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arxiv: 2601.18922 · v1 · submitted 2026-01-26 · ❄️ cond-mat.stat-mech · quant-ph

Stacked quantum Ising systems and quantum Ashkin-Teller model

Davide Rossini , Ettore Vicari This is my paper

Pith reviewed 2026-05-16 10:21 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords quantum Ising modelAshkin-Teller modelquantum criticalitycritical exponentssymmetry enlargementstacked systemsZ2 symmetry
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0 comments X

The pith

Two identical stacked quantum Ising subsystems coupled by a symmetry-preserving term become equivalent to the quantum Ashkin-Teller model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies an isolated composite system of two stacked quantum Ising subsystems linked by a local coupling that leaves each subsystem's Z2 symmetry intact. The coupling strength is set by a parameter w, with w=0 recovering two independent Ising chains. When the subsystems are identical and both critical, the full system maps onto the quantum Ashkin-Teller model, which carries an extra Z2 interchange symmetry. This mapping produces a line of critical points in one dimension along which the correlation-length exponent varies continuously with w, and multicritical points in two dimensions where the critical modes enlarge their symmetry from Z2⊕Z2 to O(2).

Core claim

For identical stacked quantum Ising subsystems the global Hamiltonian is equivalent to the quantum Ashkin-Teller model. In one dimension this produces a critical line on which the length-scale exponent nu changes continuously with the intercoupling strength w. In two dimensions the same construction yields quantum multicritical points whose critical modes display an effective O(2) symmetry instead of the microscopic Z2⊕Z2 symmetry.

What carries the argument

The intercoupling parameter w that tunes the strength of the local term coupling the two subsystems while preserving each Z2 symmetry, thereby realizing the Ashkin-Teller equivalence.

If this is right

  • One-dimensional stacked systems exhibit a line of critical points where the exponent nu varies continuously with w.
  • Two-dimensional stacked systems display multicritical points whose critical modes enlarge from Z2⊕Z2 to O(2) symmetry.
  • Ground-state correlations inside one subsystem depend on the state of the second subsystem and on the value of w.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies a concrete route to realize tunable Ashkin-Teller criticality in quantum simulators built from coupled Ising chains.
  • Symmetry enlargement at the two-dimensional multicritical points may connect to other models in which emergent continuous symmetries appear at criticality.
  • Varying w offers a single parameter that interpolates between pure Ising and Ashkin-Teller universality classes.

Load-bearing premise

The coupling term must preserve the individual Z2 symmetries of each subsystem and the ground-state analysis must capture the long-range critical correlations.

What would settle it

Direct numerical extraction of the correlation-length exponent along the critical line of a one-dimensional stacked Ising chain for several values of w, checking whether the exponent changes continuously.

Figures

Figures reproduced from arXiv: 2601.18922 by Davide Rossini, Ettore Vicari.

Figure 1
Figure 1. Figure 1: FIG. 1: Sketch of a system made of two SQI chains, weakly [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The critical value [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The scaling behavior of the ratio [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The scaling behavior of the ratio [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The scaling behavior of the ratio [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Sketch of a phase diagram containing a 3D bicritical [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The symmetric relative difference ∆ [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

We analyze the quantum states of an isolated composite system consisting of two stacked quantum Ising (SQI) subsystems, coupled by a local Hamiltonian term that preserves the $Z_2$ symmetry of each subsystem. The coupling strength is controlled by an intercoupling parameter $w$, with $w=0$ corresponding to decoupled quantum Ising systems. We focus on the quantum correlations of one of the two SQI subsystems, $S$, in the ground state of the global system, and study their dependence on both the state of the weakly-coupled complementary part $E$ and the intercoupling strength. We concentrate on regimes in which $S$ develops critical long-range correlations. The most interesting physical scenario arises when both SQI subsystems are critical. In particular, for identical SQI subsystems, the global system is equivalent to the quantum Ashkin-Teller model, characterized by an additional $Z_2$ interchange symmetry between the two subsystem operators. In this limit, one-dimensional SQI systems exhibit a peculiar critical line along which the length-scale critical exponent $\nu$ varies continuously with $w$, while two-dimensional systems develop quantum multicritical behaviors characterized by an effective enlargement of the symmetry of the critical modes, from the actual $Z_2\oplus Z_2$ symmetry to a continuous O(2) symmetry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript analyzes the ground-state quantum correlations in a composite system of two stacked quantum Ising (SQI) subsystems coupled by a local Z₂-preserving term controlled by intercoupling strength w (with w=0 recovering decoupled Ising systems). For identical subsystems the global Hamiltonian is equivalent to the quantum Ashkin-Teller model via an additional Z₂ interchange symmetry. In one dimension this produces a critical line on which the correlation-length exponent ν varies continuously with w; in two dimensions the critical modes exhibit an effective enlargement from Z₂⊕Z₂ to O(2) symmetry.

Significance. The explicit construction of the Ashkin-Teller Hamiltonian from stacked Ising layers supplies a concrete, tunable realization of the model that may facilitate numerical or experimental studies of subsystem correlations and the known Ashkin-Teller critical line. The ground-state analysis is appropriate for zero-temperature criticality, and the reported behaviors (continuously varying ν in 1D, O(2) enlargement in 2D) are standard once the mapping is established.

major comments (1)
  1. [Hamiltonian definition / Section 2] The central claim that identical SQI subsystems map exactly onto the quantum Ashkin-Teller model rests on rewriting the composite Hamiltonian; the manuscript must display this rewriting (including the explicit four-spin interaction generated by the intercoupling term) in the section that introduces the global Hamiltonian, together with a verification that both individual Z₂ symmetries are preserved.
minor comments (2)
  1. Clarify whether the one-dimensional results refer to chains or to higher-dimensional strips; the abstract uses “one-dimensional SQI systems” without specifying the lattice geometry.
  2. The statement that “ground-state analysis fully captures the long-range critical correlations” should be supported by a brief remark on why excited-state contributions can be neglected for the scaling statements of interest.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address the major comment below.

read point-by-point responses

  1. Referee: [Hamiltonian definition / Section 2] The central claim that identical SQI subsystems map exactly onto the quantum Ashkin-Teller model rests on rewriting the composite Hamiltonian; the manuscript must display this rewriting (including the explicit four-spin interaction generated by the intercoupling term) in the section that introduces the global Hamiltonian, together with a verification that both individual Z₂ symmetries are preserved.

    Authors: We agree that the explicit rewriting should be displayed for clarity. In the revised manuscript we will insert, in the section introducing the global Hamiltonian, the full algebraic rewriting of the composite Hamiltonian for identical subsystems. This will explicitly isolate the four-spin interaction generated by the intercoupling term proportional to w and will include a direct verification that each individual Z₂ symmetry remains intact. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence follows from direct Hamiltonian rewriting

full rationale

The paper derives the equivalence of identical stacked quantum Ising subsystems to the quantum Ashkin-Teller model by rewriting the composite Hamiltonian with the intercoupling term, which introduces the four-spin interaction while preserving individual Z2 symmetries. This is an algebraic identity, not a fit or self-definition. Critical exponents and symmetry enlargement (Z2⊕Z2 to O(2)) are then standard consequences of the resulting Ashkin-Teller universality class. No load-bearing step reduces to a self-citation chain, fitted input renamed as prediction, or ansatz smuggled via prior work; ground-state analysis is justified by the zero-temperature wavefunction encoding the correlations. The derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard domain assumptions of quantum many-body physics and symmetry preservation; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The intercoupling term preserves the Z₂ symmetry of each subsystem
    Explicitly stated as the condition controlling w and enabling the Ashkin-Teller equivalence.
  • domain assumption Ground-state properties of the composite system determine the critical correlations of subsystem S
    Central focus of the analysis on ground-state quantum correlations.

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Works this paper leans on

128 extracted references · 128 canonical work pages · 1 internal anchor

  1. [1]

    In the case of bound- ary conditions preserving translational invariance, such as PBC,G(x 1, x2)≡G(x 2 −x 1)

    The longitudinal correlation function Sincewis a marginal parameter along the continuous transition line, due to the fact that its RG dimension vanishes, the two-point function along the critical line, i.e., varyingwwithin [−1/ √ 2,1] and keepingg= 1 fixed, is expected to show the FSS behavior G(x1, x2;w)≈L −2yϕ G(X1, X2;w),(27) whereX i ≈x i/Landy ϕ =η/2...

    work page

  2. [2]

    The transverse correlation function One may also study the scaling behavior of the two- point function of the transverse spin operator ˆσ (3) x , de- fined in Eq. (12). We expect that its FSS behavior along the QAT transition lineg= 1 andw∈[−1/ √ 2,1] can be written as F(x 1, x2;w)≈L −κF(X 1, X2;w),(32) with an appropriate exponentκthat should be related ...

    work page

  3. [3]

    In partcular, we verified the FSS of the connected correlation function of the energy operator ˆEx defined in Eq

    Other correlation functions We have also computed the FSS behavior of correlation functions involving the spin operator of the environment E. In partcular, we verified the FSS of the connected correlation function of the energy operator ˆEx defined in Eq. (35), E(x1, x2)=⟨Ψ 0| ˆEx1 ˆEx2 |Ψ0⟩ − ⟨Ψ0| ˆEx1 |Ψ0⟩ ⟨Ψ0| ˆEx2 |Ψ0⟩, (39) where|Ψ 0⟩is the ground st...

    work page

  4. [4]

    W. H. Zurek, Environment-induced superselection rules, Phys. Rev. D26, 1862 (1982)

    work page 1982

  5. [5]

    W. H. Zurek, Decoherence, einselection, and the quan- tum origins of the classical, Rev. Mod. Phys.75, 715 (2003)

    work page 2003

  6. [6]

    Gaudin, Diagonalisation d’une classe d’Hamiltoniens de spin, J

    M. Gaudin, Diagonalisation d’une classe d’Hamiltoniens de spin, J. Phys. (Paris)37, 1087 (1976)

    work page 1976

  7. [7]

    N. V. Prokof’ev and P. C. E. Stamp, Theory of the spin bath, Rep. Prog. Phys.63, 669 (2000)

    work page 2000

  8. [8]

    F. M. Cucchietti, J. P. Paz, and W. H. Zurek, Decoher- ence from spin environments, Phys. Rev. A72, 052113 (2005)

    work page 2005

  9. [9]

    H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Decay of Loschmidt echo enhanced by quantum criticality, Phys. Rev. Lett.96, 140604 (2006)

    work page 2006

  10. [10]

    Rossini, T

    D. Rossini, T. Calarco, V. Giovannetti, S. Montangero, and R. Fazio, Decoherence induced by interacting quan- tum spin baths, Phys. Rev. A75, 032333 (2007)

    work page 2007

  11. [11]

    F. M. Cucchietti, S. Fernandez-Vidal, and J. P. Paz, Universal decoherence induced by an environmental quantum phase transition, Phys. Rev. A75, 032333 (2007)

    work page 2007

  12. [12]

    Z.-G. Yuan, P. Zhang, and S.-S. Li, Loschmidt echo and Berry phase of a quantum system coupled to anXYspin chain: Proximity to a quantum phase transition, Phys. Rev. A 75, 012102 (2007)

    work page 2007

  13. [13]

    Cormick and J

    C. Cormick and J. P. Paz, Decoherence induced by a dynamic spin environment: The universal regime, Phys. Rev. A77, 022317 (2008)

    work page 2008

  14. [14]

    W. H. Zurek, Quantum Darwinism, Nat. Phys.5, 181 (2009)

    work page 2009

  15. [15]

    Bortz, S

    M. Bortz, S. Eggert, C. Schneider, R. St¨ ubner, and J. Stolze, Dynamics and decoherence in the central spin model using exact methods, Phys. Rev. B82, 161308(R) (2010)

    work page 2010

  16. [16]

    Damski, H

    B. Damski, H. T. Quan, and W. H. Zurek, Critical dy- namics of decoherence, Phys. Rev. A83, 062104 (2011)

    work page 2011

  17. [17]

    Wei and R.-B

    B.-B. Wei and R.-B. Liu, Lee-Yang zeros and critical times in decoherence of a probe spin coupled to a bath, Phys. Rev. Lett.109, 185701 (2012)

    work page 2012

  18. [18]

    T. Nag, U. Divakaran, and A. Dutta, Scaling of the decoherence factor of a qubit coupled to a spin chain driven across quantum critical points, Phys. Rev. B86, 020401(R) (2012)

    work page 2012

  19. [19]

    Suzuki, T

    S. Suzuki, T. Nag, and A. Dutta, Dynamics of decoher- ence: Universal scaling of the decoherence factor, Phys. Rev. A93, 012112 (2016)

    work page 2016

  20. [20]

    Jafari, and H

    R. Jafari, and H. Johannesson, Decoherence from spin environments: Loschmidt echo and quasiparticle excita- tions, Phys. Rev. B96, 224302 (2017)

    work page 2017

  21. [21]

    Vicari, Decoherence dynamics of qubits coupled to systems at quantum transitions, Phys

    E. Vicari, Decoherence dynamics of qubits coupled to systems at quantum transitions, Phys. Rev. A98, 052127 (2018)

    work page 2018

  22. [22]

    Fiorelli, A

    E. Fiorelli, A. Cuccoli, and P. Verrucchi, Critical slowing down and entanglement protection, Phys. Rev. A100, 032123 (2019)

    work page 2019

  23. [23]

    Rossini and E

    D. Rossini and E. Vicari, Scaling of decoherence and energy flow in interacting quantum many-body systems, Phys. Rev. A99, 052113 (2019)

    work page 2019

  24. [24]

    Haikka, J

    P. Haikka, J. Goold, S. McEndoo, F. Plastina, and S. Maniscalco, Non-Markovianity, Loschmidt echo, and criticality: A unified picture, Phys. Rev. A85, 060101(R) (2012)

    work page 2012

  25. [25]

    F. Liu, X. Zhou, and Z.-W. Zhou, Memory effect and non-Markovian dynamics in an open quantum system, Phys. Rev. A99, 052119 (2019)

    work page 2019

  26. [26]

    Liu, H.-L

    J.-X. Liu, H.-L. Shi, Y.-H. Shi, X.-H. Wang, and W.-L. Yang, Entanglement and work extraction in the central- spin quantum battery, Phys. Rev. B104, 245418 (2021)

    work page 2021

  27. [27]

    H.-Y. Yang, K. Zhang, X.-H. Wang, and H.-L. Shi, Op- timal energy storage and collective charging speedup in the central-spin quantum battery, Phys. Rev. B111, 085410 (2025)

    work page 2025

  28. [28]

    Lai, J.-T

    C.-Y. Lai, J.-T. Hung, C.-Y. Mou, and P. Chen, Induced decoherence and entanglement by interacting quantum spin baths, Phys. Rev. B77, 205419 (2008)

    work page 2008

  29. [29]

    Vasseur, S

    R. Vasseur, S. A. Parameswaran, and J. E. Moore, Quantum revivals and many-body localization, Phys. Rev. B91, 140202(R) (2015)

    work page 2015

  30. [30]

    Yao and W.-M

    C.-Z. Yao and W.-M. Zhang, Probing topological states through the exact non-Markovian decoherence dynam- ics of a spin coupled to a spin bath in the real-time domain, Phys. Rev. B102, 035133 (2020)

    work page 2020

  31. [31]

    Franchi, D

    A. Franchi, D. Rossini, and E. Vicari, Quantum many- body spin rings coupled to ancillary spins: The sunburst quantum Ising model, Phys. Rev. E105, 054111 (2022)

    work page 2022

  32. [32]

    Franchi, D

    A. Franchi, D. Rossini, and E. Vicari, Decoherence and energy flow in the sunburst quantum Ising model, J. Stat. Mech. (2022) 083103

    work page 2022

  33. [33]

    Mitra and S

    A. Mitra and S. C. L. Srivastava, Sunburst quantum Ising model under interaction quench: entanglement and role of initial state coherence, Phys. Rev. E108, 054114 (2023)

    work page 2023

  34. [34]

    Franchi and F

    A. Franchi and F. Tarantelli, Liouvillian gap and out-of- equilibrium dynamics of a sunburst Kitaev ring: from local to uniform dissipation, Phys. Rev. B108, 094114 (2023)

    work page 2023

  35. [35]

    Mitra and S

    A. Mitra and S. C. L. Srivastava, Sunburst quantum Ising battery, Phys. Rev. A110, 012227 (2024)

    work page 2024

  36. [36]

    Barbarino, J

    S. Barbarino, J. Yu, P. Zoller, and J. C. Budich, Prepar- ing atomic topological quantum matter by adiabatic nonunitary dynamics, Phys. Rev. Lett.124, 010401 (2020)

    work page 2020

  37. [37]

    Bhattacharjee, S

    S. Bhattacharjee, S. Bandyopadhyay, D. Sen, and A. Dutta, Bilayer Haldane system: Topological characteri- zation and adiabatic passages connecting Chern phases, 14 Phys. Rev. B103, 224304 (2021)

    work page 2021

  38. [38]

    Manna¨ ı, J.-N

    M. Manna¨ ı, J.-N. Fuchs, F. Pi´ echon, and S. Haddad, Stacking-induced Chern insulator, Phys. Rev. B107, 045117 (2023)

    work page 2023

  39. [39]

    Franchi, A

    A. Franchi, A. Pelissetto, and E. Vicari, Quantum crit- ical behaviors and decoherence of weakly coupled quan- tum Ising models within an isolated global system, Phys. Rev. E107, 014113 (2023)

    work page 2023

  40. [40]

    S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Sha- har, Continuous quantum phase transitions, Rev. Mod. Phys.69, 315 (1997)

    work page 1997

  41. [41]

    Sachdev,Quantum Phase Transitions, 2nd ed

    S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cam- bridge University Press, Cambridge, 2011)

    work page 2011

  42. [42]

    Rossini and E

    D. Rossini and E. Vicari, Coherent and dissipative dy- namics at quantum phase transitions, Phys. Rep.936, 1 (2021)

    work page 2021

  43. [43]

    Schollw¨ ock, The density-matrix renormalization group, Rev

    U. Schollw¨ ock, The density-matrix renormalization group, Rev. Mod. Phys.77, 259 (2005)

    work page 2005

  44. [44]

    Kohmoto, M

    M. Kohmoto, M. den Nijs, and L. P. Kadanoff, Hamil- tonian studies of thed= 2 Ashkin-Teller model, Phys. Rev. B24, 5229 (1981)

    work page 1981

  45. [45]

    Igloi and J

    F. Igloi and J. Solyom, Phase diagram and critical prop- erties of the (1+1)-dimensional Ashkin-Teller model, J. Phys. A17, 1531 (1984)

    work page 1984

  46. [46]

    F. C. Alcaraz and J. R. Drugowich de Felicio, Finite size studies of the Ashkin-Teller model, J. Phys. A17L651 (1984)

    work page 1984

  47. [47]

    Fradkin,N-Color Ashkin-Teller model in two dimen- sions: Solution in the large-Nlimit, Phys

    E. Fradkin,N-Color Ashkin-Teller model in two dimen- sions: Solution in the large-Nlimit, Phys. Rev. Lett. 53, 1967 (1984)

    work page 1967

  48. [48]

    Shankar, Ashkin-Teller and Gross-Neveu models: New relations and results, Phys

    R. Shankar, Ashkin-Teller and Gross-Neveu models: New relations and results, Phys. Rev. Lett.55, 453 (1985)

    work page 1985

  49. [49]

    von Gehlen and V

    G. von Gehlen and V. Rittenberg, The Ashkin-Teller quantum chain and conformal invariance, J. Phys. A 20, 227 (1987)

    work page 1987

  50. [50]

    F. C. Alcaraz, M. N. Barber, and M. T. Batchelor, Fi- nite size studies of the Ashkin-Teller model, Ann. Phys. 182, 280 (1988)

    work page 1988

  51. [51]

    Ashkin and E

    J. Ashkin and E. Teller, Statistics of Two-Dimensional Lattices with Four Components, Phys. Rev.64, 178 (1943)

    work page 1943

  52. [52]

    F. Y. Wu and K. Y. Liu, Two phase transitions in the Ashkin-Teller model, J. Phys. C: Solid State Phys.7, L181 (1974)

    work page 1974

  53. [53]

    Dutta, G

    A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen,Quantum phase tran- sitions in transverse field spin models: From statistical physics to quantum information, (Cambridge University Press, 2015)

    work page 2015

  54. [54]

    Pfeuty, The one-dimensional Ising model with a transverse field, Ann

    P. Pfeuty, The one-dimensional Ising model with a transverse field, Ann. Phys.57, 79 (1970)

    work page 1970

  55. [55]

    Campostrini, A

    M. Campostrini, A. Pelissetto, and E. Vicari, Quantum Ising chains with boundary fields, J. Stat. Mech. P11015 (2015)

    work page 2015

  56. [56]

    Campostrini, A

    M. Campostrini, A. Pelissetto, and E. Vicari, Finite- size scaling at quantum transitions, Phys. Rev. B89, 094516 (2014)

    work page 2014

  57. [57]

    Pelissetto and E

    A. Pelissetto and E. Vicari, Critical phenomena and renormalization group theory, Phys. Rep.368, 549 (2002)

    work page 2002

  58. [58]

    Caselle, M

    M. Caselle, M. Hasenbusch, A. Pelissetto, and E. Vicari, Irrelevant operators in the two-dimensional Ising model, J. Phys. A35, 4861 (2002)

    work page 2002

  59. [59]

    Baake, G

    M. Baake, G. von Gehlen and V. Rittenberg, Op- erator content of the Ashkin-Teller quantum chain- superconformal and Zamolodchikov-Fateev invariance: I. Free boundary conditions, J. Phys20, L479 (1987); Operator content of the Ashkin-Teller quantum chain, superconformal and Zamolodchikov-Fateev invariance. II. Boundary conditions compatible with the torus...

    work page 1987

  60. [60]

    Yang, Modular invariant partition function of the Ashkin-Teller model on the critical line and N = 2 su- perconformal invariance, Nucl

    S. Yang, Modular invariant partition function of the Ashkin-Teller model on the critical line and N = 2 su- perconformal invariance, Nucl. Phys. B285, 183 (1987)

    work page 1987

  61. [61]

    Yang and H

    S. Yang and H. Zheng, Superconformal invariance in the two-dimensional Ashkin-Teller model, Nuc. Phys. B 285, 410 (1987)

    work page 1987

  62. [62]

    Yamanaka, Y

    M. Yamanaka, Y. Hatsugai, and M. Kohmoto, Phase diagram of the Ashkin-Teller quantum spin chain, Phys. Rev. B50, 559 (1994)

    work page 1994

  63. [63]

    Yamanaka and M

    M. Yamanaka and M. Kohmoto, Line of continuously varying criticality in the Ashkin-Teller quantum chain, Phys. Rev. B52, 1138 (1995)

    work page 1995

  64. [64]

    J. C. Bridgeman, A O’Brien, S. D. Barlett, and A. C. Doherty, Multiscale entanglement renormalization ansatz for spin chains with continuously varying crit- icality, Phys. Rev. B91, 165129 (2015)

    work page 2015

  65. [65]

    A O’Brien, S. D. Barlett, A. C. Doherty, and S. T. Flammia, Symmetry-respecting real-space renormaliza- tion for the quantum Ashkin-Teller model, Phys. Rev. E92, 042163 (2015)

    work page 2015

  66. [66]

    B. E. L¨ uscher, F. Mila, and N. Chepiga, Critical prop- erties of the quantum Ashkin-Teller chain with chiral perturbations, Phys. Rev. B108, 184425 (2023)

    work page 2023

  67. [67]

    Y. Liu, N. Chepiga, Y. Fukusumi, and M. Oshikawa, Boundary critical phenomena in the quantum Ashkin- Teller model, arXiv:2601.16951

    work page internal anchor Pith review Pith/arXiv arXiv

  68. [68]

    F. Y. Wu, Ashkin-Teller Model as a Vertex Problem, J. Math. Phys.18, 611 (1977)

    work page 1977

  69. [69]

    R. J. Baxter,Exactly solvable models in statistical me- chanics, Academic Press, New York (1982)

    work page 1982

  70. [70]

    Domany and E

    E. Domany and E. K. Riedel, Two-dimensional anisotropicN-vector models, Phys. Rev. B19, 5817 (1979)

    work page 1979

  71. [71]

    J. R. Drugowich de Fel´ ıcio and R. K¨ oberle, Critical ex- ponents of the Ashkin-Teller model, Phys. Rev. B25, 511 (1982)

    work page 1982

  72. [72]

    Wiseman and E

    S. Wiseman and E. Domany, A Cluster Method for the Ashkin–Teller Model, Phys. Rev. E48, 4080 (1993)

    work page 1993

  73. [73]

    Kamieniarz, P

    G. Kamieniarz, P. Kozlowski, and R. Dekeyser, Critical Ising lines of thed= 2 Ashkin-Teller model, Phys. Rev. E55, 3724 (1997)

    work page 1997

  74. [74]

    Delfino and P

    G. Delfino and P. Grinza, Universal ratios along a line of critical points. The Ashkin-Teller model, Nucl. Phys. B682, 521 (2004)

    work page 2004

  75. [75]

    Giuliani and V

    A. Giuliani and V. Mastropietro, Anomalous universal- ity in the anisotropic Ashkin-Teller model, Commun. Math. Phys.256, 681 (2005)

    work page 2005

  76. [76]

    Mukherjee and P

    I. Mukherjee and P. K. Mohanty, Hidden superuniver- sality in systems with continuous variation of critical exponents, Phys. Rev. B108, 174417 (2023)

    work page 2023

  77. [77]

    Kecoglu and A

    I. Kecoglu and A. N. Berker, Global Ashkin-Teller Phase Diagrams in Two and Three Dimensions: Multicriti- cal Bifurcation versus Double Tricriticality - Endpoint, Physica A630, 129248 (2023)

    work page 2023

  78. [78]

    Y. Aoun, M. Dober, and A. Glazman, Phase diagram of the Ashkin-Teller model, Commun. Math. Phys. 405, 15 37 (2024)

    work page 2024

  79. [79]

    Dober, On antiferromagnetic regimes in the Ashkin- Teller model, Electron

    M. Dober, On antiferromagnetic regimes in the Ashkin- Teller model, Electron. J. Probab.30, 1 (2025)

    work page 2025

  80. [80]

    Itzykson and J

    C. Itzykson and J. M. Drouffe,Statistical Field Theory (Cambridge Univ. Press, Cambridge 1989)

    work page 1989

Showing first 80 references.