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Nonanalyticity of circuit complexity across topological phase transitions

Zijian Xiong, Dao-Xin Yao, Zhongbo Yan

arXiv:1906.11279v2cond-mat.str-elcond-mat.quant-gashep-thquant-ph

TL;DR

The paper asks whether topological phase transitions remain visible as nonanalyticity in circuit complexity under a different quantification and across dimensions and quench protocols. It studies one- and two-dimensional topological models using an inner-product-based complexity measure. Complexity exhibits nonanalyticity at transitions, with short-time and steady-state behavior identifying pre- and post-quench transitions, respectively.

  • Problem

    Because circuit-complexity quantification is not unique, the paper examines whether nonanalyticity at topological phase transitions persists under a different measure and across dimensions and quench protocols.

    Source: §I · p. 1

  • Method

    The paper uses an inner-product-based circuit-complexity measure to study topological models in one and two dimensions, both in equilibrium and after sudden or finite-time quenches.

    Source: §I · pp. 1–2, §II · p. 2

  • Results

    Complexity is nonanalytic at topological transitions; after quenches, short-time growth reveals pre-quench transitions and the long-time steady value reveals post-quench transitions.

    Source: §I · p. 2, §VI · p. 6

  • Takeaways & Limitations

    Short-time nonanalytic behavior can be determined from very few measurements and is expected to be useful for experimental studies of topological phase transitions.

    Source: §I · p. 2, §VI · p. 6

    In higher dimensions, weakened nonanalyticity means detecting transitions with this quantity requires higher measurement precision.

    Source: §V · p. 6

Abstract

from arXiv · show

The presence of nonanalyticity in observables is a manifestation of phase transitions. Through the study of two paradigmatic topological models in one and two dimensions, in this work we show that the circuit complexity based on our specific quantification can reveal the occurrence of topological phase transitions, both in and out of equilibrium, by the presence of nonanalyticity. By quenching the system out of equilibrium, we find that the circuit complexity grows linearly or quadratically in the short-time regime if the quench is finished instantaneously or in a finite time, respectively. Notably, we find that for both the sudden quench and the finite-time quench, a topological phase transition in the pre-quench Hamiltonian will be manifested by the presence of nonanalyticity in the first-order or second-order derivative of circuit complexity with respect to time in the short-time regime, and a topological phase transition in the post-quench Hamiltonian will be manifested by the presence of nonanalyticity in the steady value of circuit complexity in the long-time regime. We also show that the increase of dimension does not remove, but only weakens the nonanalyticity of circuit complexity. Our findings can be tested in quantum simulators and cold-atom systems.

I. INTRODUCTION

Circuit complexity connects a geometric measure of state preparation to topological phase transitions, motivating a test of whether its nonanalyticity persists under a different quantification. The paper studies two topological models across dimensions and quench protocols, finding that complexity reveals transitions through nonanalytic behavior.

I. INTRODUCTION

  • Circuit complexity is the minimum number of elementary gates needed to transform a reference state into a target state, and Nielsen’s geometric formulation identifies it with a minimal geodesic length.A cost function on time-dependent Hamiltonians defines the geometry on the space of unitary operations.Source: §I · p. 1
  • Because topological invariants characterize global geometry and change at topological phase transitions, complexity between distinct ground states may also show nonanalytic behavior.Prior work found this behavior for equilibrium and dynamical transitions in the one-dimensional Kitaev model [37].Source: §I · p. 1
  • The paper tests whether nonanalyticity persists with a different cost function by studying two topological models with richer phase diagrams in different dimensions.Both sudden and finite-time quenches are considered to test the behavior out of equilibrium.Source: §I · pp. 1–2
  • The study finds nonanalyticity across topological phase transitions with a distinct complexity quantification, in equilibrium and after sudden or finite-time quenches.Short-time growth is linear for sudden quenches and quadratic for finite-time quenches; long-time steady values reveal transitions in the post-quench Hamiltonian.Source: §I · p. 2
  • Increasing dimension weakens but does not remove complexity’s nonanalyticity at topological phase transitions.The authors suggest the short-time behavior may be useful experimentally because it requires little information from the post-quench evolution.Source: §I · p. 2

II. CIRCUIT COMPLEXITY AND TOPOLOGICAL PHASE TRANSITIONS IN ONE DIMENSION

The one-dimensional generalized Kitaev model provides a setting with multiple topological transitions. Using an inner-product-based complexity measure, the section finds nonanalyticity at critical points and relates it to changes in the winding of the model’s vector.

II. CIRCUIT COMPLEXITY AND TOPOLOGICAL PHASE TRANSITIONS IN ONE DIMENSION

  • The generalized Kitaev model has phase boundaries at µ = t1−t2, µ = −t1−t2, and µ = (t1+t2)/2, with next-nearest-neighbor terms producing a richer phase diagram than the standard model.Its winding number characterizes how d(k) = (dy(k), dz(k)) winds around the origin.Source: §II · pp. 2–3
  • The adopted complexity measure is the sum of minimal angular distances between momentum-resolved states, equivalently an inner-product metric based on fidelity.It differs from the Ref.[37] measure by a square-root operation, and the analysis varies either µ or t2 while fixing the other parameters.Source: §II · pp. 2–3
  • Circuit complexity and its first derivative with respect to µT are nonanalytic at topological critical points, and the derivative is independent of the reference-state parameter µR.The passage reports divergence in dC/dµT exactly at the critical points.Source: §II · p. 3
  • At a topological transition, the winding number changes discretely, forcing the angle θk and its difference ∆θk to jump for some k; this produces nonanalytic circuit complexity.The section names γ(λ), the growth rate of complexity during adiabatic transfer to a nearby state, the adiabatical growth rate.Source: §II · p. 3
  • The adiabatical growth rate γ(µ) diverges at critical points and coincides with |dC/dµT|, reflecting the reference-state independence of that derivative under this quantification.As a state approaches a critical point, the passage says it becomes more difficult to prepare the target state adiabatically.Source: §II · p. 3

III. SUDDEN QUENCH AND CIRCUIT COMPLEXITY EVOLUTION

For an isolated system suddenly quenched from its initial ground state, complexity grows linearly at short times and later saturates with oscillations. Its short-time growth rate reveals transitions in the pre-quench Hamiltonian, while its steady value reveals transitions in the post-quench Hamiltonian.

III. SUDDEN QUENCH AND CIRCUIT COMPLEXITY EVOLUTION

  • The setup isolates a system in its initial ground state before abruptly replacing Hi with Hf, after which the state evolves unitarily under Hf.Out of equilibrium, the paper defines dynamical topological transitions through changes in the Hamiltonian invariant νH; the wave-function invariant νw remains unchanged for an isolated system.Source: §III · p. 4
  • After a sudden quench, circuit complexity first grows linearly and then saturates with oscillations; its long-time steady value is nonanalytic when the post-quench Hamiltonian changes topology.The steady value is obtained by averaging C(t) over a sufficiently long time.Source: §III · p. 4
  • The dynamical growth rate χ is nonanalytic at critical points where the pre-quench Hamiltonian changes topology, when different initial Hamiltonians are quenched to the same final Hamiltonian.For the fixed-µf example, nonanalyticity appears at every critical point.Source: §III · p. 4
  • The short-time growth rate requires little post-quench information and is presented as more convenient and accurate for detecting transitions than the steady value.The linear growth immediately after a sudden quench is specific to this quantification; Ref.[37] does not show that same linear behavior.Source: §III · p. 4

IV. FINITE-TIME QUENCH AND CIRCUIT COMPLEXITY EVOLUTION

For a finite-time quench, complexity grows quadratically at short times and its second derivative reveals transitions in the pre-quench Hamiltonian. As in a sudden quench, the long-time steady value reveals transitions in the post-quench Hamiltonian.

IV. FINITE-TIME QUENCH AND CIRCUIT COMPLEXITY EVOLUTION

  • During the finite-time ramp, µ changes from µi to µf at speed v over duration tq = |µf − µi|/|v|; the sudden-quench limit is recovered as v increases without bound.The ramp is implemented by varying the chemical potential.Source: §IV · p. 5
  • The finite-time-quench steady value is nonanalytic at post-quench topological transitions, while complexity grows quadratically immediately after the quench begins.After the quench ends, both finite-time and sudden quenches evolve under the same time-independent Hf, accounting for their similar long-time behavior.Source: §IV · p. 5
  • The time derivative of complexity grows linearly just after the ramp starts, and its slope depends on quench speed while its linear growth behavior remains robust to speed variation.The corresponding second derivative shows nonanalyticity at critical points where the pre-quench Hamiltonian changes topology.Source: §IV · p. 5
  • For finite-time quenches, short-time behavior reveals transitions in the pre-quench Hamiltonian, whereas the long-time steady value reveals transitions in the post-quench Hamiltonian.The figure caption notes that the second derivative’s nonanalyticity at critical points does not depend on quench speed.Source: §IV · p. 5, Figure 3 · p. 5

V. CIRCUIT COMPLEXITY AND TOPOLOGICAL PHASE TRANSITIONS IN TWO DIMENSIONS

In two dimensions, circuit complexity retains nonanalytic signatures of topological phase transitions, although increasing dimension smooths and weakens them. The section connects these signatures to changes in the model’s topological structure and examines them under parameter variation and a sudden quench.

V. CIRCUIT COMPLEXITY AND TOPOLOGICAL PHASE TRANSITIONS IN TWO DIMENSIONS

  • The two-dimensional model is in class D, and at its critical momenta the d-vector reverses orientation, which the authors connect to nonanalyticity in circuit complexity.The section describes the model’s topological characterization through a Chern number and the winding of d(k).Source: §V · p. 6
  • As dimension increases, C-µT curves become smoother, yet first derivatives with respect to µ and t2 remain nonanalytic at critical points and track γ.The µ and t2 variations are shown separately in Fig. 4(c) and Fig. 4(d).Source: §V · p. 6
  • After a sudden quench, the two-dimensional evolution resembles the one-dimensional case, while the Cs-µf and χ-µi curves are smoother but retain critical-point nonanalyticity visible in a first derivative.The first derivative is shown in the inset of Fig. 5(b).Source: §V · p. 6, Figure 5 · p. 6
  • Increasing dimension weakens but does not remove the complexity nonanalyticity at topological critical points, so detecting transitions in higher dimensions requires more precise measurements.The authors note that their circuit complexity can reveal transitions in higher dimensions, but the weaker signal raises the required measurement precision.Source: §V · p. 6

VI. CONCLUSIONS

The authors conclude that their circuit-complexity quantification can reveal topological phase transitions through nonanalyticity in equilibrium and quenched settings. The signal persists in higher dimensions, though it weakens, and the paper identifies fidelity measurements and quantum platforms as routes for testing its predictions.

VI. CONCLUSIONS

  • Circuit complexity based on the authors’ quantification reveals topological phase transitions through nonanalyticity in equilibrium and after sudden or finite-time quenches.Short-time growth behavior signals transitions in the pre-quench Hamiltonian, while long-time steady complexity signals transitions in the post-quench Hamiltonian.Source: §VI · p. 6
  • Increasing dimension weakens, but does not eliminate, the nonanalyticity associated with topological phase transitions.The authors state that short-time nonanalytic behavior may be useful experimentally because it can be determined with very few measurements.Source: §VI · p. 6
  • Because the quantification can be determined by measuring fidelity, the authors identify quantum simulators and cold-atom systems as platforms for testing their predictions.They give superconducting-qubit simulation and quantum state tomography as one example.Source: §VI · pp. 6–7

ACKNOWLEGEMENTS

The authors acknowledge a discussion with Chengfeng Cai and list grant and institutional support.

ACKNOWLEGEMENTS

  • The authors acknowledge Chengfeng Cai’s discussions and list support from grants, the National Supercomputer Center in Guangzhou, and the Guangdong Special Projects Leading Talent Program.Z.Y. also acknowledges Startup Grant No. 74130-18841219 and NSFC-11904417.Source: ACKNOWLEGEMENTS · p. 7

Appendix A: Derivation of the circuit complexity under the inner-product metric

The appendices derive the circuit-complexity metric for infinitesimally close states and outline the ground-state wave function used in the analysis.

Appendix A: Derivation of the circuit complexity under the inner-product metric

  • For an infinitesimal state change, the appendix minimizes the SU(2) inner-product metric and identifies circuit complexity with the resulting geodesic length.The metric is given as ds2 = dψk · dψk/4 = dθk2/4.Source: Appendix A · p. 7

Appendix B: The wave function of ground state

  • The ground-state appendix diagonalizes the Hamiltonian by a Bogoliubov transformation and treats each momentum mode independently, with each mode state a superposition of the vacuum and a paired excitation.The overlap of the reference and target states is identified as fidelity.Source: Appendix B · pp. 7–8

Appendix C: The growth rate of adiabatic circuit complexity

The appendix derives the growth rate of adiabatic circuit complexity for a small change in a model parameter. It finds logarithmic divergence at the two critical points, associated with sudden changes in θk across topological phase transitions.

Appendix C: The growth rate of adiabatic circuit complexity

  • The growth rate is defined for a reference state and a neighboring target state whose parameter differs by δλ.The appendix uses a Taylor expansion for infinitely small δλ; the linear-order term vanishes, leaving a leading term proportional to (dθk(λ)/dλ)^2(δλ)^2.Source: Appendix C · p. 8
  • γ(µ) exhibits logarithmic divergence exactly at the two critical points µ = ±1.The appendix illustrates this behavior using the special case t2 = ∆2 = 0 and t1 = ∆1 = 1.Source: Appendix C · p. 8
  • Sudden jumps in θk across a topological phase transition lead to divergence in γ(µ).The figure discussion identifies jumps at some momenta as the reason for the divergence.Source: Appendix C · p. 8

Appendix D: Post-quench circuit complexity

The appendix derives post-quench circuit complexity for a global sudden quench, using the unitary evolution of momentum states. The accompanying figure shows sudden jumps in θk at specific momenta across two critical points.

Appendix D: Post-quench circuit complexity

  • After the system starts in the ground state of Hi, a sudden quench at t = 0 switches the Hamiltonian to Hf and the state evolves unitarily.For each momentum k, the state evolves under Hf(k).Source: Appendix D · p. 8
  • Across µc = −0.2, θk jumps at k = π; across µc = 1.1, it jumps at k = 2π/3.The plotted curves use t1 = 1, t2 = 1.2, and ∆1 = ∆2 = 1.Source: Figure 6 · p. 9
  • The appendix obtains post-quench complexity from evolved momentum states and their overlaps, resolving the post-quench Hamiltonian into eigenstates with energies ±Ef(k).The expression uses the difference ∆θk = θf_k − θi_k.Source: Appendix D · pp. 8–9

Appendix E: Growth rate of post-quench circuit complexity

The appendix defines a growth rate for post-quench circuit complexity, which grows linearly right after the quench. In a special model, the rate χ is nonanalytic at the two critical points µi = ±1.

Appendix E: Growth rate of post-quench circuit complexity

  • Because complexity grows linearly right after the quench, its growth rate can be defined from any two points on the line.The appendix uses C(0) = 0 in the subsequent expression.Source: Appendix E · p. 9
  • The nonanalyticity check uses a special model and expressions involving the pre-quench parameter µi.The appendix specifies the model and expressions before stating the result for χ.Source: Appendix E · p. 9
  • χ is nonanalytic at the two critical points µi = ±1 in the special model considered.The appendix introduces this model to demonstrate the nonanalyticity.Source: Appendix E · p. 9
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