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arxiv: 2606.06605 · v1 · pith:NIRFVMBBnew · submitted 2026-06-04 · ❄️ cond-mat.str-el

Tensor network study of deconfined quantum criticality in a one-dimensional spin-phonon model

Anton Romen , Josef Willsher , David Hofmeier , Johannes Knolle , Michael Knap This is my paper

Pith reviewed 2026-06-27 23:09 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords deconfined quantum criticalityspin-phonon couplingtensor networksdouble sine-Gordon modelfour-state Potts universalityLuttinger parameterAshkin-Teller model
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The pith

In a one-dimensional spin-phonon model the deconfined quantum critical point stays continuous only above a critical phonon frequency and ends in the four-state Potts class below it.

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

The paper uses tensor network simulations on the antiferromagnetic J1-J2 chain coupled to lattice vibrations to study how phonons affect the deconfined quantum critical point between Neel antiferromagnetic and valence-bond-solid order. It finds that the critical point remains continuous for large phonon frequencies but turns strongly first-order once the frequency drops below a threshold. The mechanism is a phonon-induced lowering of the Luttinger parameter that drives the system into the double sine-Gordon regime. Because the same effective theory describes the classical Ashkin-Teller model, the endpoint of the first-order line belongs to the four-state Potts universality class. The authors also give scaling results for the phonon spectral function that could be measured experimentally.

Core claim

In the antiferromagnetic J1-J2 model coupled to lattice vibrations, tensor network simulations confirm that the deconfined quantum critical point remains stable for large phonon frequencies. Below a critical frequency the transition turns strongly first-order. The cause is a reduction of the Luttinger parameter from spin-phonon coupling, which maps the system to the double sine-Gordon model. Since the double sine-Gordon model also describes the classical Ashkin-Teller model, the critical endpoint lies in the four-state Potts universality class.

What carries the argument

The mapping of the spin-phonon instability onto the double sine-Gordon model, which identifies the critical endpoint with the four-state Potts universality class through its equivalence to the Ashkin-Teller model.

If this is right

  • The deconfined quantum critical point is stable only above a critical phonon frequency.
  • Below that frequency the transition becomes strongly first-order.
  • The endpoint of the first-order regime belongs to the four-state Potts universality class.
  • The phonon spectral function exhibits quantitative scaling that can serve as an experimental signature.

Where Pith is reading between the lines

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

  • Phonon coupling could be used to drive other candidate deconfined critical points into first-order regimes in one dimension.
  • Materials with soft phonon modes may show first-order rather than continuous deconfined transitions.
  • The four-state Potts class may appear at the termination of first-order lines in other one-dimensional quantum spin models with similar effective theories.

Load-bearing premise

That the dominant effect of the spin-phonon coupling is simply a reduction of the Luttinger parameter sufficient to enter the double sine-Gordon regime.

What would settle it

A direct numerical extraction of the Luttinger parameter versus phonon frequency that shows it remains above the threshold needed for the first-order instability even at low frequencies.

Figures

Figures reproduced from arXiv: 2606.06605 by Anton Romen, David Hofmeier, Johannes Knolle, Josef Willsher, Michael Knap.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. a) We obtain a precise estimate of the critical point [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Effects of the phonon cutoff [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Phase diagram of the spin-phonon model with [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. a,b) Critical exponents [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Critical exponents [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
read the original abstract

Deconfined quantum critical points (DQCPs) describe continuous transitions between ordered phases beyond the Landau paradigm. A simple example is the N\'eel antiferromagnet (AFM) to valence bond solid (VBS) transition in a 1D antiferromagnetic $J_1-J_2$ model. In analogy to the spin-Peierls instability of critical spin chains, DQCPs are predicted to be unstable towards lattice distortions below a critical phonon frequency. In this work, we use tensor network simulations to investigate this instability in the antiferromagnetic $J_1-J_2$ model coupled to lattice vibrations. We confirm the stability of DQCP for large phonon frequencies and demonstrate that the transition turns strongly first-order below a critical frequency. The instability is caused by a reduction of the Luttinger parameter due to spin-phonon interactions and we identify the effective theory of the behavior as the double sine-Gordon model. The same effective theory is known to describe the classical Ashkin-Teller model, which enables us to show that the critical endpoint is in the four-state Potts universality class. Furthermore, we provide quantitative numerical scaling results for the phonon spectral function, offering an experimental signature to probe DQCP-phonon coupling in low-dimensional materials.

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 uses tensor-network simulations of the antiferromagnetic J1-J2 spin chain coupled to phonons to study the stability of the deconfined quantum critical point (DQCP) separating Néel and valence-bond-solid phases. It reports that the DQCP remains stable at large phonon frequencies but the transition becomes strongly first-order below a critical frequency; the instability is traced to a phonon-induced reduction of the Luttinger parameter K, which maps the low-energy theory onto the double sine-Gordon model (equivalent to the Ashkin-Teller model) whose endpoint lies in the four-state Potts universality class. Quantitative scaling of the phonon spectral function is also presented as an experimental signature.

Significance. If the numerical evidence and effective-theory identification hold, the work supplies a concrete, falsifiable mechanism for the phonon-induced destabilization of a 1D DQCP and locates the critical endpoint in a known universality class. The tensor-network treatment of the coupled spin-phonon system and the reported spectral-function scaling constitute clear strengths that could guide material searches.

major comments (1)
  1. [Effective-theory discussion (near the identification of the double sine-Gordon model)] The central mapping to the double sine-Gordon model (and thereby to the four-state Potts class) rests on the claim that spin-phonon coupling renormalizes only the Luttinger parameter K while leaving all other phonon-induced operators irrelevant. No explicit derivation or numerical check is provided showing that the phonon coupling does not generate additional marginal or relevant operators, nor is K(ω) extracted and shown to cross its critical value exactly where the transition order changes. This step is load-bearing for both the first-order instability and the universality-class conclusion.
minor comments (2)
  1. [Abstract and results section on spectral function] The abstract states that 'quantitative numerical scaling results' for the phonon spectral function are provided, but the main text should explicitly compare the extracted exponents to the known four-state Potts values.
  2. [Numerical methods and results sections] Convergence data (bond-dimension dependence, truncation error, and finite-size scaling) for the order-parameter and correlation-length diagnostics used to distinguish continuous versus first-order behavior should be shown in a dedicated supplementary figure or table.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater clarity in the effective-theory section. The comment is constructive, and we will revise the manuscript to address it directly.

read point-by-point responses

  1. Referee: [Effective-theory discussion (near the identification of the double sine-Gordon model)] The central mapping to the double sine-Gordon model (and thereby to the four-state Potts class) rests on the claim that spin-phonon coupling renormalizes only the Luttinger parameter K while leaving all other phonon-induced operators irrelevant. No explicit derivation or numerical check is provided showing that the phonon coupling does not generate additional marginal or relevant operators, nor is K(ω) extracted and shown to cross its critical value exactly where the transition order changes. This step is load-bearing for both the first-order instability and the universality-class conclusion.

    Authors: We agree that an explicit derivation and a direct numerical extraction of K(ω) would strengthen the central claim. The mapping follows from standard bosonization of the J1-J2 chain with linear spin-phonon coupling: the phonon mode integrates out to a renormalization of the Luttinger parameter K (and velocity) while symmetry and momentum conservation keep additional cosine operators irrelevant or marginal only at the known Ashkin-Teller line. In the revised manuscript we will add a concise derivation (including the relevant operator content) either in the main text or as an appendix. We will also extract K(ω) from the decay of spin and dimer correlations in the tensor-network data and plot it versus phonon frequency, demonstrating that the continuous-to-first-order change occurs when K crosses the value at which the double sine-Gordon model flows to the four-state Potts fixed point. These additions directly address the load-bearing step identified by the referee. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent numerical simulation and external effective-theory mappings

full rationale

The paper's central claims rest on tensor-network simulations of the microscopic J1-J2 spin-phonon Hamiltonian, direct observation of the transition order change, and identification of the effective theory as the double sine-Gordon model (known independently to describe the Ashkin-Teller model and four-state Potts class). No step reduces a prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness theorem, or defines quantities in terms of each other. The Luttinger-parameter reduction is extracted from the numerics rather than assumed; the mapping to known models is presented as an external identification rather than a self-derived result. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; the paper relies on numerical tensor-network simulations of a microscopic Hamiltonian whose parameters (J1, J2, phonon frequency, coupling strength) are not enumerated here. No new entities are postulated.

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discussion (0)

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Reference graph

Works this paper leans on

83 extracted references · 1 canonical work pages

  1. [1]

    The parameters ∆ 1,2 are XXZ anisotropy parameters, which we take to be equal ∆≡ ∆1 = ∆2 unless specified otherwise

    Spin Hamiltonian The spin-part of our model is defined by the following spin-1/2 chain Hs =J 1 X j [Sj ·S j+1 + (∆1 −1)S z j Sz j+1] +J2 X j [Sj ·S j+2 + (∆2 −1)S z j Sz j+2] (2) with nearest neighborJ 1 and next-nearest neighborJ 2 exchange couplings. The parameters ∆ 1,2 are XXZ anisotropy parameters, which we take to be equal ∆≡ ∆1 = ∆2 unless specifie...

  2. [2]

    (2) as a sine-Gordon model by using a standard Jordan-Wigner transformation and bosonization [22], as briefly outlined in Appendix A

    Continuum field theory We can describe the long-wavelength behavior of the model introduced in Eq. (2) as a sine-Gordon model by using a standard Jordan-Wigner transformation and bosonization [22], as briefly outlined in Appendix A. It produces a low-energy description in terms of a compact bosonic fieldϕ(x)∈[0,2π], with the action SSG = Z dxdτ 1 2πK (∂xϕ...

  3. [3]

    Numerical method Throughout this work we simulate the model intro- duced in Eq. (1) using infinite-system Density Matrix Renormalization Group (iDMRG) simulations [39, 40] implemented within the TeNPy library [41] and optimize over infinite Matrix Product States (iMPS) in the ther- modynamic limit [42, 43]. We provide further details on the numerical impl...

  4. [4]

    Phonons We start by introducing the distortions of lattice sites around their equilibrium positionsu j =x j −¯xj, which we assume to be small. Within the harmonic approxima- tion,u j may then be modeled by dissipationless, uncou- pled Einstein phonons at some frequencyω0 via a phonon Hamiltonian Hp = X j p2 j 2m + 1 2 mω2 0u2 j .(7) By introducing appropr...

  5. [5]

    The corresponding ex- change constants are naturally bond-length dependent and we may for example modelJ i,j ∼J 0e−˜g|xi−xj | in terms of the atomic positionsxj

    Spin-phonon coupling We first note that the origin of Heisenberg-type spin- spin interactions is through exchange interactions of neighboring spinful orbitals. The corresponding ex- change constants are naturally bond-length dependent and we may for example modelJ i,j ∼J 0e−˜g|xi−xj | in terms of the atomic positionsxj. This spatial dependence is used imp...

  6. [6]

    Senthil, A

    T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. Fisher, Deconfined quantum critical points, Science 303, 1490 (2004)

    2004

  7. [7]

    Senthil, L

    T. Senthil, L. Balents, S. Sachdev, A. Vish- wanath, and M. P. Fisher, Quantum criticality beyond the landau-ginzburg-wilson paradigm, Physical Review B—Condensed Matter and Materials Physics70, 144407 (2004)

    2004

  8. [8]

    L. D. Landau and E. M. Lifshitz,Statistical physics: vol- ume 5, Vol. 5 (Elsevier, 2013)

    2013

  9. [9]

    K. G. Wilson and J. Kogut, The renormalization group and theϵexpansion, Physics reports12, 75 (1974)

    1974

  10. [10]

    Y. Liu, Z. Wang, T. Sato, M. Hohenadler, C. Wang, W. Guo, and F. F. Assaad, Superconductivity from the condensation of topological defects in a quantum spin- hall insulator, Nature Communications10, 2658 (2019)

    2019

  11. [11]

    Khalaf, S

    E. Khalaf, S. Chatterjee, N. Bultinck, M. P. Zaletel, and A. Vishwanath, Charged skyrmions and topological ori- gin of superconductivity in magic-angle graphene, Sci- ence advances7, eabf5299 (2021)

    2021

  12. [12]

    Christos, S

    M. Christos, S. Sachdev, and M. S. Scheurer, Supercon- ductivity, correlated insulators, and wess–zumino–witten terms in twisted bilayer graphene, Proceedings of the Na- tional Academy of Sciences117, 29543 (2020)

    2020

  13. [13]

    Christos, L

    M. Christos, L. Shackleton, S. Sachdev, and Z.-X. Luo, Deconfined quantum criticality of nodal d-wave super- conductivity, n´ eel order, and charge order on the square lattice at half-filling, Physical Review Research6, 033018 (2024)

    2024

  14. [14]

    Z. H. Liu, M. Vojta, F. F. Assaad, and L. Janssen, Metal- lic and deconfined quantum criticality in dirac systems, Physical Review Letters128, 087201 (2022)

    2022

  15. [15]

    Zhang and S

    Y.-H. Zhang and S. Sachdev, Deconfined criticality and ghost fermi surfaces at the onset of antiferromagnetism in a metal, Physical Review B102, 155124 (2020)

    2020

  16. [16]

    Nikolaenko, J

    A. Nikolaenko, J. von Milczewski, D. G. Joshi, and S. Sachdev, Spin density wave, fermi liquid, and frac- tionalized phases in a theory of antiferromagnetic metals using paramagnons and bosonic spinons, Physical Review B108, 045123 (2023)

    2023

  17. [17]

    G¨ otz, M

    A. G¨ otz, M. Hohenadler, and F. F. Assaad, Phases and exotic phase transitions of a two-dimensional su- schrieffer-heeger model, Physical Review B109, 195154 (2024)

    2024

  18. [18]

    Y. Liu, T. Sato, D. Hou, Z. Wang, W. Guo, and F. F. Assaad, Edge modes of topological mott insulators and deconfined quantum critical points, arXiv:2508.04455 (2025)

    arXiv 2025

  19. [19]

    X. Chen, S. Liu, D.-c. Lu, and N. Tantivasadakarn, Spontaneous breaking of non-invertible symmetries and duality to beyond-landau transitions, arXiv:2605.27672 (2026)

    Pith/arXiv arXiv 2026

  20. [20]

    Weber, Competing dirac masses in one dimen- sion: Symmetry-enhanced pseudo-first-order transition and deconfined criticality (2025)

    M. Weber, Competing dirac masses in one dimen- sion: Symmetry-enhanced pseudo-first-order transition and deconfined criticality (2025)

    2025

  21. [21]

    Levin and T

    M. Levin and T. Senthil, Deconfined quantum critical- ity and n´ eel order via dimer disorder, Physical Review B—Condensed Matter and Materials Physics70, 220403 (2004)

    2004

  22. [22]

    Senthil and M

    T. Senthil and M. P. Fisher, Competing orders, nonlinear sigma models, and topological terms in quantum mag- nets, Physical Review B—Condensed Matter and Mate- rials Physics74, 064405 (2006)

    2006

  23. [23]

    Nahum, P

    A. Nahum, P. Serna, J. Chalker, M. Ortu˜ no, and A. So- moza, Emergent so(5) symmetry at the n´ eel to valence- bond-solid transition, arXiv:1508.06668 (2015)

    Pith/arXiv arXiv 2015

  24. [24]

    C. Wang, A. Nahum, M. A. Metlitski, C. Xu, and T. Senthil, Deconfined quantum critical points: symme- tries and dualities, Physical Review X7, 031051 (2017)

    2017

  25. [25]

    Jiang and O

    S. Jiang and O. Motrunich, Ising ferromagnet to valence bond solid transition in a one-dimensional spin chain: Analogies to deconfined quantum critical points, Physical Review B99, 075103 (2019)

    2019

  26. [26]

    Roberts, S

    B. Roberts, S. Jiang, and O. I. Motrunich, Deconfined quantum critical point in one dimension, Physical Review B99, 165143 (2019)

    2019

  27. [27]

    Giamarchi,Quantum physics in one dimension, Vol

    T. Giamarchi,Quantum physics in one dimension, Vol. 121 (Clarendon press, 2003)

    2003

  28. [28]

    Mudry, A

    C. Mudry, A. Furusaki, T. Morimoto, and T. Hikihara, Quantum phase transitions beyond landau-ginzburg the- ory in one-dimensional space revisited, Physical Review B99, 205153 (2019)

    2019

  29. [29]

    J. Y. Lee, J. Ramette, M. A. Metlitski, V. Vuleti´ c, W. W. Ho, and S. Choi, Landau-forbidden quantum criticality in rydberg quantum simulators, Physical review letters 131, 083601 (2023)

    2023

  30. [30]

    Romen, S

    A. Romen, S. Birnkammer, and M. Knap, Deconfined quantum criticality in the long-range, anisotropic heisen- berg chain, SciPost Physics Core7, 008 (2024)

    2024

  31. [31]

    Pytte, Peierls instability in heisenberg chains, Physical Review B10, 4637 (1974)

    E. Pytte, Peierls instability in heisenberg chains, Physical Review B10, 4637 (1974)

    1974

  32. [32]

    R. E. Peierls,Quantum theory of solids(Oxford univer- sity press, 1955)

    1955

  33. [33]

    Cross and D

    M. Cross and D. S. Fisher, A new theory of the spin- peierls transition with special relevance to the experi- ments on ttfcubdt, Physical Review B19, 402 (1979)

    1979

  34. [34]

    Pincus, Instability of the uniform antiferromagnetic chain, Solid State Communications9(1971)

    P. Pincus, Instability of the uniform antiferromagnetic chain, Solid State Communications9(1971)

    1971

  35. [35]

    U. F. P. Seifert, J. Willsher, M. Drescher, F. Pollmann, and J. Knolle, Spin-Peierls instability of the U(1) Dirac spin liquid, Nature Communications15, 7110 (2024). 13

    2024

  36. [36]

    Ferrari, J

    F. Ferrari, J. Willsher, U. F. Seifert, R. Valent´ ı, and J. Knolle, Stability of algebraic spin liquids coupled to quantum phonons, Physical Review Research7, L042053 (2025)

    2025

  37. [37]

    Hofmeier, J

    D. Hofmeier, J. Willsher, U. F. P. Seifert, and J. Knolle, Spin-peierls instability of deconfined quantum critical points, Physical Review B110(2024)

    2024

  38. [38]

    G¨ otz, F

    A. G¨ otz, F. F. Assaad, and N. C. Costa, Tuning the order of a deconfined quantum critical point, arXiv:2412.17215 (2024)

    arXiv 2024

  39. [39]

    Ashkin and E

    J. Ashkin and E. Teller, Statistics of two-dimensional lattices with four components, Physical Review64, 178 (1943)

    1943

  40. [40]

    Kohmoto, M

    M. Kohmoto, M. den Nijs, and L. P. Kadanoff, Hamil- tonian studies of the d= 2 ashkin-teller model, Physical Review B24, 5229 (1981)

    1981

  41. [41]

    Y. Aoun, M. Dober, and A. Glazman, Phase diagram of the ashkin–teller model, Communications in Mathemat- ical Physics405, 37 (2024)

    2024

  42. [42]

    Kohn, Image of the fermi surface in the vibration spectrum of a metal, Physical Review Letters2, 393 (1959)

    W. Kohn, Image of the fermi surface in the vibration spectrum of a metal, Physical Review Letters2, 393 (1959)

    1959

  43. [43]

    Luther and I

    A. Luther and I. Peschel, Single-particle states, Kohn anomaly, and pairing fluctuations in one dimension, Physical Review B9, 2911 (1974)

    1974

  44. [44]

    S. R. White, Density matrix formulation for quantum renormalization groups, Physical review letters69, 2863 (1992)

    1992

  45. [45]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics 326, 96 (2011)

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics 326, 96 (2011)

    2011

  46. [46]

    Hauschild, J

    J. Hauschild, J. Unfried, S. Anand, B. Andrews, M. Bintz, U. Borla, S. Divic, M. Drescher, J. Geiger, M. Hefel,et al., Tensor network python (tenpy) version 1, SciPost Physics Codebases , 041 (2024)

    2024

  47. [47]

    Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Physical review letters91, 147902 (2003)

    G. Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Physical review letters91, 147902 (2003)

    2003

  48. [48]

    Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Physical review letters98, 070201 (2007)

    G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Physical review letters98, 070201 (2007)

    2007

  49. [49]

    Perez-Garcia, F

    D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix product state representations, arXiv:quant-ph/0608197 (2006)

    Pith/arXiv arXiv 2006

  50. [50]

    R. J. Glauber, Coherent and incoherent states of the ra- diation field, Physical Review131, 2766 (1963)

    1963

  51. [51]

    Orignac and R

    E. Orignac and R. Chitra, Mean-field theory of the spin- Peierls transition, Physical Review B70, 214436 (2004)

    2004

  52. [52]

    Citro, E

    R. Citro, E. Orignac, and T. Giamarchi, Adiabatic- antiadiabatic crossover in a spin-Peierls chain, Physical Review B72, 024434 (2005)

    2005

  53. [53]

    Pollmann, S

    F. Pollmann, S. Mukerjee, A. M. Turner, and J. E. Moore, Theory of finite-entanglement scaling at one- dimensional quantum critical points, Physical review let- ters102, 255701 (2009)

    2009

  54. [54]

    Vollmayr, J

    K. Vollmayr, J. D. Reger, M. Scheucher, and K. Binder, Finite size effects at thermally-driven first order phase transitions: A phenomenological theory of the order pa- rameter distribution, Zeitschrift f¨ ur Physik B Condensed Matter91, 113 (1993)

    1993

  55. [55]

    L. P. Kadanoff, Singularities near the bifurcation point of the Ashkin-Teller model, Physical Review B22, 1405 (1980)

    1980

  56. [56]

    Mussardo, V

    G. Mussardo, V. Riva, and G. Sotkov, Semiclassical particle spectrum of double sine-gordon model, Nuclear Physics B687, 189–219 (2004)

    2004

  57. [57]

    Tak´ acs and F

    G. Tak´ acs and F. W´ agner, Double sine-gordon model re- visited, Nuclear Physics B741, 353–367 (2006)

    2006

  58. [58]

    Mouland, D

    R. Mouland, D. Tong, and B. Zan, Phases of 2d gauge theories and symmetric mass generation, arXiv:2509.12305 (2025)

    Pith/arXiv arXiv 2025

  59. [59]

    Tagliacozzo, T

    L. Tagliacozzo, T. R. de Oliveira, S. Iblisdir, and J. I. La- torre, Scaling of entanglement support for matrix prod- uct states, Physical Review B—Condensed Matter and Materials Physics78, 024410 (2008)

    2008

  60. [60]

    Enting, Critical exponents for the four-state potts model, Journal of Physics A: Mathematical and General 8, L35 (1975)

    I. Enting, Critical exponents for the four-state potts model, Journal of Physics A: Mathematical and General 8, L35 (1975)

    1975

  61. [61]

    R. B. Pearson, Conjecture for the extended potts model magnetic eigenvalue, Physical Review B22, 2579 (1980)

    1980

  62. [62]

    Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the coulomb gas, Jour- nal of Statistical Physics34, 731 (1984)

    B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the coulomb gas, Jour- nal of Statistical Physics34, 731 (1984)

    1984

  63. [63]

    M. P. den Nijs, A relation between the temperature expo- nents of the eight-vertex and q-state potts model, Journal of Physics A: Mathematical and General12, 1857 (1979)

    1979

  64. [64]

    Delfino and P

    G. Delfino and P. Grinza, Universal ratios along a line of critical points. the Ashkin-Teller model, Nuclear Physics B682, 521 (2004), arXiv:hep-th/0309129

    Pith/arXiv arXiv 2004

  65. [65]

    J. Y. Lee, Y.-Z. You, S. Sachdev, and A. Vishwanath, Sig- natures of a Deconfined Phase Transition on the Shastry- Sutherland Lattice: Applications to Quantum Critical SrCu 2 ( BO 3 ) 2, Physical Review X9, 041037 (2019)

    2019

  66. [66]

    Willsher,Dynamics and Stability of Critical Spin Liquids, Ph.D

    J. Willsher,Dynamics and Stability of Critical Spin Liquids, Ph.D. thesis, Technische Universit¨ at M¨ unchen (2025), availiable athttps://mediatum.ub.tum.de/?id= 1785165

    2025

  67. [67]

    Haegeman, J

    J. Haegeman, J. I. Cirac, T. J. Osborne, I. Piˇ zorn, H. Ver- schelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Physical Review Letters 107, 070601 (2011)

    2011

  68. [68]

    Haegeman, C

    J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimiza- tion with matrix product states, Physical Review B94, 165116 (2016)

    2016

  69. [69]

    M. F. Maghrebi, Z.-X. Gong, and A. V. Gorshkov, Con- tinuous symmetry breaking in 1d long-range interacting quantum systems, Physical Review Letters119, 023001 (2017)

    2017

  70. [70]

    Willsher and J

    J. Willsher and J. Knolle, Dynamics and stability of U(1) spin liquids beyond mean-field theory: Triangular-lattice J1-J2 Heisenberg model, arXiv:2503.13831 (2025)

    arXiv 2025

  71. [71]

    G. D. Mahan,Many-particle physics(Springer Science & Business Media, 2013)

    2013

  72. [72]

    G. L. Squires,Introduction to the theory of thermal neu- tron scattering(Courier Corporation, 1996)

    1996

  73. [73]

    Shirane, S

    G. Shirane, S. M. Shapiro, and J. M. Tranquada,Neu- tron scattering with a triple-axis spectrometer: basic tech- niques(Cambridge University Press, 2002)

    2002

  74. [74]

    B. T. M. Willis and C. J. Carlile,Experimental neutron scattering(Oxford University Press, 2017)

    2017

  75. [75]

    Y. Cui, L. Liu, H. Lin, K.-H. Wu, W. Hong, X. Liu, C. Li, Z. Hu, N. Xi, S. Li,et al., Proximate deconfined quantum critical point in srcu2 (bo3) 2, Science380, 1179 (2023)

    2023

  76. [76]

    J. Guo, P. Wang, C. Huang, B.-B. Chen, W. Hong, S. Cai, J. Zhao, J. Han, X. Chen, Y. Zhou,et al., De- confined quantum critical point lost in pressurized srcu2 14 (bo3) 2, Communications Physics8, 75 (2025)

    2025

  77. [77]

    All data and simulation codes are available upon reasonable request at https://doi.org/10.5281/zenodo.20037182

    work page doi:10.5281/zenodo.20037182

  78. [78]

    S. R. White, Density matrix renormalization group al- gorithms with a single center site, Physical Review B—Condensed Matter and Materials Physics72, 180403 (2005)

    2005

  79. [79]

    H. N. Phien, G. Vidal, and I. P. McCulloch, Infinite boundary conditions for matrix product state calcula- tions, Physical Review B—Condensed Matter and Ma- terials Physics86, 245107 (2012)

    2012

  80. [80]

    Milsted, J

    A. Milsted, J. Haegeman, T. J. Osborne, and F. Ver- straete, Variational matrix product ansatz for nonuni- form dynamics in the thermodynamic limit, Physical Re- view B—Condensed Matter and Materials Physics88, 155116 (2013)

    2013

Showing first 80 references.