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Post-Quench Evolution of Complexity and Entanglement in a Topological System

Tibra Ali, Arpan Bhattacharyya, S. Shajidul Haque, Eugene H. Kim, Nathan Moynihan

arXiv:1811.05985v2hep-thcond-mat.stat-mechcond-mat.str-elquant-ph

TL;DR

Motivated by questions about complexity and entanglement after quantum quenches, the paper compares their evolution in the SSH model. It finds rapid complexity saturation and revivals, while showing that complexity is less sensitive to topological order than entanglement measures.

  • Problem

    The paper examines how complexity and entanglement evolve after a quantum quench, against the backdrop that entanglement may not capture bulk evolution after boundary equilibration.

    Source: §1 · pp. 2–3

  • Method

    The authors compare circuit and Fubini-Study complexity and fidelity with entanglement measures after quenches in the one-dimensional SSH model.

    Source: §1 · p. 3

  • Results

    In the SSH model, complexity saturates much faster than entanglement entropy, exhibits revivals, and is less sensitive to topological order than entanglement measures.

    Source: §1 · p. 3, §3 · pp. 11–12

  • Takeaways & Limitations

    Complexity, particularly the Fubini-Study approach, is computationally less demanding than entanglement measures and captures revivals and information about topological transitions.

    Source: §3 · p. 12

    In the SSH model, complexity is a less sensitive probe of topological order than entanglement, and the circuit approach cannot distinguish the cited topological transitions.

    Source: §2 · p. 11, §3 · p. 12

Abstract

from arXiv · show

We investigate the evolution of complexity and entanglement following a quench in a one-dimensional topological system, namely the Su-Schrieffer-Heeger model. We demonstrate that complexity can detect quantum phase transitions and shows signatures of revivals; this observation provides a practical advantage in information processing. We also show that the complexity saturates much faster than the entanglement entropy in this system, and we provide a physical argument for this. Finally, we demonstrate that complexity is a less sensitive probe of topological order, compared with measures of entanglement.

1 Introduction

The introduction frames complexity as a candidate complement to entanglement for understanding quantum systems, then introduces a quench study of both measures in the SSH model. The study examines their dynamics, phase-transition signatures, and sensitivity to topological order.

1 Introduction

  • The motivation is that entanglement alone may not capture bulk geometric evolution after a boundary CFT equilibrates, while complexity has been proposed as a continuing boundary quantity.This framing draws on the AdS/CFT context.Source: §1 · p. 2
  • Complexity is the shortest distance between reference and target states, operationally the minimum number of operations needed to transform one into the other.The introduction contrasts this computational concept with entanglement measures of uncertainty.Source: §1 · p. 2
  • Prior QFT complexity studies cover several theories and approaches, but the introduction characterizes complexity research as still in its infancy.The cited work includes scalar and fermionic theories, thermofield-double states, and quenches in bosonic QFTs.Source: §1 · pp. 2–3
  • The paper compares circuit and Fubini-Study complexity, fidelity, and entanglement after SSH-model quenches, examining saturation, revivals, phase transitions, and topological order.The authors describe the SSH model as a tractable one-dimensional topological insulator with a nontrivial phase diagram.Source: §1 · p. 3

2 Compelxity and Entanglement for SSH model

The section compares entanglement and two measures of complexity during SSH quenches, finding rapid complexity saturation, revivals, and distinct sensitivity to topological phases. It also relates these dynamics to entanglement structure and finite-size effects.

Complexity methods

  • Circuit complexity is computed as the minimum cost over unitary paths connecting the initial ground state to its time-evolved state.The chosen reference state is the initial ground state, rather than a direct product state, to define a distance useful for detecting phase transitions.Source: §2 · p. 5
  • The study considers quenches among topological, non-topological, and critical SSH phases, and compares circuit complexity with the Fubini-Study geodesic distance between states.The topological phase is defined by |q| < |q′|, the non-topological phase by |q| > |q′|, and the critical point by |q| = |q′|.Source: §2 · pp. 3, 6

Entanglement dynamics

  • Entanglement entropy grows linearly before saturating and later revives; its amplitude is larger and its revival period longer between massive phases than for quenches to or from the critical point.The scaled curves S(t)/l versus t/l collapse onto universal curves, and EE is slightly larger in the topological phase.Source: §2 · p. 7, Figure 1 · p. 7
  • The topological phase has entanglement-spectrum zero modes near λn = 1/2, which remain pinned when the system starts in that phase and arise from edge states.The figure compares quenches from non-topological to topological and from topological to non-topological phases.Source: §2 · p. 8, Figure 2 · p. 8

Complexity dynamics

  • For the non-topological-to-critical quench, Hamiltonian trajectories lie near the geodesic for some momenta but far from it for others, changing which paths contribute to complexity.The paper reports qualitatively similar behavior for the other quenches.Source: Figure 3 · p. 8, §2 · p. 9
  • Both circuit and Fubini-Study complexity grow rapidly and saturate with oscillations; the fidelity behaves similarly as the target state quickly becomes orthogonal to the reference.The authors identify this rapid orthogonalization as the reason for the complexity’s rapid growth and saturation.Source: §2 · p. 10
  • Complexity shows distinct revivals for quenches to the critical point, while revivals for quenches to massive phases are more pronounced between massive phases; high-frequency oscillations slow as N increases.The high-frequency oscillation is attributed to finite system size and is reported across the quench cases.Source: §2 · pp. 10–11, Figure 6 · p. 11

Comparing complexity measures

  • Fubini-Study complexity distinguishes quenches between the topological and non-topological phases and their critical-point quenches, whereas circuit complexity cannot distinguish the latter pair.For topological-to-non-topological quenches, both approaches give identical results in either quench direction.Source: §2 · p. 11

3 Discussion

In the SSH quench, complexity saturates much faster than entanglement entropy, with rapid fidelity decay offered as the mechanism but its quantitative link unresolved. Complexity also shows revivals, while its sensitivity to topological order depends on the measure used.

3 Discussion

  • Complexity saturates much faster than entanglement entropy in the SSH model, contrary to the slower saturation expected in the black-hole scenario.The authors tie rapid saturation to rapid fidelity decay and note that its precise relationship to saturation is not quantitatively understood.Source: §3 · pp. 11–12
  • Complexity shows revivals and, particularly with the Fubini-Study approach, captures information about topological transitions at lower computational cost than entanglement measures.The authors present this as making complexity appealing as a probe of quantum phase transitions.Source: §3 · p. 12
  • Entanglement is more sensitive to topological order: the entanglement entropy is larger in the TI phase, while the entanglement spectrum exposes edge-mode structure.The Fubini-Study approach distinguishes the TI and non-TI phases through their QCP quenches, but circuit complexity does not; Fubini-Study is more sensitive than expected.Source: §3 · p. 12
  • Because the SSH model has been realized in cold-atom systems where entanglement has been measured, the authors consider experimental probing of some ideas discussed here plausible.The passage frames this as an expectation, not a reported experiment.Source: §3 · p. 12

A.1 The System

This appendix section introduces the fermionic Hamiltonian used for the SSH model and identifies its momentum-space operators. It specifies the SSH parameters and the physical roles of the conduction- and valence-band operators.

A.1 The System

  • For the SSH model, the Hamiltonian parameters are ξ_k = 0 and Δ_k = q + q′ exp(−ik).The passage says the fermionic Hamiltonian is readily diagonalized.Source: §A.1 · p. 13
  • The operators {α_k, β_k} are related to {ã_k, b̃_k} by a Bogoliubov transformation.The passage places this relation after diagonalizing the Hamiltonian.Source: §A.1 · p. 13
  • α_k and β_k annihilate momentum-k fermions in the conduction and valence bands, respectively.Source: §A.1 · p. 14

A.2.1 Quench Protocol

The quench protocol considers a quench in the stated model and relates the eigenoperators of the initial and final Hamiltonians. The final Hamiltonian is then expressed using the initial Hamiltonian’s eigenoperators.

A.2.1 Quench Protocol

  • The appendix introduces a quench in the model described above.Source: §A.2.1 · p. 14
  • The eigenoperators of the initial and final Hamiltonians are related by a unitary transformation.Source: §A.2.1 · p. 14
  • The final Hamiltonian is written in terms of the initial Hamiltonian’s eigenoperators, which obey an SU(2) algebra.Source: §A.2.1 · pp. 14–15

A.3 Reference and Target States

The initial ground state of the pre-quench Hamiltonian is the reference state, and its evolution under the post-quench Hamiltonian defines the target state. The appendix also notes a decomposition used to obtain later equations and a notation simplification.

A.3 Reference and Target States

  • The time-evolution operator is introduced for evolving the reference state into the target state.Source: §A.3 · p. 15
  • The reference state is the ground state of the initial Hamiltonian, and the target state is its evolution under the final Hamiltonian.The initial state has the initial Hamiltonian’s valence band completely filled.Source: §A.3 · p. 15
  • A decomposition is used to obtain Eqs. (A.11b) and (A.11c), with γ+ written as z to simplify notation in Eq. (A.11b).Source: §A.3 · pp. 15–16

B State Manifold: Riemannian Metric; Geodesics and Complexity

The Bures distance gives the state space a Riemannian structure, whose line elements and geodesics define complexity. For each momentum the state lies on a two-sphere, and the time-evolution operator moves it on the Bloch sphere.

B State Manifold: Riemannian Metric; Geodesics and Complexity

  • The Bures distance between nearby states yields a line element and a Riemannian structure on the space of states.The fidelity F12 is Taylor-expanded to obtain the line element.Source: B State Manifold: Riemannian Metric;… · p. 16
  • For each momentum, the state manifold is a two-sphere in the CP1 representation, with spherical coordinates set by zk = |zk| exp(iφk) and |zk| = tan θk/2.The time-evolution operator moves the system on the group manifold and, for each momentum, on the Bloch sphere.Source: B State Manifold: Riemannian Metric;… · pp. 16–17
  • The geodesics are great circles, and their arc-length Ck defines the complexity for a particular momentum.The arc-length between two points is sk; the geodesic length is given in spherical coordinates by the expression in p07.Source: B State Manifold: Riemannian Metric;… · p. 17
  • The full state manifold is S2 × S2 × · · · × S2 and is assigned a metric s.The resulting system complexity is denoted C.Source: B State Manifold: Riemannian Metric;… · p. 17

C Complexity and Entanglement from the Correlation Matrix

The correlation matrix provides a route from the system’s state to complexity and entanglement measures.

C Complexity and Entanglement from the Correlation Matrix

C.1 Complexity

The complexity procedure evolves the initial correlation matrix along a unitary path and minimizes a cost functional to select a geodesic. Entanglement measures are obtained from the reduced state and the correlation matrix restricted to subsystem A.

C.1 Complexity

  • The initial correlation matrix is evolved by a unitary operator to obtain the state’s complexity.The approach uses the correlation matrix at t = 0 and compares circuit complexity with the Fubini-Study line element.Source: §C.1 · p. 18
  • Minimizing the cost functional over control functions selects the optimal unitary path, whose geodesic arc-length is the complexity.The unitary is parameterized as a path-ordered exponential with group generators and control functions; its endpoint recovers the final correlation matrix.Source: §C.1 · p. 18
  • The geodesic solution permits either φk or χk to be solved unambiguously, but not both.Source: §C.1 · p. 19

C.2 Entanglement

  • For spatial entanglement, the reduced density matrix of subsystem A is obtained by tracing out the rest of the system, and measures use its restricted correlation matrix.The eigenvalues of the restricted correlation matrix form the entanglement spectrum; the Renyi entropy is a central measure of entanglement.Source: §C.2 · p. 20
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