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Exact strong zero modes in quantum circuits and spin chains with non-diagonal boundary conditions

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Pith's one-line read Exactly conserved boundary operators exist in the integrable brick-wall circuit and the XXZ spin chain even when boundary fields break the bulk U(1) symmetry, provided the left field lies in the x-y plane.

desk verdict First exact strong zero mode for U(1)-breaking integrable boundaries; algebraic core is solid, coherence-time claim needs one more piece of evidence. read the letter →

arxiv 2511.05490 v4 pith:5EOOCZSA submitted 2025-11-07 cond-mat.stat-mech math-phmath.MPquant-ph

Sascha Gehrmann , Fabian H. L. Essler This is my paper · ORCID
classification cond-mat.stat-mechmath-phmath.MPquant-ph MSC 82B2082B2381R12
keywords exactstrongzeromodesopenboundaryconditionsXXZspinchainintegrablequantumcircuitsedgecoherencetimesmatrixproductoperatorsasymmetricsimpleexclusionprocessU(1)symmetrybreaking
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that exact strong zero modes—operators that commute exactly with the time evolution and are exponentially localised near a boundary—exist in the integrable brick-wall quantum circuit and the spin-1/2 XXZ chain with the most general integrable open boundary conditions, provided the left boundary magnetic field lies in the x-y plane. This removes a long-standing restriction: such modes were previously known to exist only when the boundary respected a global Z2 or U(1) symmetry. The exact zero mode forces infinite-temperature autocorrelations of boundary operators such as σ^z_1 to plateau at a finite value set by their overlap with the mode, signalling infinite boundary coherence times. The authors also show that under the mapping to the asymmetric simple exclusion process the mode loses its spatial locality, so it cannot shape the stochastic dynamics of that process.

What carries the argument

The central object is the double-row transfer matrix T(u) of the six-vertex model with Sklyanin's open-boundary K-matrices, which generates the conserved charges of both the circuit and the Hamiltonian. Evaluating T at the special point u*=iπ/2 makes the right boundary operator proportional to σ^z, and fixing the left boundary parameter ξ^(L)=iπ/2 makes the resulting conserved operator exponentially localised at that edge. The operator is written as a matrix-product operator (MPO) with matrices A±; the transfer matrix à of the MPO norm turns out to be independent of the ± labels, and its largest eigenvalue d6<1, given explicitly, controls the exponential decay of the Hilbert–Schmidt norms an

What would settle it

Compute the infinite-temperature autocorrelation C_{σ^z_1}(t) for N=14,16,18 in the XXZ chain with h_1=(1,0.1,0), h_N=(0.25,0.5,1), and check whether the late-time plateau approaches |c_Ψ1|² as the system grows; alternatively, compute the autocorrelation of O' = σ^z_1 − c_1 Ψ directly and test whether its late-time value vanishes with N.

Watch

Extended reading notes

Core claim

The central claim is that coexistence of an exactly conserved boundary operator with non-diagonal boundary terms is possible. For the XXZ Hamiltonian with boundary fields, an ESZM Ψ with [Ψ,H]=0 exists precisely when the left boundary field has no z-component, h^z_1=0; the right boundary field may be arbitrary. The same statement holds for the stroboscopic evolution of the integrable brick-wall circuit obtained by Trotterising the XXZ chain. The operator is constructed from the double-row transfer matrix at the special value u*=iπ/2, where the right boundary K-matrix collapses to σ^z, and with the left boundary parameter fixed to ξ^(L)=iπ/2 to guarantee exponential localisation. The paper gi

Load-bearing premise

The 'infinite boundary coherence' claim rests on the expectation that the autocorrelation of the part of a boundary operator orthogonal to the zero mode decays to a value vanishing with system size—verified numerically only for σ^z_1 at N=6,8,10.

Editorial extensions

If this is right

  • For any left boundary field in the x-y plane and arbitrary right boundary field, the XXZ chain and its Trotter circuit carry an exactly conserved operator localised at the left edge, so σ^z_1, σ^x_1σ^z_2, and σ^y_1σ^z_2 autocorrelations saturate at values set by c_Ψ1 and c_Ψ2.
  • The brick-wall circuit requires only local gates, so the h^z_1=0 condition is straightforward to implement on a quantum simulator; the predicted plateau in edge autocorrelations is directly measurable.
  • Under the similarity transformation to the asymmetric simple exclusion process, the ESZM's Hilbert–Schmidt weight at any fixed boundary site vanishes as N→∞, so the process does not inherit boundary coherence from the spin chain.
  • Because the zero mode is exactly conserved for every finite N, there is no finite-size crossover: the plateau value is exact as soon as the boundary geometry is fixed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit decay rate d6(η,δ) gives an analytic handle on the edge-mode localisation length; one testable extension is to compare it with the correlation length of the bulk XXZ chain and check whether the localisation length diverges at the isotropic point.
  • The result that h^z_1=0 is sufficient suggests a broader design principle: an ESZM survives any boundary perturbation that preserves a single discrete symmetry at the edge, even if it breaks all continuous symmetries of the bulk.
  • The ASEP non-locality result indicates that exact boundary coherence is coordinate-dependent; for dissipative or stochastic embeddings of integrable models, one should expect similar washing-out of the edge mode.
  • The plateau prediction relies on the decay of the orthogonal part of the autocorrelation, which the paper checks only for small N; a Bethe-ansatz or transfer-matrix computation of the full autocorrelation would confirm the plateau for larger systems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs exact strong zero mode operators Ψ for an integrable brick-wall quantum circuit and, in the Trotter limit, for the spin-1/2 XXZ chain with open boundary terms, under the condition that the left boundary field lies in the x-y plane (h1^z=0) while the right boundary is general. Ψ is built from the derivative of the Sklyanin-type transfer matrix at u*=iπ/2, with the left boundary parameter fixed to ξ^(L)=iπ/2. The authors give an MPO representation (Appendices A–C), prove exponential localization in Hilbert–Schmidt norm with closed-form decay rate d6 (Eqs. (32)–(33)), and verify this numerically. They then argue that the ESZM produces non-decaying infinite-temperature autocorrelation plateaus for boundary observables σ^z_1, σ^x_1σ^z_2, σ^y_1σ^z_2 (Eqs. (42)–(45)), with numerical support shown for σ^z_1. Finally, under the similarity transformation to the asymmetric simple exclusion process, the same operator loses its boundary-localized structure, and they conclude it is not dynamically significant in ASEP.

Significance. The paper's main algebraic claim is well supported and technically impressive: exact ESZM operators are constructed in explicit MPO form for both a Floquet circuit and the XXZ chain with non-diagonal boundaries, the localization length is given by a closed-form d6, and the normalization is controlled in finite volume (Appendices A–C). The numerical data in Fig. 4 confirm the exponential localization. The discussion of the ASEP map is a useful negative result: the ESZM delocalizes under the transformation, so its dynamical role in ASEP is not inherited. If the coherence-time prediction can be placed on a firmer footing, this will be a solid contribution to the strong-zero-mode literature.

major comments (2)
  1. [§3.1, after Eq. (42)] The 'infinite boundary coherence times' claim rests on the assertion that C_{O'}(t) 'is expected to decay in time to a value that vanishes as N→∞'. This is not a consequence of [Ψ,H]=0: in an integrable chain O' can overlap with other conserved operators (the transfer-matrix charges Q^(n), and possibly quasilocal conserved operators), and those overlaps control the infinite-time average of C_{O'}. The numerical test in Fig. 5 covers only σ^z_1 for N=6,8,10 and one set of boundary fields; the plateaus predicted for σ^x_1σ^z_2 and σ^y_1σ^z_2 in Eqs. (44)–(45) are not checked. Please either prove the absence of such overlaps or provide data for all three operators at several N demonstrating that the late-time values approach (45). As written the physical conclusion is stronger than the evidence.
  2. [§2.2/Eq. (24)] The construction of Ψ is not fully self-contained. The key identity (18), the choice u*=iπ/2, the condition ξ^(L)=iπ/2, and the MPO representation are taken from Refs. [26,49], both 'to appear'. Since [Ψ,H]=0 is the basis of the paper, the authors should either derive these steps in the main text/appendix or state precisely which propositions from the forthcoming papers are used and why they apply here. The explicit matrix elements in Appendix A are helpful but do not by themselves show that the operator obtained from (24) commutes with U.
minor comments (4)
  1. [§3.1, Fig. 5] The text says the autocorrelation is shown for δ=2.5, but the XXZ model is defined after taking δ→0; the figure caption says ∆=2.5. Please correct the typo.
  2. [Abstract/§1] 'General open boundary conditions' is broader than the proven statement; the left boundary field is required to satisfy h1^z=0 (Eq. (3)). Suggest rewording to avoid overclaim.
  3. [Appendix B] The normalization expression has notational artifacts (e.g. λ_L^4, λ_L^6, dangling factors) that make it hard to read; please typeset carefully and check the formula.
  4. [Eq. (26)] Please state explicitly that each Ψ_j carries a non-identity operator on site j (r_{2j-1}, r_{2j} ∈ {x,y,z}); this justifies the contributions to Eq. (45) and aids the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ESZM construction and plateau predictions are derived analytically and verified numerically, not fitted or imported from self-citations.

full rationale

The paper's central claims—an exact conserved operator Ψ with [Ψ,H]=0, exponential localization with HS norms decaying as d6^(j-1), and plateau values |c1|^2 for boundary autocorrelations—are obtained from an explicit transfer-matrix/MPO construction. The special parameters u*=iπ/2 and ξ^(L)=iπ/2 are fixed by solving the stated conditions (18) and (21), not by fitting data. The MPO matrix elements, diagonalization of A, and eigenvalues d6 are given in appendices, so the derivation is self-contained. The autocorrelation plateaus in Eqs. (42), (45) are computed from closed-form overlap formulas and then compared with exact diagonalization for σ^z_1 (Fig. 5); this is verification, not fitting labeled as prediction. Refs. [26,49] are methodological citations to same-group work in preparation, but the manuscript supplies the needed MPO expressions and diagonalization details, and those references do not assume the target result. The only soft spot is the assertion that the orthogonal part C_{O'}(t) decays to a value vanishing with N; this is a dynamical assumption, not a circular reduction of the derived ESZM algebra. No step in the derivation is equivalent to its own input by construction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities; its claims rest on standard integrability machinery plus two hand-chosen parameters (u*, ξ^(L)) that pin down the ESZM and its localization. The normalization of the ASEP-mapped operator is a technical choice rather than an ad hoc entity.

free parameters (4)
  • u* = iπ/2
    Special transfer-matrix point chosen via condition (18) so that K^(R)_0(u*) ∝ σ^z, enabling the ESZM construction.
  • ξ^(L) = iπ/2
    Left-boundary K-matrix parameter fixed by the localization requirement (21)/(23); this corresponds to setting h^z_1=0.
  • Λ = free (e.g., 1.1 in numerics)
    Free parameter of the similarity transformation (51) mapping XXZ to ASEP; it does not affect the existence of the ESZM but enters the normalization analysis.
  • N_ASEP normalization factor = piecewise in η (Eq. (55) with two regimes)
    Chosen so that the mapped operator has finite nonzero HS norm in the thermodynamic limit; the piecewise form is derived, not fitted.
assumptions (4)
  • domain assumption Reflection-equation integrability framework (Sklyanin double-row transfer matrix)
    The commuting family of transfer matrices (16) and the identification U=T(iδ/2,iδ/2) rely on the standard integrable boundary framework from Refs [40,41].
  • domain assumption Boundary K-matrix classification (de Vega–González-Ruiz)
    The most general single-qubit boundary gates compatible with integrability are parametrized by Eq. (13); the paper assumes this classification covers its circuit.
  • standard math Analytic continuation and differentiation at u*=iπ/2
    The construction differentiates the transfer matrix at an off-axis point; this requires the transfer matrix to be analytic in u, standard for the six-vertex R-matrix.
  • domain assumption SZM localization criterion via Hilbert-Schmidt norms
    The paper equates exponential decay of ||Ψ_j||² with spatial localization of Ψ; this is a standard but non-trivial measure for operators of growing support.

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Pith. "Pith review of Exact strong zero modes in quantum circuits and spin chains with non-diagonal boundary conditions." pith.science (2026). https://pith.science/paper/5EOOCZSA

@misc{pith2026251105490,
  author       = {Pith},
  title        = {Pith review of: Exact strong zero modes in quantum circuits and spin chains with non-diagonal boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5EOOCZSA}},
  note         = {Machine review of arXiv:2511.05490}
}
read the original abstract

We construct exact strong zero mode operators (ESZM) in integrable quantum circuits and the spin-1/2 XXZ chain for general open boundary conditions, which break the bulk U(1) symmetry of the time evolution operators. We show that the ESZM is localized around one of the boundaries and induces infinite boundary coherence times. Finally, we prove that the ESZM becomes spatially non-local under the map that relates the spin-1/2 XXZ chain to the asymmetric simple exclusion process, which suggests that it does not play a significant role in the dynamics of the latter.

Figures

Figures reproduced from arXiv: 2511.05490 by the authors.

Figure 1
Figure 1. Allowed directions for the boundary magnetic fields [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Schematic illustration of the quantum circuits analyzed in this study. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Graphical representation of the double row-to-row transfer matrix. We [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Left: Hilbert-Schmidt Norm of Ψj for N = 30, η = 1.25 and δ = 0.3 while the boundary parameters are set to be ξ (R) = 0.28, s (R) = 0.32, s (L) = 0.45, φ (R) = 0.67, φ (L) = 0.97 . Blue dots are the numerical computed data points while the blue line is the predicted as…
Figure 5
Figure 5. Figure 5: We observe that C σ z 1 0 (t) does not decay to zero but approaches a finite value at late times. The limiting value is in good agreement with the one predicted by the argument presented in eqn (42) and (45). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 5
Figure 5. Figure 5: Infinite temperature autocorrelation function for ∆ = 2 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Dynamic rules for the ASEP with open boundary conditions. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Normalized Hilbert Schmidt norms of ΨASEP j for j = 1, 2, 3, 4 as functions of the system size N. We observe that the contribution of ΨASEP j becomes negligible as N increases at fixed j, implying that that (58) is not localised in the vicinity of the boundary. The sys…

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Reference graph

Works this paper leans on

74 extracted references · 36 canonical work pages · cited by 4 Pith papers

  1. [1]

    Gu and X.-G

    Z.-C. Gu and X.-G. Wen,Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order, Physical Review B80(15) (2009), doi:10.1103/physrevb.80.155131

    doi:10.1103/physrevb.80.155131 2009

  2. [2]

    Pollmann, E

    F. Pollmann, E. Berg, A. M. Turner and M. Oshikawa,Symmetry protection of topological phases in one-dimensional quantum spin systems, Physical Review B85(7) (2012), doi:10.1103/physrevb.85.075125. 17 SciPost Physics Submission

    doi:10.1103/physrevb.85.075125 2012

  3. [3]

    A. Y. Kitaev,Unpaired majorana fermions in quantum wires, Physics-uspekhi44(10S), 131 (2001), doi:10.1070/1063-7869/44/10S/S29

    doi:10.1070/1063-7869/44/10s/s29 2001

  4. [4]

    P. Fendley,Parafermionic edge zero modes in Zn-invariant spin chains, Jour- nal of Statistical Mechanics: Theory and Experiment2012(11), P11020 (2012), doi:10.1088/1742-5468/2012/11/P11020

    doi:10.1088/1742-5468/2012/11/p11020 2012

  5. [5]

    P. Fendley,Strong zero modes and eigenstate phase transitions in the XYZ/interacting majorana chain, Journal of Physics A: Mathematical and Theoretical49(30), 30LT01 (2016), doi:/10.1088/1751-8113/49/30/30LT01

    work page doi:10.1088/1751-8113/49/30/30lt01 2016

  6. [6]

    Moran, D

    N. Moran, D. Pellegrino, J. K. Slingerland and G. Kells,Parafermionic clock models and quantum resonance, Phys. Rev. B95(23), 235127 (2017), doi:10.1103/PhysRevB.95.235127

    doi:10.1103/physrevb.95.235127 2017

  7. [7]

    L. M. Vasiloiu, F. Carollo, M. Marcuzzi and J. P. Garrahan,Strong zero modes in a class of generalized Ising spin ladders with plaquette interactions, Phys. Rev. B 100(2), 024309 (2019), doi:10.1103/PhysRevB.100.024309

    work page doi:10.1103/physrevb.100.024309 2019

  8. [8]

    J. Kemp, N. Y. Yao and C. R. Laumann,Symmetry-enhanced boundary qubits at infinite temperature, Physical Review Letters125(20), 200506 (2020), doi:10.1103/PhysRevLett.125.200506

    work page doi:10.1103/physrevlett.125.200506 2020

Show all 74 references
  1. [9]

    J. Kemp, N. Y. Yao, C. R. Laumann and P. Fendley,Long coherence times for edge spins, Journal of Statistical Mechanics: Theory and Experiment2017(6), 063105 (2017), doi:/10.1088/1742-5468/aa73f0

    2017 doi

  2. [10]

    D. V. Else, P. Fendley, J. Kemp and C. Nayak,Prethermal strong zero modes and topo- logical qubits, Physical Review X7(4), 041062 (2017), doi:10.1103/PhysRevX.7.041062

    2017 doi

  3. [11]

    Thakurathi, A

    M. Thakurathi, A. A. Patel, D. Sen and A. Dutta,Floquet generation of ma- jorana end modes and topological invariants, Phys. Rev. B88, 155133 (2013), doi:10.1103/PhysRevB.88.155133

    2013 doi

  4. [12]

    Chandran, V

    A. Chandran, V. Khemani, C. R. Laumann and S. L. Sondhi,Many-body local- ization and symmetry-protected topological order, Phys. Rev. B89, 144201 (2014), doi:10.1103/PhysRevB.89.144201

    2014 doi

  5. [13]

    Bahri, R

    Y. Bahri, R. Ronen and E. Altman,Localization and topology protected quan- tum coherence at the edge of hot matter, Nature Communications6, 7341 (2015), doi:10.1038/ncomms8341

    2015 doi

  6. [14]

    Iadecola, L

    T. Iadecola, L. H. Santos and C. Chamon,Stroboscopic symmetry-protected topological phases, Phys. Rev. B92, 125107 (2015), doi:10.1103/PhysRevB.92.125107

    2015 doi

  7. [15]

    Khemani, A

    V. Khemani, A. Lazarides, R. Moessner and S. L. Sondhi,Phase struc- ture of driven quantum systems, Phys. Rev. Lett.116, 250401 (2016), doi:10.1103/PhysRevLett.116.250401

    2016 doi

  8. [16]

    G. J. Sreejith, A. Lazarides and R. Moessner,Parafermion chain with2 π/k floquet edge modes, Phys. Rev. B94, 045127 (2016), doi:10.1103/PhysRevB.94.045127

    2016 doi

  9. [17]

    Potirniche, A

    I.-D. Potirniche, A. C. Potter, M. Schleier-Smith, A. Vishwanath and N. Y. Yao, Floquet symmetry-protected topological phases in cold-atom systems, Phys. Rev. Lett. 119, 123601 (2017), doi:10.1103/PhysRevLett.119.123601. 18 SciPost Physics Submission

    2017 doi

  10. [18]

    Kumar, P

    A. Kumar, P. T. Dumitrescu and A. C. Potter,String order parameters for one- dimensional floquet symmetry protected topological phases, Phys. Rev. B97, 224302 (2018), doi:10.1103/PhysRevB.97.224302

    2018 doi

  11. [19]

    Mukherjee, R

    B. Mukherjee, R. Melendrez, M. Szyniszewski, H. J. Changlani and A. Pal,Emergent strong zero mode through local floquet engineering, Phys. Rev. B109, 064303 (2024), doi:/10.1103/PhysRevB.109.064303

    2024 doi

  12. [20]

    Vernier, H.-C

    E. Vernier, H.-C. Yeh, L. Piroli and A. Mitra,Strong zero modes in integrable quantum circuits, Phys. Rev. Lett.133, 050606 (2024), doi:10.1103/PhysRevLett.133.050606

    2024 doi

  13. [21]

    Klobas, P

    K. Klobas, P. Fendley and J. P. Garrahan,Stochastic strong zero modes and their dynamical manifestations, Phys. Rev. E107(4), L042104 (2023), doi:10.1103/PhysRevE.107.L042104

    2023 doi

  14. [22]

    C. T. Olund, N. Y. Yao and J. Kemp,Boundary strong zero modes, Phys. Rev. B 111, L201114 (2025), doi:10.1103/PhysRevB.111.L201114

    2025 doi

  15. [23]

    D. J. Yates, F. H. L. Essler and A. Mitra,Almost strong(0 , π)edge modes in clean interacting one-dimensional floquet systems, Phys. Rev. B99, 205419 (2019), doi:10.1103/PhysRevB.99.205419

    2019 doi

  16. [24]

    D. J. Yates, A. G. Abanov and A. Mitra,Lifetime of almost strong edge-mode operators in one-dimensional, interacting, symmetry protected topological phases, Phys. Rev. Lett.124, 206803 (2020), doi:/10.1103/PhysRevLett.124.206803

    2020 doi

  17. [25]

    Alicea and P

    J. Alicea and P. Fendley,Topological phases with parafermions: theory and blueprints, Ann. Rev. Condensed Matter Phys.7, 119 (2016), doi:10.1146/annurev-conmatphys- 031115-011336,1504.02476

    2016 arXiv

  18. [26]

    F. H. L. Essler, P. Fendley and E. Vernier,Strong zero modes in integrable spin-s chains with open boundaries,to appear(2025)

    2025

  19. [27]

    Gritsev and A

    V. Gritsev and A. Polkovnikov,Integrable Floquet dynamics, SciPost Phys.2, 021 (2017), doi:10.21468/SciPostPhys.2.3.021

    2017 doi

  20. [28]

    Vanicat, L

    M. Vanicat, L. Zadnik and T. Prosen,Integrable trotterization: Local con- servation laws and boundary driving, Phys. Rev. Lett.121, 030606 (2018), doi:10.1103/PhysRevLett.121.030606

    2018 doi

  21. [29]

    Ljubotina, L

    M. Ljubotina, L. Zadnik and T. Prosen,Ballistic spin transport in a period- ically driven integrable quantum system, Phys. Rev. Lett.122, 150605 (2019), doi:10.1103/PhysRevLett.122.150605

    2019 doi

  22. [30]

    Medenjak, T

    M. Medenjak, T. Prosen and L. Zadnik,Rigorous bounds on dynamical response functions and time-translation symmetry breaking, SciPost Phys.9, 003 (2020), doi:10.21468/SciPostPhys.9.1.003

    2020 doi

  23. [31]

    Miao and E

    Y. Miao and E. Vernier,Integrable Quantum Circuits from the Star-Triangle Relation, Quantum7, 1160 (2023), doi:10.22331/q-2023-11-03-1160

    2023 doi

  24. [32]

    I. L. Aleiner,Bethe ansatz solutions for certain periodic quantum circuits, Annals of Physics433, 168593 (2021), doi:/10.1016/j.aop.2021.168593

    2021

  25. [33]

    P. W. Claeys, J. Herzog-Arbeitman and A. Lamacraft,Correlations and commut- ing transfer matrices in integrable unitary circuits, SciPost Phys.12, 007 (2022), doi:10.21468/SciPostPhys.12.1.007. 19 SciPost Physics Submission

    2022 doi

  26. [34]

    Vernier, B

    E. Vernier, B. Bertini, G. Giudici and L. Piroli,Integrable digital quantum simulation: Generalized gibbs ensembles and trotter transitions, Phys. Rev. Lett.130, 260401 (2023), doi:10.1103/PhysRevLett.130.260401

    2023 doi

  27. [35]

    Takahashi,Correlation length and free energy of the s = 1/2XYZ chain, Physical Review B43(7), 5788 (1991), doi:10.1103/PhysRevB.43.5788

    M. Takahashi,Correlation length and free energy of the s = 1/2XYZ chain, Physical Review B43(7), 5788 (1991), doi:10.1103/PhysRevB.43.5788

    1991 doi

  28. [36]

    D. B. Abraham, F. H. L. Essler and F. T. Latr´ emoli` ere,Correlation functions in an exactly solvable tefface-ledge-kink model, Nucl. Phys. B556, 411 (1999), doi:10.1016/S0550-3213(99)00268-0

    1999 doi

  29. [37]

    Morvan, T

    A. Morvan, T. I. Andersen, X. Mi, C. Neill, A. Petukhov, K. Kechedzhi, D. Abanin, A. Michailidis, R. Acharya, F. Aruteet al.,Formation of robust bound states of interacting microwave photons, Nature612(7939), 240 (2022), doi:10.1038/s41586- 022-05348-y

    2022 doi

  30. [38]

    Keenan, N

    N. Keenan, N. F. Robertson, T. Murphy, S. Zhuk and J. Goold,Evidence of kardar- parisi-zhang scaling on a digital quantum simulator, npj Quantum Information9(1), 72 (2023), doi:10.1038/s41534-023-00742-4

    2023 doi

  31. [39]

    Maruyoshi, T

    K. Maruyoshi, T. Okuda, J. W. Pedersen, R. Suzuki, M. Yamazaki and Y. Yoshida, Conserved charges in the quantum simulation of integrable spin chains, Journal of Physics A: Mathematical and Theoretical56(16), 165301 (2023), doi:10.1088/1751- 8121/acc369

    2023 doi

  32. [40]

    E. K. Sklyanin,Boundary conditions for integrable quantum systems, Journal of Physics A: Mathematical and General21(10), 2375 (1988), doi:10.1088/0305-4470/21/10/015

    1988 doi

  33. [41]

    H. J. de Vega and Gonz´ alez-Ruiz,Boundary k-matrices for the XYZ, XXZ and XXX spin chains, Journal of Physics A: Mathematical and General27(18), 6129 (1994), doi:10.1088/0305-4470/27/18/021

    1994 doi

  34. [42]

    V. E. Korepin, N. M. Bogoliubov and A. G. Izergin,Quantum Inverse Scattering Method and Correlation Functions, Cambridge Monographs on Mathematical Physics. Cambridge University Press, doi:10.1017/cbo9780511628832 (1993)

    1993 doi

  35. [43]

    I. V. Cherednik,Factorizing Particles on a Half Line and Root Systems, Theor. Math. Phys.61, 977 (1984), doi:10.1007/BF01038545

    1984 doi

  36. [44]

    D’Alessio and M

    L. D’Alessio and M. Rigol,Long-time behavior of isolated periodically driven interacting lattice systems, Physical Review X4(4), 041048 (2014), doi:10.1103/PhysRevX.4.041048

    2014 doi

  37. [45]

    Lazarides, A

    A. Lazarides, A. Das and R. Moessner,Equilibrium states of generic quantum systems subject to periodic driving, Physical Review E90(1), 012110 (2014), doi:10.1103/PhysRevE.90.012110

    2014 doi

  38. [46]

    Ponte, A

    P. Ponte, A. Chandran, Z. Papi´ c and D. A. Abanin,Periodically driven ergodic and many-body localized quantum systems, Annals of Physics353, 196 (2015), doi:10.1016/j.aop.2014.11.008

    2015 doi

  39. [47]

    Lazarides, A

    A. Lazarides, A. Das and R. Moessner,Periodic thermodynamics of iso- lated quantum systems, Physical review letters112(15), 150401 (2014), doi:/10.1103/PhysRevLett.112.150401. 20 SciPost Physics Submission

    2014 doi

  40. [48]

    F. H. L. Essler and M. Fagotti,Quench dynamics and relaxation in isolated integrable quantum spin chains, Journal of Statistical Mechanics: Theory and Experiment 2016(6), 064002 (2016), doi:10.1088/1742-5468/2016/06/064002

    2016 doi

  41. [49]

    Fendley, S

    P. Fendley, S. Gehrmann, E. Vernier and F. Verstraete,XYZ integrability the easy way(2025), doi:/10.48550/arXiv.2511.04674

    2025 doi

  42. [50]

    Gwa and H

    L.-H. Gwa and H. Spohn,Six-vertex model, roughened surfaces, and an asymmetric spin hamiltonian, Phys. Rev. Lett.68, 725 (1992), doi:10.1103/PhysRevLett.68.725

    1992 doi

  43. [51]

    Gwa and H

    L.-H. Gwa and H. Spohn,Bethe solution for the dynamical-scaling exponent of the noisy burgers equation, Phys. Rev. A46, 844 (1992), doi:10.1103/PhysRevA.46.844

    1992 doi

  44. [52]

    F. C. Alcaraz, M. Droz, M. Henkel and V. Rittenberg,Reaction-diffusion processes, critical dynamics, and quantum chains, Ann. Phys. (N. Y.)230(2), 250 (1994), doi:10.1006/aphy.1994.1026

    1994

  45. [53]

    Sandow,Partially asymmetric exclusion process with open boundaries, Phys

    S. Sandow,Partially asymmetric exclusion process with open boundaries, Phys. Rev. E50, 2660 (1994), doi:10.1103/PhysRevE.50.2660

    1994 doi

  46. [54]

    F. H. L. Essler and V. Rittenberg,Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries, Journal of Physics A: Mathematical and General29(13), 3375 (1996), doi:10.1088/0305-4470/29/13/013

    1996 doi

  47. [55]

    Derrida,An exactly soluble non-equilibrium system: the asymmetric simple exclusion process, Phys

    B. Derrida,An exactly soluble non-equilibrium system: the asymmetric simple exclusion process, Phys. Rep.301(1-3), 65 (1998), doi:10.1016/S0370-1573(98)00006-4

    1998 doi

  48. [56]

    G. M. Sch¨ utz,Exactly solvable models for many-body systems far from equilib- rium, InPhase transitions and critical phenomena, vol. 19, pp. 1–251. Elsevier, doi:10.1016/S1062-7901(01)80015-X (2001)

    2001 doi

  49. [57]

    Mallick,Some exact results for the exclusion process, J

    K. Mallick,Some exact results for the exclusion process, J. Stat. Mech.2011(01), P01024 (2011), doi:10.1088/1742-5468/2011/01/P01024

    2011 doi

  50. [58]

    Derrida, M

    B. Derrida, M. R. Evans, V. Hakim and V. Pasquier,Exact solution of a 1d asym- metric exclusion model using a matrix formulation, J. Phys. A26(7), 1493 (1993), doi:10.1088/0305-4470/26/7/011

    1993 doi

  51. [59]

    Sch¨ utz and E

    G. Sch¨ utz and E. Domany,Phase transitions in an exactly soluble one- dimensional exclusion process, Journal of statistical physics72(1), 277 (1993), doi:10.1007/BF01048050

    1993 doi

  52. [60]

    T. Sasamoto,One-dimensional partially asymmetric simple exclusion process with open boundaries: orthogonal polynomials approach, Journal of Physics A: Mathematical and General32(41), 7109 (1999), doi:10.1088/0305-4470/32/41/306

    1999 doi

  53. [61]

    T. Sasamoto,Density profile of the one-dimensional partially asymmetric simple exclusion process with open boundaries, Journal of the Physical Society of Japan69(4), 1055 (2000), doi:10.1143/JPSJ.69.1055

    2000 doi

  54. [62]

    R. A. Blythe, M. R. Evans, F. Colaiori and F. H. L. Essler,Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra, Journal of Physics A: Mathematical and General33(12), 2313 (2000), doi:10.1088/0305-4470/33/12/301

    2000 doi

  55. [63]

    Uchiyama, T

    M. Uchiyama, T. Sasamoto and M. Wadati,Asymmetric simple exclusion process with open boundaries and askey–wilson polynomials, Journal of Physics A: Mathematical and General37(18), 4985 (2004), doi:10.1088/0305-4470/37/18/006. 21 SciPost Physics Submission

    2004 doi

  56. [64]

    Enaud and B

    C. Enaud and B. Derrida,Large deviation functional of the weakly asym- metric exclusion process, Journal of statistical physics114(3), 537 (2004), doi:10.1023/B:JOSS.0000012501.43746.cf

    2004

  57. [65]

    de Gier and F

    J. de Gier and F. H. L. Essler,Bethe ansatz solution of the asymmetric ex- clusion process with open boundaries, Phys. Rev. Lett.95, 240601 (2005), doi:10.1103/PhysRevLett.95.240601

    2005 doi

  58. [66]

    de Gier and F

    J. de Gier and F. H. L. Essler,Exact spectral gaps of the asymmetric exclusion process with open boundaries, J. Stat. Mech.2006(12), P12011 (2006), doi:10.1088/1742- 5468/2006/12/P12011

    2006 doi

  59. [67]

    R. A. Blythe and M. R. Evans,Nonequilibrium steady states of matrix-product form: a solver’s guide, Journal of Physics A: Mathematical and Theoretical40(46), R333 (2007), doi:10.1088/1751-8113/40/46/R01

    2007 doi

  60. [68]

    de Gier and F

    J. de Gier and F. H. L. Essler,Slowest relaxation mode of the partially asymmetric exclusion process with open boundaries, J. Phys. A: Math. Theor.41(48), 485002 (2008), doi:10.1088/1751-8113/41/48/485002

    2008 doi

  61. [69]

    Simon,Construction of a coordinate bethe ansatz for the asymmetric simple exclusion process with open boundaries, J

    D. Simon,Construction of a coordinate bethe ansatz for the asymmetric simple exclusion process with open boundaries, J. Stat. Mech.2009(07), P07017 (2009), doi:10.1088/1742-5468/2009/07/p07017

    2009 doi

  62. [70]

    de Gier and F

    J. de Gier and F. H. L. Essler,Large deviation function for the current in the open asymmetric simple exclusion process, Phys. Rev. Lett.107, 010602 (2011), doi:10.1103/PhysRevLett.107.010602

    2011 doi

  63. [71]

    Lazarescu and V

    A. Lazarescu and V. Pasquier,Bethe Ansatz and Q-operator for the open ASEP, J. Phys.A 47(29), 295202 (2014), doi:10.1088/1751-8113/47/29/295202

    2014 doi

  64. [72]

    Krug,Boundary-induced phase transitions in driven diffusive systems, Phys

    J. Krug,Boundary-induced phase transitions in driven diffusive systems, Phys. Rev. Lett.67, 1882 (1991), doi:10.1103/PhysRevLett.67.1882

    1991 doi

  65. [73]

    Henkel and G

    M. Henkel and G. Sch¨ utz,Boundary-induced phase transitions in equilibrium and non- equilibrium systems, Physica A: Statistical Mechanics and its Applications206(1-2), 187 (1994), doi:10.1016/0378-4371(94)90124-4

    1994 doi

  66. [74]

    X. Mi, M. Sonner, M. Y. Niu, K. W. Lee, B. Foxen, R. Acharya, I. Aleiner, T. I. Ander- sen, F. Arute, K. Aryaet al.,Noise-resilient edge modes on a chain of superconducting qubits, Science378(6621), 785 (2022), doi:10.1126/science.abq5769. 22

    2022 doi

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