) and $\Delta_\tau=1/8$; (ii) the renormalization-group fixed-point structure governing the defect — whether the ε-coupling is irrelevant there, or enhanced/reduced defects flow to extraordinary/ordinary interface fixed points with vanishing or surface-type exponents, including the complementary locally-ordered/locally-disordered flow claimed for ε<0; and (iii) an independently checkable derivation that establishes these exponents and the phase diagram and thereby resolves the standing contradiction between the field-theoretical (Naón, 2009) and the DMRG/Monte-Carlo (Lajkó–Iglói, 2011; Duchowney–Naón–Iucci, 2016/2022) determinations, whose proposed 2022 regularization-based reconciliation has not been independently verified.

References

1. J. Ashkin and E. Teller, 'Statistics of Two-Dimensional Lattices with Four Components', Physical Review 64, 178 (1943), doi:10.1103/PhysRev.64.178, https://doi.org/10.1103/PhysRev.64.178
2. R. J. Baxter, 'Eight-Vertex Model in Lattice Statistics', Physical Review Letters 26, 832 (1971), doi:10.1103/PhysRevLett.26.832, https://doi.org/10.1103/PhysRevLett.26.832
3. R. Z. Bariev, Zh. Eksp. Teor. Fiz. 77, 1217 (1979) [Sov. Phys. JETP 50, 613 (1979)], http://www.jetp.ac.ru/cgi-bin/e/index/e/50/3/p613
4. B. M. McCoy and J. H. H. Perk, 'Two-Spin Correlation Functions of an Ising Model with Continuous Exponents', Physical Review Letters 44, 840 (1980), doi:10.1103/PhysRevLett.44.840, https://doi.org/10.1103/PhysRevLett.44.840
5. F. Iglói, I. Peschel and L. Turban, 'Inhomogeneous systems with unusual critical behaviour', Advances in Physics 42, 683 (1993), doi:10.1080/00018739300101544, https://doi.org/10.1080/00018739300101544
6. Péter Lajkó, Loïc Turban and Ferenc Iglói, 'Scaling properties at the interface between different critical subsystems: The Ashkin-Teller model', Physical Review B 76, 224423 (2007), arXiv:0709.1322, doi:10.1103/PhysRevB.76.224423, https://doi.org/10.1103/PhysRevB.76.224423
7. Carlos Naón, 'Critical behavior of the spin correlation function in the Ashkin-Teller and Baxter models with a line defect', Physical Review E 79, 051112 (2009), arXiv:0812.4263, doi:10.1103/PhysRevE.79.051112, https://arxiv.org/abs/0812.4263
8. Carlos Naón and Marta Trobo, 'The spin correlation function in 2D statistical mechanics models with inhomogeneous line defects', Journal of Statistical Mechanics (2011) P02021, arXiv:1012.5139, doi:10.1088/1742-5468/2011/02/P02021, https://arxiv.org/abs/1012.5139
9. Péter Lajkó and Ferenc Iglói, 'Numerical study of the critical behavior of the Ashkin–Teller model at a line defect', Journal of Statistical Mechanics (2011) P05025, arXiv:1105.1687, doi:10.1088/1742-5468/2011/05/P05025, https://arxiv.org/abs/1105.1687
10. G. Duchowney, C. Naón and A. Iucci, 'Critical behavior of the Ashkin-Teller model with a line defect: a Montecarlo study', arXiv:1612.08876 (2016), https://arxiv.org/abs/1612.08876
11. G. Duchowney, C. Naón and A. Iucci, 'Critical behavior of the Ashkin–Teller model with a line defect: Towards reconciliation between numerical and analytical results', Physica A: Statistical Mechanics and its Applications 597, 127287 (2022), doi:10.1016/j.physa.2022.127287, https://doi.org/10.1016/j.physa.2022.127287