Pith. sign in

REVIEW 4 major objections 6 minor 1 cited by

Entanglement Holography in Quantum Phases via Twisted R\'enyi-N Correlators

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read By replicating the reduced density matrix of a symmetry-protected topological state along an extra dimension, the twisted Rényi-N correlator of that matrix reproduces the bulk strange correlator and reveals long-range (or…

desk verdict RDM-only SPT diagnostic is real in 1d, plausible in 2d; the 'exact correspondence' claim overshoots the proof. read the letter →

arxiv 2506.10076 v1 pith:Q4YL4ZE3 submitted 2025-06-11 cond-mat.str-el quant-ph

Pablo Sala , Frank Pollmann , Masaki Oshikawa , Yizhi You This is my paper
classification cond-mat.str-elquant-ph
keywords symmetry-protectedtopologicalorderentanglementHamiltoniantwistedRényi-Ncorrelatorstrangereplicadirectionreduceddensitymatrixmixed-stateSPTphases
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

The paper claims that, in symmetry-protected topological (SPT) phases with area-law entanglement, the reduced density matrix of a spatial cut can be treated as a lower-dimensional mixed state, and that replicating this density matrix along an extra 'replica direction' reconstructs the original higher-dimensional SPT wavefunction. The central discovery is an exact correspondence: the twisted Rényi-N correlator built from the reduced density matrix equals the bulk strange correlator of the reconstructed SPT state. As a result, the reduced density matrix alone carries a universal signature of topological order — its twisted Rényi-N correlator shows long-range (or quasi-long-range) order along the replica direction, even though ordinary correlation functions of the RDM are short-ranged. This gives a practical tool to detect SPT phases, locate phase transitions, and distinguish different SPT phases from entanglement data alone, with extensions to thermal states, open quantum systems, and mixed-state SPT phases.

What carries the argument

The central object is the dimensional-extension (replica) construction: each copy of the reduced density matrix ρ = Σ_ν λ_ν |ν⟩⟨ν| is vectorized to an entangled pair |ν⟩_L|ν⟩_R, and a rank-four tensor T_{ijkl} = δ_{ij}δ_{kl} inserts an intra-unit-cell coupling between the bra of copy i and the ket of copy i+1. This produces a matrix-product-state-like fixed-point wavefunction |Ψ_SPT⟩ whose unit cells run along the replica index. The twisted Rényi-N correlator $C^{{(N)}}$ = Tr[ρ^N M_O ρ^N M_O^†]/Tr[$ρ^{{2N}}$] is the observable that carries the argument: it is shown to be the overlap of |Ψ_SPT⟩ with a trivial product state with operators inserted at positions separated by N, i.e., the bulk strange correlator. The mechanism for long-range order is that the projective representation of the symmetry group forces a degenerate leading eigenspace of ρ; in the large-N limit only that degenerate space survives, so a charge-flipping insertion M_O yields a non-vanishing (and quantized) correlator. In 2d, ρ^N maps to a chiral-boson path integral on a torus, so the correlator becomes a temporal two-point function with algebraic decay.

What would settle it

Take a 2d Chern-insulator reduced density matrix, fix a finite spatial length l_x, and increase the replica number N beyond l_x. The paper predicts exponential decay C(N) ∼ $e^{{-N/l_x}}$ in this order of limits; observing a power-law instead would falsify the order-of-limits claim. Conversely, for a genuinely finite-correlation-length interacting SPT state, compute C(N) after sending l_x → ∞ first: if the correlator decays exponentially rather than algebraically, the phase-correspondence assumption behind the holographic mapping fails.

Watch

Extended reading notes

Core claim

Working at the level of a spatial bipartition, the authors start with the reduced density matrix ρ of an SPT wavefunction and build a pure state by taking N copies of ρ, vectorizing each copy into an entangled pair, and gluing the pairs together with a fixed rank-four tensor. The resulting state |Ψ_SPT⟩ is the fixed-point SPT wavefunction in one higher dimension. The paper's central identity is that the twisted Rényi-N correlator $C^{{(N)}}$ = Tr[ρ^N M_O ρ^N M_O^†]/Tr[$ρ^{{2N}}$] equals the strange correlator ⟨Ψ_trivial|O_1^† O_N|Ψ_SPT⟩/⟨Ψ_trivial|Ψ_SPT⟩, with the operator insertions separated by N copies along the replica axis. Because the SPT strange correlator is (quasi) long-ranged, the twisted correlator of the RDM must be (quasi) long-ranged in the replica direction: in 1d it saturates to a quantized value determined by the degeneracy of the entanglement spectrum, while in 2d Chern and quantum spin Hall systems it decays algebraically with an exponent controlled by the interaction parameter K, provided the spatial length is taken to infinity before N. The authors verify the correspondence numerically for 1d spin chains and a coupled-ladder model, and use it to reformulate the half-integer-spin constraint for closed and open systems and to propose diagnostics for mixed-state SPT order.

Load-bearing premise

The replication is exact only for zero-correlation-length fixed-point states; for generic finite-correlation-length SPT wavefunctions the paper assumes that the replicated state still lies in the same SPT phase as the original, so that its strange correlator keeps the universal (quasi) long-range order, an assumption supported by heuristic cut-and-glue and field-theory arguments rather than by a proof.

Editorial extensions

If this is right

  • The twisted Rényi-N correlator of the half-chain reduced density matrix is a stand-alone SPT diagnostic: it saturates to a finite quantized value in an SPT phase, falls to zero in the trivial phase, and numerically tracks the spin-1 SPT-to-trivial transition as the anisotropy D is tuned.
  • The same correlator distinguishes distinct SPT phases: in the spin-2 chain with (Z2 × Z2)^2 symmetry it takes the values 2, 1, and 0 in the SO(5)-symmetric, intermediate, and trivial phases respectively.
  • For 2d Chern and quantum spin Hall states the correlator decays algebraically in the replica index, with the exponent set by the interaction parameter K, reproducing the quasi-long-range strange correlator of the bulk.
  • As a corollary, the thermal density matrix of a half-integer-spin chain with translation and spin-rotation symmetries must exhibit (quasi) long-range order in its twisted Rényi-N correlator, giving a strengthened form of the half-integer-spin constraint for both closed and open systems.
  • In mixed-state SPT phases, the twisted Rényi-N correlator built from the surgery operator ρ_v ρ_t is proposed as a diagnostic that detects and distinguishes mSPT topology.

Reading between the lines

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

  • If the correspondence holds beyond the fixed point, the quantized large-N value of the twisted correlator becomes a direct readout of the entanglement-spectrum degeneracy, which could be measured in experiments through randomized-measurement schemes for nonlinear functionals of the density matrix.
  • The replica construction can be read as a tensor-network prescription: it converts a mixed-state preparation into a higher-dimensional pure-state preparation, suggesting a route to synthesize SPT states from lower-dimensional mixed states on near-term quantum devices.
  • The thermal-state corollary implies that the anomaly constraint leaves a fingerprint not just in the energy spectrum but in the replica-direction correlations of the Gibbs state, which might be probed by measuring Rényi-N correlation functions in ultracold atomic gases.
  • For mixed-state SPT order, the surgery-operator correlator may be less sensitive to the choice of trivial reference state than existing diagnostic correlators, since it directly constructs the fixed-point state whose strange correlator is the target; this is a conjecture to be tested.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes a holographic framework in which the reduced density matrix (RDM) of a d-dimensional symmetry-protected topological (SPT) state is replicated along a new 'replica direction' to reconstruct a (d+1)-dimensional fixed-point SPT wavefunction. The central object is the twisted Rényi-N correlator (TRNC) of the RDM, Eq. (9), which is claimed to be exactly dual to the bulk strange correlator of the reconstructed SPT state, Eq. (10). The authors present a fixed-point construction (Sec. II C), numerical demonstrations in 1d Haldane and spin-2 SO(5) chains (Sec. III), analytical and numerical studies of 2d Chern and quantum spin Hall states via the cut-and-glue approach (Sec. IV), and corollaries for the Lieb-Schultz-Mattis theorem and mixed-state SPTs (Secs. V-VI).

Significance. If the central claim is appropriately qualified, the paper provides a useful new diagnostic: the TRNC is an entanglement-spectrum-derived quantity that can distinguish SPT order from trivial order using only the RDM, without the full wavefunction. The fixed-point identity in Sec. II C is clean, and the DMRG results for the Haldane chain, the SO(5) chain, and the ladder system are convincing evidence for universal long-range or quasi-long-range order of the TRNC. The extension to thermal states and mixed-state SPTs is suggestive. However, the strongest advertised statement--an exact correspondence between the TRNC and the bulk strange correlator of the original SPT wavefunction--is proven only for the fixed-point state built from the RDM, not for generic SPT ground states; the manuscript's own limitations in Sec. IV B confirm that the 2d extension rests on approximations. The paper is a solid contribution to the quantum-information characterization of SPT phases, but it needs careful revision of the claims' scope.

major comments (4)
  1. [Sec. II C / Abstract] The abstract and Sec. II C claim an 'exact correspondence' between the bulk strange correlator of the (d+1)-dimensional SPT state and the twisted Rényi-N correlator of the d-dimensional RDM. The derivation in Eqs. (4)-(10) establishes this equality only for the fixed-point state |Ψ_SPT⟩ constructed by replicating the RDM, which is a zero-correlation-length state built from the eigenvectors and eigenvalues of ρ. For a generic finite-correlation-length SPT ground state, the bulk strange correlator is not generally equal to the TRNC. A concrete example is the Haldane chain away from the AKLT point: Eq. (18) gives C_X(N→∞)=1 purely from the degeneracy of the entanglement spectrum, while the actual strange correlator of the original wavefunction depends on D and on the choice of trivial reference. The manuscript therefore establishes universal long-range (or quasi-long-range) order of the TRNC, not the exact reproduction of the bulk strange correlator. Please either qualify the 'exact' claim throughout, including the abstract, or provide a proof that the replicated state lies in the same phase as the original wavefunction and that nonuniversal magnitudes do not affect the claimed duality.
  2. [Sec. IV A-B / Sec. III B] The extension to 2d systems relies on the cut-and-glue entanglement Hamiltonian and on a phenomenological WZW mapping. Specifically, Eq. (23) introduces a coupling U between ket and bra fields, and the text then extends U m_L·m_R as (∂_x m)^2 locally in Eq. (25); this is an approximation, not a derivation from the original wavefunction. The manuscript itself states in Sec. IV B that 'our analysis relies on a simplified model of a non-interacting Chern insulator with zero correlation length' and that the chiral-boson form breaks down in more general settings, with a non-universal exponent η. Since the central 2d prediction of power-law decay along the replica direction rests on this approximate mapping, please provide a concrete criterion for when the free-boson answer is stable (e.g., under relevant perturbations, or with explicit verification that the exponent remains finite) or state clearly that the 2d quasi-long-range order is a conjecture supported by the ladder numerics.
  3. [Sec. V, Corollary A-B] The 'strengthened LSM theorem' is not established by the arguments given. Corollary A asserts that the thermal density matrix e^{-H_{1d}} of a 1d spin-1/2 chain 'can always be viewed' as the RDM of a 2d SPT wavefunction, but this identification is not proven; a generic local Hamiltonian's thermal state is a specific mixed state that need not coincide with the RDM obtained from any 2d SPT wavefunction. Similarly, Corollary B for mixed states assumes that any short-range-correlated state with weak symmetries maps to the same 2d WZW description. These are strong assumptions that go beyond the fixed-point identity. Please either supply a precise class of Hamiltonians or states for which the identification holds, or present the LSM statements as conjectures rather than theorems.
  4. [Sec. IV A / Sec. IV B] The order-of-limits issue is load-bearing for the 2d claim. The paper correctly notes that one must take l_x→∞ before N→∞ to obtain power-law decay, and that for finite l_x the correlator decays exponentially as e^{-N/l_x}. Since any numerical or experimental probe has finite system size, the practical content of the 2d fingerprint depends on the separation of scales N≪l_x. The manuscript should state this limitation explicitly in the main text of Sec. IV and provide guidance on how large l_x must be relative to N for the power-law to be observable, given that the ladder numerics in Fig. 9 use N≤L.
minor comments (6)
  1. [Abstract] The abstract contains a typo ('colloary') and a grammatical error ('we generalized'); please correct these.
  2. [Eq. (8) and surrounding text] The definition of M_O is unclear: the phrase 'M_O = O_{ij} δ_{ij}' seems to have an index mismatch. Please define M_O explicitly as a tensor with its indices and specify the contraction with ρ in Eq. (9).
  3. [Eq. (13)] The notation Σx and Σy in the strange-correlator expression is ambiguous, as x and y appear both as spatial labels and as operators. Please use distinct symbols for the operators and the positions.
  4. [Figs. 5 and 7] The figures for the Haldane and SO(5) chains do not show convergence of C_X(N) with respect to the MPS bond dimension χ; please add a convergence check analogous to Fig. 9(b), or state that the values are converged at χ=60 (Fig. 5) and χ=50 (Fig. 7).
  5. [Sec. IV B] The text says that the algebraic decay 'is in quantitative agreement' with the strange correlator of the QSH state, but the exponent is stated to be non-universal. Please clarify which quantity is in quantitative agreement, given that both exponents are non-universal.
  6. [Sec. VII / Ref. [106]] The independent study mentioned in the conclusion should be discussed in the introduction or conclusion to clarify the incremental contribution of this work relative to Ref. [106].

Circularity Check

1 steps flagged · score 4.0 of 10

The exact TRNC–strange-correlator equality is a construction-level identity (Eq. (9) = Eq. (10) via the replica-built state of Eq. (4)); the paper's quantitative SPT diagnostics are nonetheless supported independently by Eq. (18) and DMRG.

  1. self definitional [Sec. II B, Eq. (4); Sec. II C, Eq. (9)–Eq. (10); abstract]
    "By replicating ρ, we reconstruct the fixed-point SPT wavefunction, establishing an exact correspondence between the bulk strange correlator of the (d+1)-dimensional SPT state and the twisted Rényi-N operator of the d-dimensional reduced density matrix."

    The |Ψ_SPT⟩ of Eq. (4) is built by vectorizing ρ=Σ_ν λ_ν|ν⟩⟨ν| and gluing bra of copy i to ket of copy i+1. Substituting that definition into the ratio ⟨Ψ_trivial|O†_1 O_N|Ψ_SPT⟩/⟨Ψ_trivial|Ψ_SPT⟩ yields Tr[ρ^N M_O ρ^N M_O†]/Tr[ρ^{2N}] identically, so Eq. (10) is a rewrite of the same trace, not an independent bulk prediction. Thus the exact correspondence is an identity for the replica-constructed fixed-point state; for a generic finite-correlation-length SPT ground state it is assumed, not derived, and the bulk state has been manufactured from the RDM.

full rationale

The central 'holographic' equality is a definitional identity: the bulk state is defined from the RDM, so the strange correlator of that state is the TRNC of the RDM by construction. This is a genuine circular step in the paper's framing of an 'exact correspondence' between the bulk SPT state and the RDM. However, the paper's physical conclusions do not rest on that identity alone. The 1d SPT diagnostic is independently derived from the degeneracy structure of the leading entanglement levels (Eq. (18): C_X(N→∞)=1 in the Haldane phase, 0 in the trivial phase) and confirmed by iDMRG; the 2d power-law prediction follows from the chiral-boson/thermal-state representation of the RDM; and the LSM corollary is a separate spectral argument. Ref. [50] is cited for the correlator and the dimensional-extension mapping, but the relevant equality is re-derived in this paper, so the self-citation is not the load-bearing element. The main circularity is the construction-level status of Eq. (9)=Eq. (10), not an empirically fitted parameter disguised as a prediction. Accordingly, a moderate score of 4 reflects partial circularity in the central correspondence while recognizing the independent content of the phase diagnostics.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim depends primarily on the standard MPS/projective representation classification of 1d SPT phases and on the cut-and-glue description of 2d entanglement Hamiltonians from prior literature. The paper itself adds the phenomenological Choi action WZW mapping and the order-of-limits choice. No new physical entities are introduced.

free parameters (2)
  • U (edge-edge coupling in Choi action) = U → ∞ (equivalently g → 0)
    Introduced by hand in Eq. (23) to glue ket/bra fields; argued to be relevant so the final universal result is independent of its value.
  • K (Luttinger parameter) = 1 for free theory; tunable in interacting theories
    Sets the power-law exponent N^{-1/K} in Eq. (42); not fitted to data in this paper, but the predicted exponent depends on it.
assumptions (6)
  • standard math 1d SPT phases are classified by projective representations of the symmetry group on MPS bond indices, giving degenerate entanglement spectra.
    Used in Sec. III to infer the degeneracy structure of ρ that controls C_X(N→∞).
  • domain assumption Fixed-point SPT wavefunctions have long-range or quasi-long-range strange correlators.
    Borrowed from Refs. [38,42] and used in Sec. II C and IV to predict LRO along the replica direction.
  • domain assumption The RDM of an area-law SPT state is a lower-dimensional mixed state localized near the entanglement cut.
    Central to the holographic mapping in Secs. II B and III A; holds for gapped area-law states but is not proven for all cases.
  • domain assumption The entanglement Hamiltonian of a Chern insulator or QSH state is a 1d chiral/helical boson obtained by cut-and-glue.
    Taken from Refs. [43-45] and used in Secs. IV A and IV B to compute the power-law decay of the TRNC.
  • ad hoc to paper The Choi action for the RDM can be written with a coupling U between ket and bra fields, and then extended locally as (∂_x m)^2.
    Introduced phenomenologically in Sec. III B, Eqs. (23)-(25). The universal claim is argued to be independent of U because U is relevant.
  • domain assumption The thermodynamic limit l_x → ∞ is taken before the replica index N → ∞.
    Required in Sec. IV A/B for algebraic decay; the opposite order gives exponential decay and is stated explicitly.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Entanglement Holography in Quantum Phases via Twisted R\'enyi-N Correlators." pith.science (2026). https://pith.science/paper/Q4YL4ZE3

@misc{pith2026250610076,
  author       = {Pith},
  title        = {Pith review of: Entanglement Holography in Quantum Phases via Twisted R\'enyi-N Correlators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q4YL4ZE3}},
  note         = {Machine review of arXiv:2506.10076}
}
abstract

We introduce a holographic framework for the entanglement Hamiltonian in symmetry-protected topological (SPT) phases with area-law entanglement, whose reduced density matrix $\rho \propto e^{-H_e}$ can be treated as a lower-dimensional mixed state. By replicating $\rho$, we reconstruct the fixed-point SPT wavefunction, establishing an exact correspondence between the bulk strange correlator of the (d+1)-dimensional SPT state and the twisted R\'enyi-N operator of the d-dimensional reduced density matrix. Notably, the reduced density matrix exhibits long-range or quasi-long-range order along the replica direction, revealing a universal entanglement feature in SPT phases. As a colloary, we generalized the framework of twisted R\'enyi-N correlator to thermal states and open quantum systems, providing an alternative formulation of the Lieb-Schultz-Mattis theorem, applicable to both closed and open systems. Finally, we extend our protocol to mixed-state SPT phases and introduce new quantum information metrics -- twisted R\'enyi-N correlators of the surgery operator -- to characterize the topology of mixed states.

Figures

Figures reproduced from arXiv: 2506.10076 by the authors.

Figure 1
Figure 1. FIG. 1. a) Spatial bipartition of a 1d SPT into regions [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. a) The 1d SPT wavefunction is constructed by taking [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The inner product between the SPT wavefunction [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The strange correlator of the dual wavefunction, [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Euclidean path-integral view [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. a) The mixed-state density matrix, with the [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Graphical representation of the twisted R´enyi- [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Diagnosing 2D symmetry protected topological states via mixed state anomaly

    cond-mat.str-el 2025-06 conditional novelty 6.0 of 10

    The RDM of a 2D Z2 SPT state acts as a 1D anomalous mixed state whose twisted disorder parameter contains a quantized topological constant (D=4 with time reversal) and supports symmetry-breaking-type long-range order.

Reference graph

Works this paper leans on

110 extracted references · 61 canonical work pages · cited by 1 Pith paper

  1. [1]

    Can the salient properties of the bulk wavefunc- tion be inferred from measurements of its reduced density matrix?

  2. [2]

    Is it possible to map nonlinear observables of the reduced density matrix to corresponding bulk cor- relation functions?

  3. [3]

    This dimensional-extension mapping, originally proposed in Ref

    When the boundary of an SPT state undergoes de- coherence and becomes a mixed state, does it retain its distinctive topological or symmetry-protected features? Motivated by these questions, we show that ad- dimensional density matrixρwith an anomaly under a weak symmetryGcan be extended to a (d+ 1)- dimensional SPT wavefunction Ψspt protected byGsym- metr...

  4. [4]

    This means that physical observables throughout the bulk can be accessed and simulated via nonlinear operators on the reduced density ma- trix

    The bulk strange correlator of a fixed-point SPT wavefunction in (d+ 1)-dimensions can be obtained by evaluating the twisted R´ enyi-Noperator on the reduced density matrix, which is ad-dimensional mixed state. This means that physical observables throughout the bulk can be accessed and simulated via nonlinear operators on the reduced density ma- trix

  5. [5]

    Previous approaches often relied on the entangle- ment spectrum as a fingerprint for SPT wavefunc- tions, which is easily obtained in one dimension but numerically challenging in higher dimensions. In contrast, our mapping demonstrates that simpler to compute nonlinear observables derived from the entanglement Hamiltonian can effectively charac- terize th...

  6. [6]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys.80, 517 (2008)

    2008

  7. [7]

    Calabrese and A

    P. Calabrese and A. Lefevre, Physical Review A78, 032329 (2008)

    2008

  8. [8]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of Physics A: Math- ematical and Theoretical42, 504005 (2009)

    2009

Show all 110 references
  1. [9]

    Peschel and V

    I. Peschel and V. Eisler, Journal of Physics A: Mathe- matical and Theoretical42, 504003 (2009). 18

    2009

  2. [10]

    Pollmann, E

    F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Physical review b85, 075125 (2012)

    2012

  3. [11]

    Verstraete, J

    F. Verstraete, J. Cirac, and V. Murg, Adv. Phys.57, 143 (2008)

    2008

  4. [12]

    Schollw¨ ock, Ann

    U. Schollw¨ ock, Ann. Phys.326, 96 (2011)

    2011

  5. [13]

    Alexandradinata, T

    A. Alexandradinata, T. L. Hughes, and B. A. Bernevig, Phys. Rev. B84, 195103 (2011)

    2011

  6. [14]

    Prodan, T

    E. Prodan, T. L. Hughes, and B. A. Bernevig, Phys. Rev. Lett.105, 115501 (2010)

    2010

  7. [15]

    Fidkowski, Phys

    L. Fidkowski, Phys. Rev. Lett.104, 130502 (2010)

    2010

  8. [16]

    Chandran, V

    A. Chandran, V. Khemani, and S. L. Sondhi, Physical review letters113, 060501 (2014)

    2014

  9. [17]

    M. P. Zaletel, R. S. K. Mong, and F. Pollmann, Phys. Rev. Lett.110, 236801 (2013)

    2013

  10. [18]

    Zhang, T

    Y. Zhang, T. Grover, A. Turner, M. Oshikawa, and A. Vishwanath, Physical Review B85, 235151 (2012)

    2012

  11. [19]

    Kitaev and J

    A. Kitaev and J. Preskill, Phys. Rev. Lett.96, 110404 (2006)

    2006

  12. [20]

    Levin and X.-G

    M. Levin and X.-G. Wen, Phys. Rev. Lett.96, 110405 (2006)

    2006

  13. [21]

    D. J. Luitz, N. Laflorencie, and F. Alet, Journal of Statistical Mechanics: Theory and Experiment2014, P08007 (2014)

    2014

  14. [22]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, and X.-G. Wen, Physical review b 82, 155138 (2010)

    2010

  15. [23]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Science 338, 1604 (2012)

    2012

  16. [24]

    Chen, Z.-X

    X. Chen, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B Con- dens. Matter84, 235141 (2011)

    2011

  17. [25]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B Con- dens. Matter83, 035107 (2011)

    2011

  18. [27]

    Pollmann, E

    F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Phys. Rev. B Condens. Matter85, 075125 (2012)

    2012

  19. [28]

    A. M. Turner, F. Pollmann, and E. Berg, Phys. Rev. B Condens. Matter83, 075102 (2011)

    2011

  20. [29]

    Li and F

    H. Li and F. D. M. Haldane, Phys. Rev. Lett.101, 010504 (2008)

    2008

  21. [30]

    E. Wybo, F. Pollmann, S. Sondhi, and Y. You, Physical Review B103, 115120 (2021)

    2021

  22. [31]

    Y. O. Nakagawa and S. Furukawa, Physical Review B 96, 205108 (2017)

    2017

  23. [32]

    Y. You, E. Wybo, F. Pollmann, and S. Sondhi, Physical Review B106, L161104 (2022)

    2022

  24. [33]

    H.-H. Tu, Y. Zhang, and X.-L. Qi, Physical Re- view B—Condensed Matter and Materials Physics88, 195412 (2013)

    2013

  25. [34]

    Matsuura, X

    S. Matsuura, X. Wen, L.-Y. Hung, and S. Ryu, Physical Review B93, 195113 (2016)

    2016

  26. [35]

    Marvian, Physical Review B95, 045111 (2017)

    I. Marvian, Physical Review B95, 045111 (2017)

    2017

  27. [36]

    Y. You, J. Bibo, and F. Pollmann, Physical Review Research2, 033192 (2020)

    2020

  28. [37]

    L. H. Santos, J. Cano, M. Mulligan, and T. L. Hughes, Physical Review B98, 075131 (2018)

    2018

  29. [38]

    Lundgren, J

    R. Lundgren, J. Blair, M. Greiter, A. L¨ auchli, G. A. Fiete, and R. Thomale, Physical review letters113, 256404 (2014)

    2014

  30. [39]

    Sohal, B

    R. Sohal, B. Han, L. H. Santos, and J. C. Teo, Physical Review B102, 045102 (2020)

    2020

  31. [40]

    G. J. Henderson, G. Sreejith, and S. H. Simon, Physical Review B104, 195434 (2021)

    2021

  32. [41]

    Nehra, D

    R. Nehra, D. S. Bhakuni, A. Ramachandran, and A. Sharma, Physical Review Research2, 013175 (2020)

    2020

  33. [42]

    Mollabashi, N

    A. Mollabashi, N. Shiba, and T. Takayanagi, Journal of High Energy Physics2014, 1 (2014)

    2014

  34. [43]

    Y.-Z. You, Z. Bi, A. Rasmussen, K. Slagle, and C. Xu, Physical review letters112, 247202 (2014)

    2014

  35. [44]

    Vanhove, M

    R. Vanhove, M. Bal, D. J. Williamson, N. Bultinck, J. Haegeman, and F. Verstraete, Physical review letters 121, 177203 (2018)

    2018

  36. [45]

    Wu, Y.-Y

    H.-Q. Wu, Y.-Y. He, Y.-Z. You, C. Xu, Z. Y. Meng, and Z.-Y. Lu, Physical Review B92, 165123 (2015)

    2015

  37. [46]

    Zhou, M.-Y

    C. Zhou, M.-Y. Li, Z. Yan, P. Ye, and Z. Y. Meng, Physical Review B106, 214428 (2022)

    2022

  38. [47]

    Lepori, M

    L. Lepori, M. Burrello, A. Trombettoni, and S. Pa- ganelli, Physical Review B108, 035110 (2023)

    2023

  39. [48]

    X.-L. Qi, H. Katsura, and A. W. Ludwig, Physical re- view letters108, 196402 (2012)

    2012

  40. [49]

    Chen and E

    X. Chen and E. Fradkin, Journal of Statistical Mechan- ics: Theory and Experiment2013, P08013 (2013)

    2013

  41. [50]

    Lundgren, Y

    R. Lundgren, Y. Fuji, S. Furukawa, and M. Oshikawa, Physical Review B88, 245137 (2013)

    2013

  42. [51]

    Murciano, P

    S. Murciano, P. Sala, Y. Liu, R. S. Mong, and J. Alicea, Physical Review X13, 041042 (2023)

    2023

  43. [52]

    S. J. Garratt and E. Altman, PRX Quantum5, 030311 (2024)

    2024

  44. [53]

    Tang and X

    Q. Tang and X. Wen, arXiv preprint arXiv:2411.13705 (2024)

    2024 arXiv

  45. [54]

    Z. Yang, D. Mao, and C.-M. Jian, Physical Review B 108, 165120 (2023)

    2023

  46. [55]

    Sun, J.-H

    S. Sun, J.-H. Zhang, Z. Bi, and Y. You, arXiv preprint arXiv:2410.08205 (2024)

    2024 arXiv

  47. [56]

    P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Physical Review B110, 155150 (2024)

    2024

  48. [57]

    Y.-N. Zhou, X. Li, H. Zhai, C. Li, and Y. Gu, National Science Review12, nwae287 (2025)

    2025

  49. [58]

    Huang and S

    Z.-M. Huang and S. Diehl, Physical Review Letters134, 053002 (2025)

    2025

  50. [59]

    Ding, H.-R

    K. Ding, H.-R. Zhang, B.-T. Liu, and S. Yang, arXiv preprint arXiv:2412.07563 (2024)

    2024 arXiv

  51. [60]

    Hsin, Z.-X

    P.-S. Hsin, Z.-X. Luo, and H.-Y. Sun, Journal of High Energy Physics2024, 1 (2024)

    2024

  52. [61]

    Y. Bao, R. Fan, A. Vishwanath, and E. Altman, arXiv preprint arXiv:2301.05687 (2023)

    2023 arXiv

  53. [62]

    R. Fan, Y. Bao, E. Altman, and A. Vishwanath, arXiv preprint arXiv:2301.05689 (2023)

    2023 arXiv

  54. [63]

    J. Y. Lee, Y.-Z. You, and C. Xu, arXiv preprint arXiv:2210.16323 (2022)

    2022 arXiv

  55. [64]

    J. Y. Lee, arXiv preprint arXiv:2402.16937 (2024)

    2024

  56. [65]

    Zhang, Y

    J.-H. Zhang, Y. Qi, and Z. Bi, arXiv preprint arXiv:2210.17485 (2022)

    2022

  57. [66]

    L. A. Lessa, M. Cheng, and C. Wang, arXiv preprint arXiv:2401.17357 (2024)

    2024 arXiv

  58. [67]

    Y. Zang, Y. Gu, and S. Jiang, Physical Review Letters 133, 106503 (2024)

    2024

  59. [68]

    Wang and L

    Z. Wang and L. Li, arXiv preprint arXiv:2403.14533 (2024)

    2024 arXiv

  60. [69]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett.59, 799 (1987)

    1987

  61. [70]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B83, 035107 (2011)

    2011

  62. [71]

    Schuch, D

    N. Schuch, D. P´ erez-Garc ´ ıa, and I. Cirac, Physical Re- view B—Condensed Matter and Materials Physics84, 165139 (2011)

    2011

  63. [72]

    F. D. M. Haldane, Physics letters a93, 464 (1983)

    1983

  64. [73]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, inCondensed matter physics and exactly soluble mod- 19 els(Springer, 1988) pp. 253–304

    1988

  65. [74]

    W. Chen, K. Hida, and B. C. Sanctuary, Physical Re- view B67(2003), 10.1103/physrevb.67.104401

    2003 doi

  66. [75]

    Pollmann, A

    F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B81, 064439 (2010)

    2010

  67. [76]

    S. R. White, Phys. Rev. Lett.69, 2863 (1992)

    1992

  68. [77]

    I. P. McCulloch, arXiv preprint arXiv:0804.2509 (2008)

    2008 arXiv

  69. [78]

    Xu and T

    C. Xu and T. Senthil, Physical Review B87, 174412 (2013)

    2013

  70. [79]

    Z. Bi, A. Rasmussen, K. Slagle, and C. Xu, Physical Review B91, 134404 (2015)

    2015

  71. [80]

    Xu and A

    C. Xu and A. W. Ludwig, Physical review letters110, 200405 (2013)

    2013

  72. [81]

    F. D. M. Haldane, Physical review letters50, 1153 (1983)

    1983

  73. [82]

    Altland and B

    A. Altland and B. D. Simons,Condensed matter field theory(Cambridge university press, 2010)

    2010

  74. [83]

    Tu and R

    H.-H. Tu and R. Or´ us, Physical Review B—Condensed Matter and Materials Physics84, 140407 (2011)

    2011

  75. [84]

    Tu, G.-M

    H.-H. Tu, G.-M. Zhang, and T. Xiang, Physical Re- view B—Condensed Matter and Materials Physics78, 094404 (2008)

    2008

  76. [85]

    Scalapino, S.-C

    D. Scalapino, S.-C. Zhang, and W. Hanke, Physical Review B58, 443 (1998)

    1998

  77. [86]

    Okamoto, T

    K. Okamoto, T. Tonegawa, H. Nakano, T. Sakai, K. No- mura, and M. Kaburagi, inJournal of Physics: Confer- ence Series, Vol. 320 (IOP Publishing, 2011) p. 012018

    2011

  78. [87]

    Schollw¨ ock and T

    U. Schollw¨ ock and T. Jolicœur, Europhysics Letters30, 493 (1995)

    1995

  79. [88]

    Oshikawa, Journal of Physics: Condensed Matter4, 7469 (1992)

    M. Oshikawa, Journal of Physics: Condensed Matter4, 7469 (1992)

    1992

  80. [89]

    J. C. Y. Teo and C. L. Kane, Phys. Rev. B89, 085101 (2014)

    2014

  81. [90]

    Peschel and V

    I. Peschel and V. Eisler, Journal of physics a: mathe- matical and theoretical42, 504003 (2009)

    2009

  82. [91]

    Peschel, Journal of Physics A: Mathematical and Gen- eral36, L205 (2003)

    I. Peschel, Journal of Physics A: Mathematical and Gen- eral36, L205 (2003)

    2003

  83. [92]

    Verlinde and H

    E. Verlinde and H. Verlinde, Nuclear Physics B288, 357 (1987)

    1987

  84. [93]

    Sarma, Y

    A. Sarma, Y. Bao, N. Myerson-Jain, T. Kiely, and C. Xu, arXiv preprint arXiv:2503.11788 (2025)

    2025 arXiv

  85. [94]

    Poilblanc, Physical review letters105, 077202 (2010)

    D. Poilblanc, Physical review letters105, 077202 (2010)

    2010

  86. [95]

    Chatterjee, S

    A. Chatterjee, S. D. Pace, and S.-H. Shao, Physical Review Letters134, 021601 (2025)

    2025

  87. [96]

    Miao, SciPost Physics11, 066 (2021)

    Y. Miao, SciPost Physics11, 066 (2021)

    2021

  88. [97]

    Vernier, E

    E. Vernier, E. O’Brien, and P. Fendley, Journal of Statistical Mechanics: Theory and Experiment2019, 043107 (2019)

    2019

  89. [98]

    Lu, Annals of Physics470, 169806 (2024)

    Y.-M. Lu, Annals of Physics470, 169806 (2024)

    2024

  90. [99]

    C.-M. Jian, Z. Bi, and C. Xu, Physical Review B97, 054412 (2018)

    2018

  91. [100]

    D. V. Else and C. Nayak, Physical Review B90, 235137 (2014)

    2014

  92. [101]

    E. Lieb, T. Schultz, and D. Mattis, Annals of Physics 16, 407 (1961)

    1961

  93. [102]

    Affleck and E

    I. Affleck and E. H. Lieb, letters in mathematical physics 12, 57 (1986)

    1986

  94. [103]

    Oshikawa, Physical review letters84, 1535 (2000)

    M. Oshikawa, Physical review letters84, 1535 (2000)

    2000

  95. [104]

    G. Y. Cho, C.-T. Hsieh, and S. Ryu, Physical Review B96, 195105 (2017)

    2017

  96. [105]

    H. C. Po, H. Watanabe, C.-M. Jian, and M. P. Zaletel, Physical review letters119, 127202 (2017)

    2017

  97. [106]

    Notice that the opposite implication is not true, as one can always fine-tune a degenerate mode that has noth- ing to do with symmetry charge

  98. [107]

    Kawabata, R

    K. Kawabata, R. Sohal, and S. Ryu, Physical review letters132, 070402 (2024)

    2024

  99. [108]

    You and M

    Y. You and M. Oshikawa, Physical Review B110, 165160 (2024)

    2024

  100. [109]

    Ma and A

    R. Ma and A. Turzillo, arXiv preprint arXiv:2403.13280 (2024)

    2024 arXiv

  101. [110]

    Ma, J.-H

    R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, arXiv preprint arXiv:2305.16399 (2023)

    2023 arXiv

  102. [111]

    C. Xu, Y. Zang, Y. Ma, Y. Gu, and S. Jiang, to appear (2025)

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.