PaperlayerView exact revision on arXiv

Source-linked AI summary

Quantum complexity and topological phases of matter

Pawel Caputa, Sinong Liu

arXiv:2205.05688v3hep-thcond-mat.stat-mechquant-ph

TL;DR

Whether quantum complexity can reliably distinguish topological phases remains an open question. The paper analyzes spread complexity in the SSH model, including ground states and quantum quenches, and finds phase-sensitive behavior, including a constant ground-state value in the topological regime.

  • Problem

    Whether complexity is a universal probe correlated with topological phases remains open, and existing Nielsen-type studies have yielded differing conclusions.

    Source: INTRODUCTION · p. 1

  • Method

    The authors analytically study spread complexity in the SSH model, computing its ground-state behavior and analyzing exactly solvable quench protocols.

    Source: INTRODUCTION · p. 1, COMPLEXITY DURING QUANTUM QUENCH · p. 4

  • Results

    In the SSH ground state, complexity depends on the parameter ratio in the trivial phase and is constant in the topological regime; quench evolution also shows phase-sensitive features.

    Source: CONCLUSIONS AND DISCUSSION · p. 5

  • Takeaways & Limitations

    The SSH and related Kitaev-model analyses motivate spread complexity as a possible probe of topological phases, while further tests in more general interacting models are needed.

    Source: CONCLUSIONS AND DISCUSSION · p. 5

    The SSH analysis assumes t1,t2 ≥ 0 and μs = 0, with the phase boundary at t1 = t2.

    Source: SPREAD COMPLEXITY IN THE SSH MODEL · p. 2

Abstract

from arXiv · show

In this work, we find that the complexity of quantum many-body states, defined as a spread in the Krylov basis, may serve as a new probe that distinguishes topological phases of matter. We illustrate this analytically in one of the representative examples, the Su-Schrieffer-Heeger model, finding that spread complexity becomes constant in the topological phase. Moreover, in the same setup, we analyze exactly solvable quench protocols where the evolution of the spread complexity shows distinct dynamical features depending on the topological vs non-topological phase of the initial state as well as the quench Hamiltonian.

INTRODUCTION

The introduction frames topological phases as a promising setting for testing quantum complexity, while noting that the role of complexity as a phase probe remains unresolved. The paper addresses this question by studying spread complexity in the SSH model.

INTRODUCTION

  • Earlier Nielsen-type studies found differing conclusions, with some arguing phase distinctions may appear as discontinuities under suitable choices of ambiguities [22] [23] [24].The cited framework involves choices such as gates, penalties, and cost functions.Source: INTRODUCTION · p. 1
  • Whether complexity is a universal probe correlated with topological phases remains an open question.Studies of state-complexity measures in topological models are still in their infancy.Source: INTRODUCTION · p. 1
  • The paper analyzes spread complexity [25] in the SSH model, finding it sensitive to the model's phases and constant in the topological phase.The introduction also reports that the measure distinguishes phases following quantum quenches.Source: INTRODUCTION · p. 1

SPREAD COMPLEXITY OF QUANTUM STATES

Spread complexity measures how a state spreads from a reference state under unitary evolution. Its computation uses the Krylov basis, where the state amplitudes determine the complexity.

SPREAD COMPLEXITY OF QUANTUM STATES

  • Spread complexity estimates state complexity from the spread of a unitarily evolved reference state across an orthonormal basis, minimizing a cost over bases that include the reference state.The paper takes the evolution parameter as arbitrary and allows it to represent physical time.Source: SPREAD COMPLEXITY OF QUANTUM STATES · pp. 1–2
  • The minimizing basis is the Krylov basis, formed by orthogonalizing the states generated by successive powers of the circuit Hamiltonian acting on the reference state.The orthogonalization uses the Lanczos algorithm; in this basis the Hamiltonian is generally tridiagonal.Source: SPREAD COMPLEXITY OF QUANTUM STATES · p. 2
  • Lanczos coefficients encode information also contained in moments of the return amplitude.The return amplitude is the overlap between the evolved state and the reference state.Source: SPREAD COMPLEXITY OF QUANTUM STATES · p. 2
  • Solving the discrete Schrödinger equation for the Krylov-basis amplitudes gives the probabilities used to compute complexity, with unitarity ensuring their sum is one.The initial condition places the state at the reference vector, and the return amplitude is related to the first coefficient.Source: SPREAD COMPLEXITY OF QUANTUM STATES · p. 2
  • The paper applies this state-based measure to test whether it can detect topological phases in integrable systems.The paper describes spread complexity as a generalization of operator K-complexity to quantum states.Source: SPREAD COMPLEXITY OF QUANTUM STATES · p. 2

SPREAD COMPLEXITY IN THE SSH MODEL

The paper computes spread complexity for the SSH ground state using an SU(2) coherent-state representation. The result is phase-dependent: it varies with the coupling ratio in the non-topological phase and is constant in the topological phase.

SPREAD COMPLEXITY IN THE SSH MODEL

  • The SSH model is topological for t2 > t1 and non-topological for t1 > t2, with the phases separated by t1 = t2.The calculation assumes t1,t2 ≥ 0 and μs = 0.Source: SPREAD COMPLEXITY IN THE SSH MODEL · p. 2
  • For each positive momentum mode, the authors write the ground state as a circuit from a reference ground state and use SU(2) coherent-state techniques to expand it in the Krylov basis.Because j = 1/2, each mode has two Krylov basis vectors and two amplitudes.Source: SPREAD COMPLEXITY IN THE SSH MODEL · p. 3
  • The total ground-state complexity density is obtained by integrating the mode contributions over momentum and including the negative-momentum sector.The continuum result is rescaled to remove the volume factor.Source: SPREAD COMPLEXITY IN THE SSH MODEL · p. 3
  • Spread complexity is constant for t2 > t1, while for t1 > t2 it depends linearly on the ratio t2/t1.The authors give an analytical expression for the ground-state complexity density; Fig. 1 presents its behavior across the phases.Source: SPREAD COMPLEXITY IN THE SSH MODEL · p. 3, Figure 1 · p. 3

COMPLEXITY DURING QUANTUM QUENCH

In SSH quenches, spread complexity distinguishes protocols that swap the initial and final Hamiltonian parameters, and its time dependence depends on both Hamiltonians and the initial state's phase. Analytical results identify phase-sensitive early-time growth and a late-time constant.

COMPLEXITY DURING QUANTUM QUENCH

  • The quench starts from the ground state of Hi and evolves it under a different SSH Hamiltonian Hf; each mode's complexity is derived using coherent-state techniques and the Loschmidt amplitude.The mode contributions are obtained by extracting Lanczos coefficients from the return amplitude.Source: COMPLEXITY DURING QUANTUM QUENCH · p. 4
  • For each momentum mode, complexity is periodic with periods set by Hf, while its amplitude depends on both Hamiltonians and is not symmetric under swapping them.When one final-Hamiltonian parameter vanishes, the oscillation amplitude distinguishes the initial state's two phases.Source: COMPLEXITY DURING QUANTUM QUENCH · p. 4
  • Swapping the initial and final Hamiltonian parameters produces distinguishable quench-protocol evolutions of spread complexity.The comparison is shown for initial and final Hamiltonians in opposite topological phases.Source: Figure 2 · p. 4, COMPLEXITY DURING QUANTUM QUENCH · p. 4
  • The early-time regime has universal quadratic growth that distinguishes whether the initial state is topological or non-topological.The phase boundary is parametrized by α = 1 for the initial Hamiltonian ratio.Source: COMPLEXITY DURING QUANTUM QUENCH · p. 4
  • At late times, spread complexity approaches a constant depending on α and β, symmetrically under their exchange and identically for the two swapped protocols.The constant results from summing otherwise periodic contributions from different momenta and can probe late-time quench-state physics.Source: COMPLEXITY DURING QUANTUM QUENCH · p. 5

CONCLUSIONS AND DISCUSSION

The SSH ground-state spread complexity depends on t1/t2 in the trivial phase but becomes constant in the topological regime. Quench dynamics also show features sensitive to phase, while further work is needed to connect spread complexity to established topological order parameters.

CONCLUSIONS AND DISCUSSION

  • In the SSH ground state, spread complexity depends on t1/t2 in the trivial phase and becomes constant in the topological insulator regime.The authors expect this behavior to be universal based on related analysis in the Kitaev model, and call for tests in more general interacting models.Source: CONCLUSIONS AND DISCUSSION · p. 5
  • Early- and late-time quench features are sensitive to topological phases, and the authors identify spread complexity’s advantage over geometric measures studied in [22].They propose further study of slow or fast quenches, time-dependent Hamiltonians, Floquet driving, and dynamical quantum phase transitions [47].Source: CONCLUSIONS AND DISCUSSION · p. 5
  • The relation between spread complexity and Berry phases or Chern numbers remains an important open connection, including order parameters extractable experimentally with certain quench protocols.The authors suggest these links may sharpen spread complexity as a tool in condensed matter and many-body physics.Source: CONCLUSIONS AND DISCUSSION · p. 5

APPENDIX A: DETAILS OF THE SSH MODEL

The appendix reformulates the SSH model as a free-fermion Hamiltonian in momentum space and expresses its ground state using SU(2) coherent states. It also identifies a parameter choice corresponding to the simplified main-text setup.

APPENDIX A: DETAILS OF THE SSH MODEL

  • A Fourier transform and Pauli-matrix rotation yield a free-fermion momentum-space Hamiltonian, which is then diagonalized using orthonormal eigenvectors.The Hamiltonian is also decomposed into parts analogous to left- and right-moving continuum Dirac Hamiltonians.Source: Appendix A · p. 7
  • The SSH ground state is written in terms of the free-fermion Fock vacuum and as a product of generalized SU(2) coherent states.The appendix relates the ground state to the left- and right-sector ground states through a Bogoliubov transformation.Source: Appendix A · p. 7
  • In the main text, setting μs = 0 corresponds to R2 = ψk = 0 in the appendix’s parameterization.Source: Appendix A · p. 8

APPENDIX B: SU(2) COHERENT STATES

The appendix introduces SU(2) coherent states and uses them to describe evolution in the Krylov basis. This representation supplies spread-complexity expressions for the quench calculation and a restricted relation to the return amplitude.

APPENDIX B: SU(2) COHERENT STATES

  • Generalized coherent states are defined by acting with a displacement operator on the lowest-weight state |j, −j⟩.The appendix also introduces the su(2) commutator algebra and its ladder operators.Source: Appendix B · p. 8
  • The coherent-state evolution is represented by amplitudes in Krylov basis vectors, and those amplitudes satisfy a discrete Schrödinger equation with Lanczos coefficients.For the quench calculation, this mapping yields the per-mode spread-complexity formula referenced as equation (15) in the main text.Source: Appendix B · pp. 8–9
  • For j = 1/2, the spread complexity is |ψ1(t)|^2 and can be related to the return amplitude by unitarity.The appendix notes that this relation does not hold for more general evolutions requiring more Krylov basis vectors.Source: Appendix B · p. 9

APPENDIX C: EARLY AND LATE TIMES

The appendix examines early- and late-time spread complexity after a quench. At late times, oscillatory contributions become negligible, yielding a parameter-dependent constant, with certain parameter values requiring separate treatment.

APPENDIX C: EARLY AND LATE TIMES

  • At t ∼ 0+, the early-time spread complexity is expressed using α and β, the initial and final hopping-parameter ratios.Source: Appendix C · p. 9
  • At late times, the oscillatory cosine contribution to the integral is negligible, giving a constant equivalent to replacing sin^2(Rf t) with 1/2.The resulting expression is the late-time constant shown in equation (23) of the main text.Source: Appendix C · p. 10
  • The expression for the late-time constant is naively singular at αβ = 1, α = 0, and β = 0; those cases require analysis from equation (20) before integrating over momenta.Source: Appendix C · p. 10

APPENDIX D: ONE-DIMENSIONAL KITAEV MODEL

Using the Kitaev chain as an additional example, the authors compute spread complexity with their general formalism and find a plateau in the topological phase. There it is independent of µ, while its constant depends on ∆.

APPENDIX D: ONE-DIMENSIONAL KITAEV MODEL

  • The calculation selects the ground state of HCP, the Cooper-pair vacuum, as the reference and uses SU(2) coherent states.The authors derive the total spread complexity in the continuum limit.Source: Appendix D · p. 11
  • In the topological phase, |µ| < 1, spread complexity is constant and independent of µ.The analytical constant depends on ∆.Source: Appendix D · p. 11
  • As |µ| → ∞, the spread complexities for the plotted ∆ values approach C = 1/2.The figure also marks analytical results for ∆ = ±1 within |µ| < 1.Source: Figure 4 · p. 11
  • Spread complexity has a discontinuity in its second derivative with respect to ∆.The passage introduces this feature after discussing the dependence of the topological-phase constant on ∆.Source: Appendix D · p. 11

APPENDIX E: RELATION TO TOPOLOGICAL ORDER PARAMETERS

The authors explore whether spread complexity relates to known topological-order probes by examining a position-space expectation value and its connection to entanglement. They conclude that a precise mapping to established probes remains unsettled.

APPENDIX E: RELATION TO TOPOLOGICAL ORDER PARAMETERS

  • A proposed connection places Lie-algebra K-complexity near Berry connections and Fubini–Study volume, with Zak’s phase as a comparison, but leaves the link for future work.The passage attributes the Berry-connection and Fubini–Study observations to [28] and the Zak-phase example to.Source: Appendix E · p. 12
  • The authors approximately relate ground-state spread complexity to an expectation value of a position-space operator.They use the operator expectation to develop intuition about the complexity measure.Source: Appendix E · p. 12
  • In the trivial phase, the reduced density matrix is pure, with trivial entanglement entropy, and the expectation value of Pj is correlated with that structure.The discussion compares this with a topological-phase reduction in which only one of three sectors contributes, retrieving only part of the ground-state information.Source: Appendix E · p. 13
  • For simpler integrable models, the authors say it is too early to determine whether spread complexity can be expressed through known topological-phase probes.They suggest a possible correlation with entanglement-like probes but say the details remain to be spelled out.Source: Appendix E · p. 13
Loading 2205.05688v3…