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arxiv: 2408.00597 · v3 · submitted 2024-08-01 · ❄️ cond-mat.stat-mech

Topological Classification of Dynamical Quantum Phase Transitions in the 1D XY model via Critical Mode Analysis

Bao-Ming Xu This is my paper

Pith reviewed 2026-05-23 22:32 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords dynamical quantum phase transitionstopological classificationXY modelwinding numbercritical modesboundary modesinterior modesquench dynamics
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0 comments X

The pith

Critical interior modes produce integer winding DQPTs while boundary modes produce half-integer ones in the 1D XY model.

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

The paper shows that dynamical quantum phase transitions in the one-dimensional XY model under quenches fall into distinct topological classes depending on the character of their critical modes. Modes located inside the Brillouin zone always produce an integer jump in the winding number, while modes at the zone boundary always produce a half-integer jump. Counting the combinations of these two kinds of modes yields exactly six possible topological types, three of which had not been reported before. The same mode-based rule is stated to hold for any one-dimensional two-band model.

Core claim

The topological diversity of DQPTs originates from the nature of the critical modes, which are classified into boundary modes and interior modes. Critical interior modes always lead to DQPTs with an integer winding number, while critical boundary modes always result in DQPTs characterized by a half-integer winding number. By analyzing the number and classification of critical modes, DQPTs in the one-dimensional XY model are categorized into six types according to their distinct topological features, with corresponding dynamical phase diagrams provided for each.

What carries the argument

Classification of critical modes into interior (integer winding) versus boundary (half-integer winding) types, which fixes the topological class of each DQPT.

If this is right

  • DQPTs fall into six topological categories determined by the possible combinations of interior and boundary critical modes.
  • Three of the six categories have not been identified in earlier work on the XY model.
  • The same interior-boundary distinction supplies the winding number for any quench in the SSH model, Kitaev chain, Rice-Mele model, or Creutz model.
  • Dynamical phase diagrams can be drawn for each of the six types by locating the quenches that activate the corresponding mode combinations.

Where Pith is reading between the lines

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

  • The mode classification may allow prediction of winding numbers in other two-band chains without computing the full Loschmidt echo.
  • If the rule survives weak interactions, it could classify DQPTs in models that are not strictly free fermions.
  • Numerical checks on finite but large chains could test whether the integer versus half-integer distinction remains sharp when the critical mode sits very close to the zone boundary.

Load-bearing premise

The topological class of a DQPT is fixed solely by whether its critical modes are interior or boundary, with no further dependence on quench details or system size.

What would settle it

A single explicit quench in the XY model that produces a DQPT whose winding-number jump is half-integer when driven only by interior critical modes, or integer when driven only by boundary critical modes.

Figures

Figures reproduced from arXiv: 2408.00597 by Bao-Ming Xu.

Figure 1
Figure 1. Figure 1: FIG. 1: (Color online) The schematic drawing of the Bloch [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (Color online) Lines of Fisher zeros (a1 and a2), the t [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (Color online) Lines of Fisher zeros (a), the time evo [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (Color online) Lines of Fisher zeros (a), the time evo [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (Color online) Lines of Fisher zeros (a1 and a2), the t [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: (Color online) Lines of Fisher zeros (a), the time evo [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: (Color online) Lines of Fisher zeros (a), the time evo [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: (Color online) The dynamical phase diagrams. Note th [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

Dynamical quantum phase transitions (DQPTs), which serve as a theoretical framework for understanding far-from-equilibrium physics in quantum many-body systems, have recently been observed experimentally. Their topological properties are typically characterized by the winding number, which acts as an order parameter. While DQPTs exhibiting both integer and half-integer jumps in the winding number have been reported, the underlying mechanisms behind these distinct topological behaviors, as well as the potential existence of other topological classes, remain open questions. To address this, we investigate DQPTs in the one-dimensional XY model under a quench protocol. We show that the observed topological diversity originates from the nature of the critical modes, which we classify into two categories: boundary modes and interior modes. Specifically, critical interior modes always lead to DQPTs with an integer winding number, while critical boundary modes always result in DQPTs characterized by a half-integer winding number. By analyzing the number and classification of critical modes, we provide a classification of the topological properties of DQPTs in the one-dimensional XY model. According to their distinct topological features, we categorize DQPTs into six types, three of which have not been previously identified in the literature. We discuss in detail the conditions associated with each type and present the corresponding dynamical phase diagrams. Our framework is not restricted to the XY model; it is applicable to other two-band models in one-dimensional systems, including the SSH model, Kitaev chain, Rice-Mele model, and Creutz model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that DQPTs in the 1D XY model under quenches arise from critical modes that fall into two classes—boundary and interior—with interior modes always producing integer winding-number jumps and boundary modes always producing half-integer jumps. Counting and labeling these modes yields a six-type topological classification of DQPTs (three previously unreported), together with associated dynamical phase diagrams; the framework is asserted to extend to other 1D two-band models.

Significance. If the mode-type rule is shown to be robust, the work supplies a concrete, predictive taxonomy that accounts for the coexistence of integer and half-integer winding numbers and identifies new classes, thereby organizing a body of numerical and experimental observations in integrable 1D systems.

major comments (2)
  1. [abstract and critical-mode classification section] The central claim that the winding-number jump is fixed solely by whether a critical mode is boundary or interior (abstract and the classification section) is load-bearing. The manuscript must demonstrate explicitly that this assignment is insensitive to the concrete quench trajectory in (h,γ) space, the k-location of the critical point inside the Brillouin zone, and simultaneous passage through multiple critical points; otherwise the six-type taxonomy does not follow.
  2. [section deriving the winding jump from the mode product] The Loschmidt amplitude is a product over k-modes. The derivation that a boundary-mode contribution cannot be altered by the dispersion relation or by finite-size corrections (thereby preserving the strict half-integer jump) is not shown to be general; a counter-example or analytic argument ruling out such dependence is required.
minor comments (2)
  1. [mode classification paragraph] Notation for “boundary” versus “interior” modes should be defined with an explicit k-interval or dispersion criterion rather than left implicit.
  2. [phase-diagram figures] The dynamical phase diagrams would benefit from an overlay of the quench paths used to realize each of the six types.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism. We agree that the central claims require explicit demonstrations of insensitivity and generality to fully support the proposed classification. Below we respond to each major comment and indicate the planned revisions.

read point-by-point responses

  1. Referee: [abstract and critical-mode classification section] The central claim that the winding-number jump is fixed solely by whether a critical mode is boundary or interior (abstract and the classification section) is load-bearing. The manuscript must demonstrate explicitly that this assignment is insensitive to the concrete quench trajectory in (h,γ) space, the k-location of the critical point inside the Brillouin zone, and simultaneous passage through multiple critical points; otherwise the six-type taxonomy does not follow.

    Authors: We agree that an explicit demonstration of insensitivity is necessary to substantiate the load-bearing claim. In the revised manuscript we will insert a dedicated subsection immediately after the critical-mode classification. This subsection will (i) consider representative quench trajectories spanning different paths in (h,γ) space that cross the same critical mode, (ii) vary the Brillouin-zone location of the critical point while keeping its boundary/interior character fixed, and (iii) examine simultaneous crossings of multiple critical modes of mixed types. Analytic arguments based on the Loschmidt-echo product structure together with numerical scans will show that the winding-number jump remains determined exclusively by mode type, thereby justifying the six-type taxonomy. revision: yes

  2. Referee: [section deriving the winding jump from the mode product] The Loschmidt amplitude is a product over k-modes. The derivation that a boundary-mode contribution cannot be altered by the dispersion relation or by finite-size corrections (thereby preserving the strict half-integer jump) is not shown to be general; a counter-example or analytic argument ruling out such dependence is required.

    Authors: We concur that the generality of the boundary-mode half-integer contribution must be established more rigorously. In the revised derivation section we will augment the product-form analysis with an explicit analytic argument: near a boundary critical point the phase accumulation contributed by that mode is shown to be exactly π (mod 2π) independent of the concrete dispersion relation, because the relevant matrix element vanishes linearly while the gap closes quadratically. Finite-size corrections are addressed by demonstrating that any 1/L corrections to the Loschmidt amplitude vanish in the thermodynamic limit and do not alter the topological jump; this is supported by analytic bounds and numerical checks across system sizes up to several hundred sites for representative dispersions. revision: yes

Circularity Check

0 steps flagged

No circularity; classification derived from explicit mode analysis of Loschmidt amplitude

full rationale

The paper's central derivation classifies DQPTs by counting and labeling critical modes (boundary vs. interior) in the product-form Loschmidt amplitude of the XY model, then assigns integer vs. half-integer winding jumps on that basis. No quoted step reduces a prediction to a fitted parameter, renames a known result, or loads the uniqueness claim on a self-citation whose content is itself unverified. The six-type taxonomy follows directly from enumerating mode combinations under the stated quench protocol; external applicability to SSH, Kitaev, etc., is asserted without circular bootstrap. This is the normal non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification rests on standard domain assumptions of topological band theory and quench dynamics; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The winding number functions as a topological order parameter that jumps at DQPTs
    Invoked throughout the abstract as the quantity whose integer or half-integer character defines the classes
  • domain assumption The 1D XY model and similar systems are two-band models whose critical modes can be classified as boundary or interior
    Stated as the basis for the six-type taxonomy

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