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arxiv: 2605.26932 · v1 · pith:MVXQTNY5new · submitted 2026-05-26 · ❄️ cond-mat.stat-mech · hep-th· quant-ph

Long-range deformations in Gaussian States

Francisco Pereira , Nandagopal Manoj , Sara Murciano This is my paper

Pith reviewed 2026-07-01 16:16 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-thquant-ph
keywords Kitaev chainMajorana fermionslong-range interactionsGaussian statesimaginary time evolutiontopological phasespower-law decayentanglement entropy
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The pith

The exponent of power-law couplings in imaginary-time deformations of the Kitaev chain controls three distinct long-distance regimes without finite-strength transitions.

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

This paper starts from the ground state of the Kitaev Majorana chain and applies an imaginary-time evolution generated by a quadratic Hamiltonian whose couplings decay as 1/r to the alpha. The resulting deformed states remain Gaussian, permitting exact calculation of their properties. The value of the exponent alpha divides the infrared behavior into three cases: for alpha greater than 1 only smooth changes occur until infinite deformation, at alpha equal to 1 an infinitesimal change reaches the topological regime possibly with emergent symmetry, and for alpha less than 1 the state is critical-like for any nonzero deformation. A reader would care because these results show how the range of interactions can alter the constraints on phase transitions in one-dimensional systems under imaginary-time evolution.

Core claim

Starting from the Kitaev chain ground state, imaginary-time evolution under a quadratic Hamiltonian with power-law couplings of the form 1/r^α produces deformed Gaussian states whose long-distance properties fall into three regimes governed by α. For α > 1 the topological regime appears only as the deformation strength tends to infinity, with finite strength yielding smooth crossovers. At α = 1 an infinitesimal deformation drives the system topological, and in special cases an emergent Kramers-Wannier symmetry produces Ising scaling. For α < 1 the state exhibits critical-like behavior at all nonzero deformation strengths. Even arbitrarily long-range interactions yield no sharp phase transiti

What carries the argument

The power-law exponent α in the couplings of the imaginary-time evolution Hamiltonian, which dictates the infrared regime of the deformed Gaussian state.

If this is right

  • For α greater than 1, only asymptotic infinite deformation reaches the topological regime with finite strength producing smooth crossovers.
  • At α equal to 1, an infinitesimal deformation drives the system to the topological regime, sometimes with emergent Kramers-Wannier symmetry and Ising scaling.
  • For α less than 1, the state shows critical-like behavior for all nonzero deformation strength.
  • Arbitrarily long-range interactions still produce no sharp phase transition at nonzero finite deformation strength.

Where Pith is reading between the lines

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

  • One could test these regimes by preparing such deformed states in cold-atom or superconducting qubit arrays with tunable long-range couplings.
  • The results suggest that the no-go theorem for local imaginary-time evolution inducing transitions in 1D can be circumvented by long-range terms in a controlled, exponent-dependent way.
  • Future work might examine whether these three regimes persist when the deformation Hamiltonian is non-quadratic.

Load-bearing premise

The imaginary-time evolution generated by the quadratic power-law Hamiltonian preserves the Gaussian nature of the initial state.

What would settle it

Computing or measuring the two-point correlation functions or entanglement spectrum for a fixed α=1 and small deformation strength to verify if topological signatures appear immediately rather than at a finite threshold.

Figures

Figures reproduced from arXiv: 2605.26932 by Francisco Pereira, Nandagopal Manoj, Sara Murciano.

Figure 1
Figure 1. Figure 1: Phase diagram of a deformation of a trivial state. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Contour plot showing the eight sectors used in the analytic continuation of η±. Regions 1–4 correspond to λ > 0, while regions 5–8 correspond to λ < 0. The two tables below list, for each numbered sector, which branch of η± is selected depending on whether the contour is taken on the upper or lower side of the cut. 1 + |g| 2 λ2 − 1 Side of contour η± expression 1 : > 0 Top (a ∓ ib) 2 : > 0 Bottom (a ± ib) … view at source ↗
Figure 3
Figure 3. Figure 3: Entanglement entropy for α > 1 and initial critical state. Entangle￾ment entropy in the state (8) following a non-unitary evolution with long-range hopping in eq. (2), α = 2, µ = 0, and initial state with h = 1 = γ in Eq. (1). The left panel shows the analytical behavior of the effective central charge (31) after the non-unitary evolution. In the right panel, the data points were obtained using Eq. (21) fo… view at source ↗
Figure 4
Figure 4. Figure 4: Entanglement entropy for α = 1. Entanglement entropy in the state (8) following a non-unitary evolution with short-range hopping in eq. (2), α = 1, µ = 0, and initial state with γ = 1 and h = 1.5 (left panel) and h = 0.5 (right panel) in Eq. (1).The data points were obtained using Eq. (21) for β ∈ [0, 1]. For all β ̸= 0 shown, we obtain an altered effective central charge that agrees well with the analytic… view at source ↗
Figure 5
Figure 5. Figure 5: Entanglement entropy for α < 1. Entanglement entropy in the state (8) following the non-unitary evolution in eq. (2), with β = 2, µ = 0, and initial state with γ = 1 and h = 1.5 in Eq. (1). The data points were obtained using Eq. (21) for α < 1. For all α > 0 shown, we obtain an altered logarithmic scaling of the entanglement entropy that agrees well with the analytical predictions obtained by plugging Eq.… view at source ↗
Figure 6
Figure 6. Figure 6: Correlation functions for α > 1. Correlations in the state (8) following a non-unitary evolution with long-range hopping in eq. (2), α = 2 and initial critical state, h = γ = 1 in Eq. (1). Data points were obtained using Eqs. (38) and (40) for β ∈ [0, 0.5]. For all β ̸= 0 shown, we obtain altered power-law correlations with exponents that agree well with the analytical predictions in Eqs. (45) and (46) (da… view at source ↗
Figure 7
Figure 7. Figure 7: Correlation functions vs measurement strength [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Correlation functions for α = 1. Correlations in the state (8) following a non-unitary evolution with short-range hopping in eq. (2), t = −1, α = 1 and initial state with h = 1.5, γ = 1 in Eq. (1). The left panel shows the analytical behavior of the power￾law exponent of the x and string-correlators after the non-unitary evolution. For large values of the measurement strength, they tend to the one of the e… view at source ↗
Figure 9
Figure 9. Figure 9: Correlation functions for α < 1. Correlations in the state (8) following a non-unitary evolution with long-range (left panel) or short-range (right panel) hopping in eq. (2), β = 2 and initial state with h = 1.5, γ = 1 in Eq. (1). Data points were obtained using Eq. (38) for α ∈ [0.1, 0.7]. For all values of α shown, we obtain altered power-law correlations with exponents that continuously change as a func… view at source ↗
Figure 10
Figure 10. Figure 10: Entanglement spectrum Left panel: entanglement spectrum for differ￾ent values of β, α = 1.5, starting from the initial state with h = 1.5, γ = 1. Although the lowest-lying levels may appear nearly degenerate at first sight, the inset shows the difference between the first two eigenvalues of ρA as a function of system size. Its satu￾ration to a finite nonzero value demonstrates that these levels are not tr… view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of the correlator symbol across [PITH_FULL_IMAGE:figures/full_fig_p032_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of the correlator symbol across [PITH_FULL_IMAGE:figures/full_fig_p033_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Evolution of the correlator symbol across [PITH_FULL_IMAGE:figures/full_fig_p034_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the correlator symbol across [PITH_FULL_IMAGE:figures/full_fig_p035_14.png] view at source ↗
read the original abstract

Imaginary-time evolution by a local Hamiltonian cannot induce a phase transition in one dimension, but longer-range interactions may subvert such constraints. Starting from the ground state of the Kitaev Majorana chain, we modify the wave function by an imaginary-time evolution generated by a quadratic Hamiltonian with power-law couplings that enhance pairing correlations, typically of the form $1/r^{\alpha}$, where $r$ is the distance between two sites. As the state remains Gaussian, entanglement and correlation functions can be computed analytically. We find that the decay exponent $\alpha$ controls three distinct infrared regimes: for $\alpha>1$, the deformation produces only smooth crossovers at finite deformation strength, while the topological regime is reached only asymptotically as the deformation strength tends to infinity. At $\alpha=1$, the deformation induces an immediate flow to the topological phase: an infinitesimal deformation strength drives the system to a topological regime, and in a particular case, an emergent Kramers-Wannier symmetry enforces Ising-like scaling at long distances. For $\alpha<1$, the deformed state shows the same critical-like behavior for all non-zero deformation strength. We observe that even with arbitrarily long-range interactions, these models do not display a sharp phase transition at non-zero deformation strength.

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

0 major / 2 minor

Summary. The manuscript examines imaginary-time evolution of the Kitaev Majorana chain ground state under a quadratic Hamiltonian with power-law couplings decaying as 1/r^α. Because the evolution preserves Gaussianity, the covariance matrix admits an exact closed-form transformation, from which entanglement and correlation functions are computed analytically. The decay exponent α is shown to partition the infrared behavior into three regimes: for α>1 only smooth crossovers occur at finite deformation strength, with the topological phase reached only as strength → ∞; at α=1 an infinitesimal deformation drives the system into the topological regime, and a special case exhibits emergent Kramers-Wannier symmetry with Ising scaling; for α<1 the state remains critical-like for any nonzero strength. The work concludes that arbitrarily long-range interactions still produce no sharp phase transition at finite deformation.

Significance. If the analytic expressions for the covariance matrix hold, the paper supplies an exact, parameter-free classification of long-range deformations in one-dimensional Gaussian topological states. The explicit link between the singularity structure of the Fourier-transformed power-law kernel and the three infrared regimes constitutes a concrete, falsifiable prediction that can be checked numerically or experimentally. The absence of any hidden fitting parameters or self-referential definitions strengthens the result.

minor comments (2)
  1. [Abstract] The abstract refers to “a particular case” in which emergent Kramers-Wannier symmetry appears; the main text should state explicitly which value of α or which form of the deformation Hamiltonian realizes this case (e.g., §3 or Eq. (12)).
  2. Figure captions should indicate the system size and the precise definition of the deformation strength used in each panel so that the claimed crossovers and scaling can be reproduced directly from the plotted data.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment. We are pleased that the referee recognizes the exact solvability via the covariance matrix and the concrete classification of the three infrared regimes controlled by α.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claims follow from direct analytic computation of the covariance matrix under imaginary-time evolution generated by a quadratic power-law Hamiltonian. Gaussianity is preserved by the quadratic form of H (standard property, not a fitted or self-defined assumption), and the three infrared regimes are read off from the small-k singularity of the Fourier transform of the 1/r^α kernel, which changes character at α=1. No parameters are fitted to data and then relabeled as predictions, no self-citations bear the load of the uniqueness or derivation steps, and no ansatz is smuggled in. The derivation is self-contained against the explicit equations for the deformed state.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the preservation of Gaussianity under the specified deformation, which is treated as a domain property of quadratic Hamiltonians rather than derived.

axioms (1)
  • domain assumption The deformed wave function remains Gaussian
    This is invoked to justify analytical computation of entanglement and correlations.

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Works this paper leans on

74 extracted references · 54 canonical work pages

  1. [1]

    Topological order and conformal quantum critical points.Annals of Physics, 310(2):493–551, 2004

    Eddy Ardonne, Paul Fendley, and Eduardo Fradkin. Topological order and conformal quantum critical points.Annals of Physics, 310(2):493–551, 2004. doi:10.1016/j.aop.2004.01.004. 20

    work page doi:10.1016/j.aop.2004.01.004 2004

  2. [2]

    Stochastic Matrix Form

    Claudio Castelnovo, Claudio Chamon, Christopher Mudry, and Pierre Pujol. From quan- tum mechanics to classical statistical physics: Generalized Rokhsar–Kivelson Hamiltoni- ans and the “Stochastic Matrix Form” decomposition.Annals of Physics, 318(2):316–344, August 2005. ISSN 0003-4916. URLhttp://dx.doi.org/10.1016/j.aop.2005.01.006

    work page doi:10.1016/j.aop.2005.01.006 2005

  3. [3]

    A quantum topological phase transition at the microscopic level.Physical Review B, 77:054433, 2008

    Claudio Castelnovo and Claudio Chamon. A quantum topological phase transition at the microscopic level.Physical Review B, 77:054433, 2008. doi:10.1103/PhysRevB.77.054433

    work page doi:10.1103/physrevb.77.054433 2008

  4. [4]

    Rokhsar and Steven A

    Daniel S. Rokhsar and Steven A. Kivelson. Superconductivity and the Quantum Hard- Core Dimer Gas.Phys. Rev. Lett., 61(20):2376–2379, November 1988. ISSN 0031-9007. URLhttp://dx.doi.org/10.1103/PhysRevLett.61.2376

    work page doi:10.1103/physrevlett.61.2376 1988

  5. [5]

    From classical to quantum dynamics at Rokhsar–Kivelson points.Journal of Physics: Condensed Matter, 16(11):S891–S898, March 2004

    C L Henley. From classical to quantum dynamics at Rokhsar–Kivelson points.Journal of Physics: Condensed Matter, 16(11):S891–S898, March 2004. ISSN 1361-648X. URL http://dx.doi.org/10.1088/0953-8984/16/11/045

    work page doi:10.1088/0953-8984/16/11/045 2004

  6. [6]

    Topological order and quantum criticality

    Claudio Castelnovo, Simon Trebst, and Matthias Troyer. Topological order and quantum criticality. pages 169–194, 2010. doi:10.1201/b10273-10

    work page doi:10.1201/b10273-10 2010

  7. [7]

    S. V. Isakov, P. Fendley, A. W. W. Ludwig, S. Trebst, and M. Troyer. Dynam- ics at and near conformal quantum critical points.Phys. Rev. B, 83:125114, Mar

  8. [8]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevB.83.125114

    doi:10.1103/PhysRevB.83.125114. URLhttps://link.aps.org/doi/10.1103/ PhysRevB.83.125114

    work page doi:10.1103/physrevb.83.125114

  9. [9]

    Unpaired Majorana fermions in quantum wires.Physics-Uspekhi, 44 (10S):131–136, October 2001

    A Yu Kitaev. Unpaired Majorana fermions in quantum wires.Physics-Uspekhi, 44 (10S):131–136, October 2001. ISSN 1468-4780. URLhttp://dx.doi.org/10.1070/ 1063-7869/44/10S/S29

    2001

  10. [10]

    New directions in the pursuit of Majorana fermions in solid state systems

    Jason Alicea. New directions in the pursuit of Majorana fermions in solid state systems. Reports on Progress in Physics, 75(7):076501, June 2012. ISSN 1361-6633. URLhttp: //dx.doi.org/10.1088/0034-4885/75/7/076501

    work page doi:10.1088/0034-4885/75/7/076501 2012

  11. [11]

    Long-range interacting quantum systems.Rev

    Nicolò Defenu, Tobias Donner, Tommaso Macrì, Guido Pagano, Stefano Ruffo, and An- drea Trombettoni. Long-range interacting quantum systems.Rev. Mod. Phys., 95:035002, Aug 2023. doi:10.1103/RevModPhys.95.035002. URLhttps://link.aps.org/doi/10. 1103/RevModPhys.95.035002

    work page doi:10.1103/revmodphys.95.035002 2023

  12. [12]

    Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions.Physics Reports, 1074:1– 92, 2024

    Nicolò Defenu, Alessio Lerose, and Silvia Pappalardi. Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions.Physics Reports, 1074:1– 92, 2024. ISSN 0370-1573. doi:https://doi.org/10.1016/j.physrep.2024.04.005. URL https://www.sciencedirect.com/science/article/pii/S0370157324001406. Out- of-equilibrium dynamics of quantum ma...

    work page doi:10.1016/j.physrep.2024.04.005 2024

  13. [13]

    Gorshkov, and Guido Pupillo

    Davide Vodola, Luca Lepori, Elisa Ercolessi, Alexey V. Gorshkov, and Guido Pupillo. Kitaev chains with long-range pairing.Phys. Rev. Lett., 113:156402, Oct

  14. [14]

    URLhttps://link.aps.org/doi/10

    doi:10.1103/PhysRevLett.113.156402. URLhttps://link.aps.org/doi/10. 1103/PhysRevLett.113.156402

    work page doi:10.1103/physrevlett.113.156402

  15. [15]

    Long-range Ising and Kitaev models: phases, correlations and edge modes.New J

    Davide Vodola, Luca Lepori, Elisa Ercolessi, and Guido Pupillo. Long-range Ising and Kitaev models: phases, correlations and edge modes.New J. Phys., 18(1): 015001, December 2015. ISSN 1367-2630. doi:10.1088/1367-2630/18/1/015001. URL https://doi.org/10.1088/1367-2630/18/1/015001

    work page doi:10.1088/1367-2630/18/1/015001 2015

  16. [16]

    Z.-X. Gong, M. F. Maghrebi, A. Hu, M. L. Wall, M. Foss-Feig, and A. V. Gorshkov. Topological phases with long-range interactions.Phys. Rev. B, 93(4):041102, January

  17. [17]

    URLhttp://dx.doi.org/10.1103/PhysRevB.93.041102

    ISSN 2469-9969. URLhttp://dx.doi.org/10.1103/PhysRevB.93.041102. 21

    work page doi:10.1103/physrevb.93.041102

  18. [18]

    Fate of Measurement-Induced Phase Transition in Long-Range Interactions.Phys

    Takaaki Minato, Koudai Sugimoto, Tomotaka Kuwahara, and Keiji Saito. Fate of Measurement-Induced Phase Transition in Long-Range Interactions.Phys. Rev. Lett., 128(1):010603, January 2022. ISSN 1079-7114. URLhttp://dx.doi.org/10.1103/ PhysRevLett.128.010603

    2022

  19. [19]

    Maxwell Block, Yimu Bao, Soonwon Choi, Ehud Altman, and Norman Y. Yao. Measurement-Induced Transition in Long-Range Interacting Quantum Circuits.Phys. Rev. Lett., 128(1):010604, January 2022. ISSN 1079-7114. URLhttp://dx.doi.org/ 10.1103/PhysRevLett.128.010604

    work page doi:10.1103/physrevlett.128.010604 2022

  20. [20]

    Müller, S

    T. Müller, S. Diehl, and M. Buchhold. Measurement-Induced Dark State Phase Transi- tions in Long-Ranged Fermion Systems.Phys. Rev. Lett., 128(1):010605, January 2022. ISSN 1079-7114. URLhttp://dx.doi.org/10.1103/PhysRevLett.128.010605

    work page doi:10.1103/physrevlett.128.010605 2022

  21. [21]

    Extended kitaev chain with longer-range hopping and pairing.Phys

    Antonio Alecce and Luca Dell’Anna. Extended kitaev chain with longer-range hopping and pairing.Phys. Rev. B, 95:195160, May 2017. doi:10.1103/PhysRevB.95.195160. URL https://link.aps.org/doi/10.1103/PhysRevB.95.195160

    work page doi:10.1103/physrevb.95.195160 2017

  22. [22]

    Esteve, Fernando Falceto, and Amilcar R

    Filiberto Ares, José G. Esteve, Fernando Falceto, and Amilcar R. de Queiroz. En- tanglement entropy in the long-range kitaev chain.Phys. Rev. A, 97:062301, Jun

  23. [23]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevA.97.062301

    doi:10.1103/PhysRevA.97.062301. URLhttps://link.aps.org/doi/10.1103/ PhysRevA.97.062301

    work page doi:10.1103/physreva.97.062301

  24. [24]

    Esteve, Fernando Falceto, and Amilcar R

    Filiberto Ares, Jose G. Esteve, Fernando Falceto, and Amilcar R. de Queiroz. Entangle- ment in fermionic chains with finite range coupling and broken symmetries.Phys. Rev. A, 92(4):042334, 2015. doi:10.1103/PhysRevA.92.042334

    work page doi:10.1103/physreva.92.042334 2015

  25. [25]

    Current transport properties and phase diagram of a kitaev chain with long-range pairing.Physical Review B, 97(15): 155113, 2018

    Domenico Giuliano, Simone Paganelli, and Luca Lepori. Current transport properties and phase diagram of a kitaev chain with long-range pairing.Physical Review B, 97(15): 155113, 2018. doi:10.1103/PhysRevB.97.155113

    work page doi:10.1103/physrevb.97.155113 2018

  26. [26]

    Jäger, Luca Dell’Anna, and Giovanna Morigi

    Simon B. Jäger, Luca Dell’Anna, and Giovanna Morigi. Edge states of the long- range kitaev chain: An analytical study.Physical Review B, 102(3):035152, 2020. doi:10.1103/PhysRevB.102.035152

    work page doi:10.1103/physrevb.102.035152 2020

  27. [27]

    Disordered kitaev chains with long-range pairing.Journal of Physics: Condensed Matter, 29(11):115401, 2017

    Xiaoming Cai. Disordered kitaev chains with long-range pairing.Journal of Physics: Condensed Matter, 29(11):115401, 2017. doi:10.1088/1361-648X/aa5a39. URLhttps: //doi.org/10.1088/1361-648X/aa5a39

    work page doi:10.1088/1361-648x/aa5a39 2017

  28. [28]

    Logarithmic, fractal and volume-law entanglement in a kitaev chain with long-range hopping and pairing

    Andrea Solfanelli, Stefano Ruffo, Sauro Succi, and Nicolò Defenu. Logarithmic, fractal and volume-law entanglement in a kitaev chain with long-range hopping and pairing. Journal of High Energy Physics, 2023(5):66, 2023. doi:10.1007/JHEP05(2023)066

    work page doi:10.1007/jhep05(2023)066 2023

  29. [29]

    Quantum heat engine with long-range advantages.New J

    Andrea Solfanelli, Guido Giachetti, Michele Campisi, Stefano Ruffo, and Nicolò Defenu. Quantum heat engine with long-range advantages.New J. Phys., 25(3):033030, March

  30. [30]

    doi:10.1088/1367-2630/acc04e

    ISSN 1367-2630. doi:10.1088/1367-2630/acc04e. URLhttps://doi.org/10.1088/ 1367-2630/acc04e

    work page doi:10.1088/1367-2630/acc04e

  31. [31]

    Cinnirella, Andrea Nava, Gabriele Campagnano, and Domenico Giuliano

    Emmanuele G. Cinnirella, Andrea Nava, Gabriele Campagnano, and Domenico Giuliano. Phase diagram of the disordered kitaev chain with long-range pairing connected to ex- ternal baths.Phys. Rev. B, 111:155149, Apr 2025. doi:10.1103/PhysRevB.111.155149. URLhttps://link.aps.org/doi/10.1103/PhysRevB.111.155149. 22

    work page doi:10.1103/physrevb.111.155149 2025

  32. [32]

    Baghran, R

    R. Baghran, R. Jafari, and A. Langari. Competition of long-range interactions and noise atarampedquenchdynamicalquantumphasetransition: Thecaseofthelong-rangepair- ingkitaevchain.Phys. Rev. B,110:064302, Aug2024. doi:10.1103/PhysRevB.110.064302. URLhttps://link.aps.org/doi/10.1103/PhysRevB.110.064302

    work page doi:10.1103/physrevb.110.064302

  33. [33]

    Agarwal and Yogesh N

    Kaustubh S. Agarwal and Yogesh N. Joglekar. Pt-symmetry breaking in a kitaev chain with one pair of gain-loss potentials.Physical Review A, 104(2):022218, 2021. doi:10.1103/PhysRevA.104.022218

    work page doi:10.1103/physreva.104.022218 2021

  34. [34]

    Localization and multifractal properties of the long-range kitaev chain in the presence of an aubry-andré-harper modulation.Physical Review B, 106(2):024204, 2022

    Joana Fraxanet, Utso Bhattacharya, Tobias Grass, Maciej Lewenstein, and Alexandre Dauphin. Localization and multifractal properties of the long-range kitaev chain in the presence of an aubry-andré-harper modulation.Physical Review B, 106(2):024204, 2022. doi:10.1103/PhysRevB.106.024204

    work page doi:10.1103/physrevb.106.024204 2022

  35. [35]

    Correlations, long-range entanglement, and dynamics in long-range kitaev chains.Physical Review B, 106(15):155126, 2022

    Gianluca Francica and Luca Dell’Anna. Correlations, long-range entanglement, and dynamics in long-range kitaev chains.Physical Review B, 106(15):155126, 2022. doi:10.1103/PhysRevB.106.155126

    work page doi:10.1103/physrevb.106.155126 2022

  36. [36]

    Disordered Kitaev chain with long-range pairing: Loschmidt echo revivals and dynamical phase transitions.J

    Utkarsh Mishra, R Jafari, and Alireza Akbari. Disordered Kitaev chain with long-range pairing: Loschmidt echo revivals and dynamical phase transitions.J. Phys. A: Math. Theor., 53(37):375301, August 2020. ISSN 1751-8121. doi:10.1088/1751-8121/ab97de. URLhttps://doi.org/10.1088/1751-8121/ab97de

    work page doi:10.1088/1751-8121/ab97de 2020

  37. [37]

    Ignacio Cirac, and Antonio Acín

    Senaida Hernández-Santana, Christian Gogolin, J. Ignacio Cirac, and Antonio Acín. Correlation Decay in Fermionic Lattice Systems with Power-Law Inter- actions at Nonzero Temperature.Phys. Rev. Lett., 119(11):110601, September

  38. [38]

    URLhttps://link.aps.org/doi/10

    doi:10.1103/PhysRevLett.119.110601. URLhttps://link.aps.org/doi/10. 1103/PhysRevLett.119.110601

    work page doi:10.1103/physrevlett.119.110601

  39. [39]

    Multipartite En- tanglement in Topological Quantum Phases.Phys

    Luca Pezzè, Marco Gabbrielli, Luca Lepori, and Augusto Smerzi. Multipartite En- tanglement in Topological Quantum Phases.Phys. Rev. Lett., 119(25):250401, Decem- ber 2017. doi:10.1103/PhysRevLett.119.250401. URLhttps://link.aps.org/doi/10. 1103/PhysRevLett.119.250401

    work page doi:10.1103/physrevlett.119.250401 2017

  40. [40]

    Maghrebi, Zhe-Xuan Gong, Michael Foss-Feig, and Alexey V

    Mohammad F. Maghrebi, Zhe-Xuan Gong, Michael Foss-Feig, and Alexey V. Gorshkov. Causality and quantum criticality in long-range lattice models.Phys. Rev. B, 93(12): 125128, March 2016. doi:10.1103/PhysRevB.93.125128. URLhttps://link.aps.org/ doi/10.1103/PhysRevB.93.125128

    work page doi:10.1103/physrevb.93.125128 2016

  41. [41]

    Universal dynam- ical scaling of long-range topological superconductors.Phys

    Nicolò Defenu, Giovanna Morigi, Luca Dell’Anna, and Tilman Enss. Universal dynam- ical scaling of long-range topological superconductors.Phys. Rev. B, 100(18):184306, November 2019. doi:10.1103/PhysRevB.100.184306. URLhttps://link.aps.org/doi/ 10.1103/PhysRevB.100.184306

    work page doi:10.1103/physrevb.100.184306 2019

  42. [42]

    PhilippUhrich, Nicolò Defenu, RouhollahJafari, andJad C. Halimeh. Out-of-equilibrium phase diagram of long-range superconductors.Phys. Rev. B, 101(24):245148, June

  43. [43]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevB.101.245148

    doi:10.1103/PhysRevB.101.245148. URLhttps://link.aps.org/doi/10.1103/ PhysRevB.101.245148

    work page doi:10.1103/physrevb.101.245148

  44. [44]

    Information propagation and equilibration in long-range Kitaev chains.Phys

    Mathias Van Regemortel, Dries Sels, and Michiel Wouters. Information propagation and equilibration in long-range Kitaev chains.Phys. Rev. A, 93(3):032311, March

  45. [45]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevA.93.032311

    doi:10.1103/PhysRevA.93.032311. URLhttps://link.aps.org/doi/10.1103/ PhysRevA.93.032311. 23

    work page doi:10.1103/physreva.93.032311

  46. [46]

    CriticalphenomenaandKibble–Zurekscalinginthelong-rangequantumIsingchain.New J

    Daniel Jaschke, Kenji Maeda, Joseph D Whalen, Michael L Wall, and Lincoln D Carr. CriticalphenomenaandKibble–Zurekscalinginthelong-rangequantumIsingchain.New J. Phys., 19(3):033032, March 2017. ISSN 1367-2630. doi:10.1088/1367-2630/aa65bc. URLhttps://doi.org/10.1088/1367-2630/aa65bc

    work page doi:10.1088/1367-2630/aa65bc 2017

  47. [47]

    Singular dynamics and emergence of nonlocality in long-range quantum models.J

    L Lepori, A Trombettoni, and D Vodola. Singular dynamics and emergence of nonlocality in long-range quantum models.J. Stat. Mech., 2017(3):033102, March 2017. ISSN 1742-

    2017

  48. [48]

    URLhttps://doi.org/10.1088/1742-5468/ aa569d

    doi:10.1088/1742-5468/aa569d. URLhttps://doi.org/10.1088/1742-5468/ aa569d

    work page doi:10.1088/1742-5468/aa569d

  49. [49]

    Entanglementinthexyspinchain.Journal of Physics A: Mathematical and General, 38(13):2975, mar 2005

    ARIts, B-QJin, andVEKorepin. Entanglementinthexyspinchain.Journal of Physics A: Mathematical and General, 38(13):2975, mar 2005. doi:10.1088/0305-4470/38/13/011. URLhttps://doi.org/10.1088/0305-4470/38/13/011

    work page doi:10.1088/0305-4470/38/13/011 2005

  50. [50]

    Percy Deift, Alexander Its, and Igor Krasovsky. Toeplitz matrices and toeplitz de- terminants under the impetus of the ising model: Some history and some recent results.Communications on Pure and Applied Mathematics, 66(9):1360–1438, 2013. doi:10.1002/cpa.21467

    work page doi:10.1002/cpa.21467 2013

  51. [51]

    Reduced density matrices and entanglement entropy in free lattice models.Journal of Physics A: Mathematical and Theoretical, 42(50):504003,

    Ingo Peschel and Viktor Eisler. Reduced density matrices and entanglement entropy in free lattice models.Journal of Physics A: Mathematical and Theoretical, 42(50):504003,

  52. [52]

    doi:10.1088/1751-8113/42/50/504003

    work page doi:10.1088/1751-8113/42/50/504003

  53. [53]

    Calculation of reduced density matrices from correlation functions.J

    Ingo Peschel. Calculation of reduced density matrices from correlation functions.J. Phys. A, 36(14):L205, 2003. doi:10.1088/0305-4470/36/14/101

    work page doi:10.1088/0305-4470/36/14/101 2003

  54. [54]

    Pasquale Calabrese and John L. Cardy. Entanglement entropy and quantum field theory. J. Stat. Mech., 0406:P06002, 2004. doi:10.1088/1742-5468/2004/06/P06002

    work page doi:10.1088/1742-5468/2004/06/p06002 2004

  55. [55]

    Biao-Qing Jin and Vladimir E. Korepin. Quantum spin chain, toeplitz determinants and the fisher-hartwig conjecture.Journal of Statistical Physics, 116(1-4):79–95, 2004. doi:10.1023/B:JOSS.0000037230.37166.42

    work page doi:10.1023/b:joss.0000037230.37166.42 2004

  56. [56]

    Toeplitz determinants with singular generating functions.Advances in Chemical Physics, 15:333–353, 1968

    Michael E Fisher and R E Hartwig. Toeplitz determinants with singular generating functions.Advances in Chemical Physics, 15:333–353, 1968

    1968

  57. [57]

    R. E. Hartwig and M. E. Fisher. Asymptotic behavior of toeplitz matrices and determi- nants.Archive for Rational Mechanics and Analysis, 32(3):190–225, 1969

    1969

  58. [58]

    Aspects of toeplitz determinants

    Igor Krasovsky. Aspects of toeplitz determinants. In D. Lenz, F. Sobieczky, and W. Woess, editors,Random Walks, Boundaries and Spectra, pages 305–324. Springer, Basel, 2011

    2011

  59. [59]

    Filiberto Ares, M. A. Rajabpour, and Jacopo Viti. Exact full counting statistics for the staggered magnetization and the domain walls in the xy spin chain.Phys. Rev. E, 103(4): 042107, 2021. doi:10.1103/PhysRevE.103.042107. URLhttps://arxiv.org/abs/2012. 14012

    work page doi:10.1103/physreve.103.042107 2021

  60. [60]

    Asymptotics of toeplitz, hankel, and toeplitz+hankel determinants with fisher-hartwig singularities.Annals of Mathematics, 174(2):1243–1299, 2011

    Percy Deift, Alexander Its, and Igor Krasovsky. Asymptotics of toeplitz, hankel, and toeplitz+hankel determinants with fisher-hartwig singularities.Annals of Mathematics, 174(2):1243–1299, 2011. doi:10.4007/annals.2011.174.2.12

    work page doi:10.4007/annals.2011.174.2.12 2011

  61. [61]

    Majorana chain and ising model – (non-invertible) translations, anomalies, and emanant symmetries.SciPost Physics, 16:064, 2024

    Nathan Seiberg and Shu-Heng Shao. Majorana chain and ising model – (non-invertible) translations, anomalies, and emanant symmetries.SciPost Physics, 16:064, 2024. doi:10.21468/SciPostPhys.16.3.064. 24

    work page doi:10.21468/scipostphys.16.3.064 2024

  62. [62]

    Entanglementspectrumoftopologicalinsulatorsandsuperconductors

    LukaszFidkowski. Entanglementspectrumoftopologicalinsulatorsandsuperconductors. Physical Review Letters, 104(13):130502, 2010. doi:10.1103/PhysRevLett.104.130502

    work page doi:10.1103/physrevlett.104.130502 2010

  63. [63]

    HuiLiandF.D.M.Haldane. Entanglementspectrumasageneralizationofentanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states.Physical Review Letters, 101:010504, 2008. doi:10.1103/PhysRevLett.101.010504

    work page doi:10.1103/physrevlett.101.010504 2008

  64. [64]

    Turner, and Masaki Oshikawa

    Frank Pollmann, Erez Berg, Ari M. Turner, and Masaki Oshikawa. Entanglement spec- trum of a topological phase in one dimension.Physical Review B, 81:064439, 2010. doi:10.1103/PhysRevB.81.064439

    work page doi:10.1103/physrevb.81.064439 2010

  65. [65]

    Entanglement hamiltonians in two-dimensional conformal field theory.Journal of Statistical Mechanics: Theory and Experiment, 2016(12):123103,

    John Cardy and Erik Tonni. Entanglement hamiltonians in two-dimensional conformal field theory.Journal of Statistical Mechanics: Theory and Experiment, 2016(12):123103,

    2016

  66. [66]

    doi:10.1088/1742-5468/2016/12/123103

    work page doi:10.1088/1742-5468/2016/12/123103 2016

  67. [67]

    Entanglement hamiltonian of the 1+1-dimensional free compact boson.Journal of Statistical Mechanics: Theory and Ex- periment, 2020(12):123103, 2020

    Ananda Roy, Frank Pollmann, and Hubert Saleur. Entanglement hamiltonian of the 1+1-dimensional free compact boson.Journal of Statistical Mechanics: Theory and Ex- periment, 2020(12):123103, 2020. doi:10.1088/1742-5468/aba498. URLhttps://doi. org/10.1088/1742-5468/aba498

    work page doi:10.1088/1742-5468/aba498 2020

  68. [68]

    Work in preparation

    Yue Liu, Nandagopal Manoj, and Jason Alicea. Work in preparation

  69. [69]

    Potter, and Norman Y

    Michael Foss-Feig, Guido Pagano, Andrew C. Potter, and Norman Y. Yao. Progress in Trapped-Ion Quantum Simulation.Ann. Rev. Condensed Matter Phys., 16(1):145–172,

  70. [70]

    doi:10.1146/annurev-conmatphys-032822-045619

    work page doi:10.1146/annurev-conmatphys-032822-045619

  71. [71]

    Sara Murciano, Pablo Sala, Yue Liu, Roger S. K. Mong, and Jason Alicea. Measurement-altered ising quantum criticality.Phys. Rev. X, 13:041042, Dec

  72. [72]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevX.13.041042

    doi:10.1103/PhysRevX.13.041042. URLhttps://link.aps.org/doi/10.1103/ PhysRevX.13.041042

    work page doi:10.1103/physrevx.13.041042

  73. [73]

    Dissipative Preparation of Many-Body Quantum States: Towards Practical Quantum Advantage.Physics, 1:010901, 2025

    Lin Lin. Dissipative Preparation of Many-Body Quantum States: Towards Practical Quantum Advantage.Physics, 1:010901, 2025. doi:10.1063/5.0283315. A Emergent Kramers-Wannier Symmetry UnderaKramers-Wannier(KW)transformation, theMajoranafermionssimplygettranslated by one Majorana site in real space. In momentum space, this corresponds toγk,e 7→γ k,o and γk,o...

    work page doi:10.1063/5.0283315 2025

  74. [74]

    (63) The critical value ofβis β∗ = 2√ 4 +π 2 arctanh √ 4 +π 2 2 +π ! ≈0.492.(64) 26 If we compute Eq

    Using (60), u0±(β) = cosh 1 2 p 4 +π 2β − 2 sinh 1 2 √ 4 +π 2β √ 4 +π 2 , v 0±(β) =± πsinh 1 2 √ 4 +π 2β √ 4 +π 2 . (63) The critical value ofβis β∗ = 2√ 4 +π 2 arctanh √ 4 +π 2 2 +π ! ≈0.492.(64) 26 If we compute Eq. (59) and we plot them for an arbitrary value ofh,β∗(h)as a function ofk, we observe that they are not equal ton(k, β)andg(k, β), respective...