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Dynamically Induced Topology and Quantum Monodromies in a Proximity Quenched Gapless Wire

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Pith's one-line read A sudden quench coupling a topologically trivial gapless wire to fractional or Majorana bound states turns those bound states into mobile anyons whose fractional charge, fermion parity, and exchange statistics remain measurable over many…

desk verdict Useful extension of Majorana quench interferometry with solid exact solutions, but the statistical-phase extraction rests on an unproven cancellation. read the letter →

arxiv 1908.06111 v1 pith:A4QFK63Y submitted 2019-08-16 cond-mat.mes-hall cond-mat.str-el

Daniel Dahan (BGU) , Eytan Grosfeld (BGU) , Babak Seradjeh (IUB , MPI-PKS) This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.str-el
keywords quantumquenchtopologicalboundstatesMajoranafermionsfractionalchargeexchangestatisticsboundarychangingoperatorsgaplesswireentanglemententropy
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

After a sudden quench couples a topologically trivial one-dimensional gapless wire to topological bound states, the bound states do not simply dissolve: they leak into the wire, propagate through it, and retain measurable signatures of the parent topological phase over many traversal times. The paper claims this is a temporal proximity effect, with topology dynamically induced in the wire even though the wire itself is topologically trivial in equilibrium. For fractional solitons the signature is a moving charge $1/2$ with near-vanishing charge fluctuations at its center; for Majorana states it is a zero subsystem fermion parity between partners and a doubling of the revival period. These signatures survive moderate disorder and short-range interactions, and the revival phases carry the Abelian and non-Abelian exchange statistics of the bound states. If correct, the work turns a quench-and-measure protocol in one dimension into a probe of fractional statistics without braiding particles in two dimensions.

What carries the argument

The engine of the argument is the boundary-changing operator (BCO). In the low-energy limit the quench acts like a $\mathbb{Z}_2$ branch cut on the chiral field of the wire, and the post-quench state is $\sigma(\ell,t)\sigma(0,t)|\Omega_0\rangle$, where $\sigma$ is the twist field of the Ising conformal field theory (conformal dimension $h_\sigma=1/16$). The fidelity and the statistical phases are read from the four-point correlator of these twist fields, whose analytic continuation produces the quantum monodromies. Exchange phases are isolated through the ratio-of-overlaps phase in Eq. (1), $\theta = \arg[O_{q_Lq_R,p_Lp_R}(t)/O_{q'_Lq'_R,p'_Lp'_R}(t)]$, which is asserted to remove the dynamical phase. The exact single- and two-bound-state evolutions are carried by functions $g_{ab}(t)$ built from Laguerre polynomials, with peak amplitudes decaying as $n^{-1/3}$ after $n$ revivals, which gives the long coherence time.

What would settle it

Compute the full time-resolved overlaps $O_{q_Lq_R,p_Lp_R}(t)$ in a finite wire and test whether $\theta_s^{SSH}(t)=\frac{1}{2}\arg[O_{11,11}(t)/O_{10,10}(t)]$ stays flat over successive revival windows once the non-interacting evolution is exactly accounted for; any drift with the revival number $n$ would show that a non-statistical dynamical contribution has leaked into the extracted phase.

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Extended reading notes

Core claim

The central claim is that a sudden coupling of a gapless wire to topological bound states—fractional SSH solitons or Kitaev Majoranas—dynamically induces topological properties in the wire that remain observable over a long coherence time. After the quench the bound states enter the wire as mobile anyons: the SSH soliton propagates as a charge-$1/2$ excitation with suppressed local charge fluctuations, while a propagating Majorana forces the fermion parity of every subsystem lying between it and its remote partner to zero. The paper further claims that exchange statistics are visible in the revival dynamics: for two Majoranas the non-Abelian Ising statistics appears as a doubled period of fidelity revivals, and for two SSH solitons the Abelian statistical phase $\theta_s^{SSH}=\frac{1}{2}\theta_{11,11}^{10,10}$ is accumulated in the relative phase between parity-resolved overlaps, with a corresponding Majorana phase $\theta_s^K=\theta_{11,00}^{00,00}$. Analytical low-energy results based on the Ising conformal field theory agree with exact diagonalization, and time-dependent density matrix renormalization group calculations show that short-range interactions can sharpen the propagating soliton charge. The parent topology is therefore imprinted on the wire in time, not in space.

Load-bearing premise

The extraction of exchange statistics rests on the assumption that the relative phase between two overlap amplitudes cancels the dynamical phase exactly, so that what remains is purely the statistical phase; the paper does not prove that the two processes acquire identical dynamical phases.

Editorial extensions

If this is right

  • A single quench lets a fractional SSH soliton enter the wire as a mobile charge-$1/2$ excitation; at its center the local charge fluctuations nearly vanish, so the fractional charge behaves as a good quantum number while moving.
  • With two Kitaev chains, the non-Abelian Ising statistics of the Majorana pair is visible as a doubling of the fidelity revival period, and the fermion parity of every subsystem between a propagating Majorana and its remote partner is zero.
  • The statistical phases $\theta_s^{SSH}$ and $\theta_s^K$ are encoded in the relative phases of parity-resolved overlaps and remain coherent over many return times even though the revival amplitude decays as $n^{-1/3}$.
  • Short-range interactions in the wire can reduce the decay of the soliton charge and sharpen its revival peaks; modest disorder leaves the first few fidelity, parity, and charge peaks sharp.
  • A double quench creates, at the moment of disconnection, a particle-hole pair for the SSH case and a Majorana pair for the Kitaev case, with one member trapped near the interface and the other propagating through the wire.

Reading between the lines

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

  • The one-dimensional revival monodromy is treated as a stand-in for two-dimensional braiding; a direct comparison of these phases with a genuine braiding protocol in the same microscopic model would test that equivalence, but the paper does not carry it out.
  • If the same boundary-changing-operator structure holds for parafermions, fidelity revivals should exhibit richer non-Abelian monodromies than the Ising case; the authors only gesture at this extension.
  • The two-copy interference idea sketched for measuring entanglement could be adapted to measure the relative statistical phase directly, making the protocol testable in nanowire or cold-atom arrays.
  • Because interactions sharpen the soliton charge, tuning interactions might be used to increase the visibility of the statistical phase, a consequence the paper does not discuss.
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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

3 major / 5 minor

Summary. This paper considers a one-dimensional gapless wire that is suddenly coupled at its ends to topological bound states from SSH chains (fractional solitons) or Kitaev chains (Majorana zero modes). The authors claim that after the quench the bound states leak into the wire, propagate as mobile anyons, and leave measurable signatures of the parent topology: quantized fractional charge, zero subsystem fermion parity, entanglement-entropy patterns, and relative phases between parity-resolved overlaps that are interpreted as Abelian and non-Abelian exchange-statistics phases. The analytical framework combines exact equations-of-motion solutions (Eqs. (8)-(10), Appendix C) with a conformal-field-theory description in which the quench inserts Ising twist fields acting as boundary-changing operators, and the overlap of Eq. (7) is expressed through four-twist correlators. The analytical results are compared with exact diagonalization and tDMRG simulations for clean, disordered, and interacting wires.

Significance. The proposed effect is conceptually interesting and, if established, would generalize the topological proximity effect to the time domain and provide experimentally accessible signatures of fractional statistics in a 1D quench setup. Strengths of the paper include the parameter-free analytical solutions (given the model), the explicit derivation of the equations-of-motion propagators and the Pfaffian/Onishi overlap formulas, and the independent numerical checks by exact diagonalization and tDMRG. The main caveat is that the central phase-extraction procedure in Eq. (1) and the identification of 1D revival monodromies with braiding statistics are asserted rather than derived; unless these are supplied, the quantitative claims about exchange statistics remain a conjecture.

major comments (3)
  1. [Section III, Eq. (1)] The statement that the overlap ratio 'encodes various non-Abelian statistical phases by removing the dynamical phase' is the central load-bearing assumption and is not proved. After the quench the Hamiltonian is time-independent, so each overlap is a sum of oscillating eigenstate contributions, O_{q,p}(t)=Σ_n ⟨Ω_q(0)|n⟩⟨n|Ω_p(0)⟩e^{-iE_nt}; the ratio of two such sums does not in general factor as a common dynamical phase times a statistical phase. The exact solutions in Appendix C and the four-twist correlator (C25) may imply a factorization, but the paper does not show it, nor does it demonstrate that the sinh and cutoff-dependent prefactors of (C25) cancel in the parity-specific ratio. This matters because the plateau values θ_SSH^s and θ_K^s are the quantitative evidence for statistics; if the cancellation fails, these phases contain non-universal contributions. Please add a derivation of Eq. (1) from the overlap formulas, or explicitly state and justify the conditions under which the dynamical phase cancels.
  2. [Section IV and Appendix C, Eqs. (7), (C18)-(C25)] The connection between the CFT overlap (7) and the parity-resolved overlaps entering Eq. (1) is not spelled out. Equation (7) is the fidelity amplitude of the full many-body state, whereas Eq. (1) involves overlaps between states with different initial/final parities (p_L,p_R) and (q_L,q_R). The text mentions that the SSH case uses conformal blocks D_± = B_+^2 ± B_-^2 and the Kitaev case uses B_±, but it never derives which block corresponds to which overlap O_{q_L q_R,p_L p_R}(t), nor how the ratio in Eq. (1) follows from (C25). Without this mapping the analytical curves in Fig. 2(b),(d) are not independently checkable from the printed equations. Please provide the explicit parity-resolved overlap expressions and the resulting formula for θ.
  3. [Section III (paragraph after Eq. (7)) and Section VI] The identification of the revival-phase monodromies with 'exchange statistics' or 'braiding' of mobile anyons in one dimension is under-motivated. In 2+1D braiding is defined by worldlines that avoid each other, while in a 1D wire two TBSs propagating through the same channel must meet unless a precise chiral-separation mechanism is specified; the phrase 'they effectively braid, keeping their chirality intact' is not a definition. The paper should either supply a concrete mapping from the time evolution of Eqs. (6)-(10) to the braid group (even in a restricted sense), or clearly state the weaker claim that the extracted phases equal the BCO monodromies of the conformal blocks, and that this is a signature rather than a proof of exchange statistics.
minor comments (5)
  1. [Eq. (8)] The k=0 term is not well defined because the binomial coefficient (j-1 choose k-1) is used for k=0 and j≥0; the expression should either follow the Laguerre representation in Eq. (C9) or state the convention for the k=0 term.
  2. [Fig. 6 caption] The caption contains the typo 'Soltion charge'; please also specify the charge window and the smoothing length used in the figure.
  3. [Appendix D heading] The heading 'formulea' should read 'formulae'.
  4. [Section V.A, Eq. (11)] The smoothing length l is a free parameter; the figures show a single value (l=20), and a convergence statement or a short discussion of the dependence of Q^{(s)} on l would help the reader assess the robustness of the fractional-charge claim.
  5. [Fig. 2 and Eqs. (2)-(3)] The superscript/subscript notation for θ_SSH^s and θ_K^s in the caption differs from Eqs. (2)-(3); please unify the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the overlap phases and monodromies are computed from the quench Hamiltonian, checked by exact diagonalization, and the self-citation to Ref. [9] is not load-bearing.

full rationale

The paper's central derivation is self-contained rather than circular. The statistical phases θ_SSH_s and θ_K^s are not fitted parameters; they are obtained from exact diagonalization of the lattice model (Appendix A) and from independent analytic solutions of the effective equations of motion (Appendix C, Eqs. (8)-(10)), which are then compared in Fig. 2 with no adjustable target. The BCO/twist-field overlap calculation, Eq. (7) and Eqs. (C18)-(C25), follows from the stated low-energy limit λ ≫ v_F/√ℓ and is validated against exact diagonalization (Fig. 3), so the monodromy result is a derived consequence of the model, not an input. The only self-citation of note is Ref. [9] for the Majorana period-doubling interpretation; that prior result is disclosed, and the present paper independently derives the relevant revival dynamics and parity formulas, so the citation is not load-bearing. The assertion in Eq. (1) that the overlap-ratio phase 'encodes' exchange statistics by removing the dynamical phase is an interpretive step whose dynamical-phase cancellation is not fully proven; that is a correctness/justification concern, not a circular reduction, since the value of the ratio is computed rather than inserted by definition. I therefore find no circular step under the stated criteria.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No parameter is fitted to the central observables; the analytical curves in Figs. 2-3 use the model parameters directly. The only hand-chosen extraction parameter is the charge-smoothing length l=20. The central claim relies on the low-energy mapping to a chiral field with a Z2 branch cut, the identification of the quench with Ising twist fields, and the phase-extraction assumption of Eq. (1). These are domain or paper-specific assumptions rather than standard math. No new particles, forces, or conserved quantities are introduced.

free parameters (1)
  • smoothing length l = 20 (lattice sites)
    Hand-chosen in Eq. (11) and used in Figs. 4 and 6 to define the smoothed charge; it controls the width of the charge profile without changing the quantized value of the fractional charge.
assumptions (6)
  • domain assumption The low-energy linearized chiral field theory (Eq. 4) with local coupling to TBSs (Eq. 5) accurately describes the lattice SSH/Kitaev chains coupled to the gapless wire at half filling.
    Invoked in Section IV and Appendix C when replacing lattice models by the effective chiral Hamiltonian and coupling; this is a standard continuum-limit assumption, not separately proven for each parameter set.
  • ad hoc to paper In the strong-coupling limit lambda >> v_F/sqrt(l), the quench is equivalent to inserting a Z2 branch cut implemented by Ising twist fields sigma(0,t) sigma(l,t) acting as boundary-changing operators.
    Stated in Section IV ('the quench introduces a Z2 branch cut') and used in Eqs. (7) and (C18)-(C25); if this mapping fails, the predicted monodromies are not guaranteed.
  • ad hoc to paper The relative phase between two overlap processes in Eq. (1) cancels the dynamical phase and isolates the exchange statistics phase.
    Assumed in Section III to define theta_SSH_s and theta_K_s; the cancellation is asserted, not derived, and is the basis for claiming statistical phases in one dimension.
  • standard math Standard CFT correlators for the Ising model and the conformal map from strip to plane apply to the non-interacting unfolded chiral fermion and give the correct monodromies.
    Used in Appendix C3; these are standard results from conformal field theory, not re-derived in the paper.
  • domain assumption SSH solitons carry charge 1/2 with Abelian statistics and Kitaev Majorana modes obey Ising non-Abelian statistics.
    Background imported from Refs. [2,3,18,19] without derivation; needed to interpret the observed quantum numbers and phases.
  • standard math The Onishi/Pfaffian overlap formula and the Nambu-Green's function trace formulas correctly compute overlaps, parity, and entanglement for the noninteracting system.
    Used in Appendices B and D to obtain numerical observables; standard results supplied with derivations in the appendices.

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Pith. "Pith review of Dynamically Induced Topology and Quantum Monodromies in a Proximity Quenched Gapless Wire." pith.science (2026). https://pith.science/paper/A4QFK63Y

@misc{pith2026190806111,
  author       = {Pith},
  title        = {Pith review of: Dynamically Induced Topology and Quantum Monodromies in a Proximity Quenched Gapless Wire},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4QFK63Y}},
  note         = {Machine review of arXiv:1908.06111}
}
read the original abstract

We study the quench dynamics of a topologically trivial one-dimensional gapless wire following its sudden coupling to topological bound states. We find that as the bound states leak into and propagate through the wire, signatures of their topological nature survive and remain measurable over a long lifetime. Thus, the quench dynamically induces topological properties in the gapless wire. Specifically, we study a gapless wire coupled to fractionally charged solitons or Majorana fermions and characterize the dynamically induced topology in the wire, in the presence of disorder and short-range interactions, by analytical and numerical calculations of the dynamics of fractional charge, fermion parity, entanglement entropy, and fractional exchange statistics. In a dual effective description, this phenomenon is described by correlators of boundary changing operators, which, remarkably, generate topologically non-trivial monodromies in the gapless wire, both for abelian and non-abelian quantum statistics of the bound states.

Figures

Figures reproduced from arXiv: 1908.06111 by the authors.

Figure 1
Figure 1. FIG. 1. Setup: a gapless wire of length [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Fidelity and statistical phase. Panels (a) and (b) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison between the exact diagonalization (solid [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Local soliton charge (a) and excess fluctuations (b) [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Entanglement entropy (left) and charge fluctuations [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Soltion charge [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The fermion parity Π to the left of a cut in the [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Fidelity, parity and soliton charge in the presence of disorder in the GW. Panels (a) and (b): a sudden quench of a [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

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