Discrete power-law decay of subsystem distance after a quantum quench

AI summary

Overview

Research area: Quantum many-body non-equilibrium dynamics, specifically local relaxation and equilibration after a global quantum quench in an exactly solvable integrable model (the infinite one-dimensional transverse-field Ising chain).

Technical level: Advanced. The paper assumes familiarity with the generalized Gibbs ensemble (GGE), Gaussian fermionic states, Majorana two-point functions, and quantum distance measures such as Uhlmann fidelity and Bures distance.

Scope: A numerical study showing that the Bures distance between a subsystem's time-evolved reduced density matrix and its GGE stationary value decays as a power law whose exponent takes only discrete values, classified by the pre- and post-quench fields and by analytic properties of the initial-state excitation fraction.

What This Paper Is About

When an isolated integrable quantum system is driven out of equilibrium by a sudden change of Hamiltonian (a quench), a local subsystem does not thermalize in the ordinary sense but instead relaxes toward a generalized Gibbs ensemble. This paper asks how fast that relaxation happens: it tracks a mathematically rigorous distance between the subsystem's reduced density matrix and its final GGE value, and asks what controls the decay rate.

The goal is to replace an earlier, narrower result—a t^(-3/2) decay found only for ground-state initial conditions using a distance measure that is not a proven metric—with a systematic classification covering broad quench parameters and general excited Gaussian initial states.

Key Contributions

1. Replacement of a non-rigorous measure with a true metric. The authors adopt the Bures distance, built from the Uhlmann fidelity, which satisfies non-negativity, symmetry, identity of indiscernibles, and the triangle inequality. This fixes the third limitation the authors identify in prior work, where the distance was defined from the trace of the squared difference of density matrices and had not been proven to be a metric.

2. Extension from ground states to general excited Gaussian eigenstates. The initial state is specified by a momentum-dependent excitation fraction m(φ) ∈ [0,1], so the analysis covers arbitrary Gaussian eigenstates of the pre-quench Hamiltonian rather than only its ground state.

3. A classification scheme for decay exponents. The exponent λ is shown to be fixed jointly by the transverse field pair (h₀, h) and by properties of the symmetric function m_S(φ) = m(φ) + m(−φ) − 1 on φ ∈ [0, π]—its continuity, its values at φ = 0 and φ = π, and the behavior of its first derivative at boundaries and interior points.

4. Systematic coverage of the quench parameter space. Five representative field pairs—(−2, 2), (2, 1), (1, 2), (−1, 1), and (−1, 2)—are identified as sufficient to represent all qualitatively distinct regimes, including quenches from and to critical points and across ferromagnetic and paramagnetic phases.

Main Findings

• Discrete exponent set. At late times the subsystem Bures distance follows B_A(t) ∼ t^(−λ), with λ confined to discrete values: 1, 5/4, 3/2, 7/4, 2, 5/2, and potentially further values.

• Explicit example. For the initial state m(φ) = (1/2) sin²(φ/2), which gives m_S(φ) = −cos²(φ/2), the exponents for the five representative field pairs (−2, 2), (2, 1), (1, 2), (−1, 1), (−1, 2) are 3/2, 5/2, 1, 2, and 3/2 respectively. These numerical examples used subsystem length ℓ = 16.

• Discontinuous m_S gives λ = 1 unconditionally. When m_S(φ) is discontinuous on [0, π], the exponent is 1 regardless of the pre- and post-quench fields.

• Continuous m_S with finite derivative. Here λ depends on whether m_S vanishes at φ = 0 and φ = π, whether it is C¹-smooth at those boundaries (encoded as (s/n, s/n)), and—for two special configurations—whether it is smooth across the interior interval φ ∈ (0, π). Symmetry forces m_S′(0) = 0 when the function is C¹ at φ = 0, with the same holding at φ = π.

• Continuous m_S with divergent derivative. When m_S′ diverges at one or more points, the exponent can in principle depend on the antisymmetric part m_A(φ) = m(φ) − m(−φ); the authors restrict to m_A(φ) = 0 because the general dependence is too involved. Boundary behavior is encoded as (f/i, f/i), and no complete closed-form rule is established; a flowchart covers the remaining three cases.

• Ground-state result recovered. The previously reported t^(−3/2) decay for ground-state initial conditions emerges as a special case, since m(φ) = 0 identically implies m_S(φ) = −1.

• Robustness across distance measures. The same discrete classification holds for the Bures distance, the measure used in the earlier work, and a "relative distance" R(ρ, σ) = sqrt(2S(ρ‖σ)) built from the quantum relative entropy.

• Relation to magnetization decay. Prior work reported magnetization exponents λ̃ = 1, 3/2, 5/2. Using Hölder's inequality together with the trace-distance/Bures-distance inequality D ≤ B sqrt(1 − B²/4), the authors derive λ̃ ≥ λ, and numerically find λ̃ = λ in all cases studied. They note that magnetization oscillates more strongly than B_A(t) at moderately large subsystem sizes, making the subsystem distance numerically more reliable for extracting exponents.

• Subsystem-size scaling. Fitting B_A ∼ ℓ^α / t^λ in the regime t ≫ ℓ ≫ 1, they find α = 2 for most examples (consistent with the ground-state result of prior work), α = 1 in various cases, and tentative evidence for α = 3/2 that could not be firmly confirmed because numerical errors grow substantially at large t.

Methodology in Plain English

The authors study a chain of interacting quantum spins (the transverse-field Ising model) that can be solved exactly by transforming the spins into non-interacting fermionic modes, each labeled by momentum. They prepare the system in a known eigenstate of one Hamiltonian, then suddenly switch the transverse field and let the system evolve under the new Hamiltonian.

Because the initial states are Gaussian and the evolution preserves that property, the entire state—and any subsystem of it—can be described by a small set of two-point correlation functions rather than an exponentially large wavefunction. The subsystem is a finite interval of length ℓ embedded in an infinite chain, so there are no finite-size effects from the total system.

For each time, they compute the reduced density matrix of that interval and its long-time GGE limit, then evaluate the Bures distance between them using a recently developed algorithm for fermionic Gaussian states. Plotting this distance against time on logarithmic axes reveals straight lines whose slopes are the decay exponents. By scanning many field pairs and many choices of the excitation fraction m(φ), and by organizing the results into decision flowcharts based on continuity and boundary behavior of m_S(φ), they map out which exponent occurs when.

The work is numerical fitting rather than analytical derivation; the paper states this explicitly and lists a rigorous derivation as future work.

Why This Matters

Impact on research. The paper turns an isolated observation into a classification: rather than one known exponent for one initial condition, it gives a discrete menu of exponents with a decision procedure for which applies. It also upgrades the diagnostic tool from an unproven distance measure to the Bures distance, which carries metric axioms and connects cleanly to fidelity, trace distance, and relative entropy. The bound λ̃ ≥ λ links two previously separate literatures—subsystem-distance decay and order-parameter decay—and the five representative field pairs give other researchers a compact set of cases to test.

Potential applications (the paper itself does not claim technological applications, but names experimental platforms where such quenches are studied):

• Ultracold atomic gases, where sudden changes of lattice or interaction parameters realize quenches.
• Trapped ion systems, used for controlled simulation of spin-chain dynamics.
• Superconducting qubit arrays, where programmable coupling quenches can probe relaxation.
• Benchmarking and validation of classical numerical methods for non-equilibrium quantum matter, since exact results in this model serve as reference data.

Industry relevance. The relevance is indirect and methodological: quantum hardware platforms named in the paper (cold atoms, ions, superconducting circuits) are the settings where quench dynamics are measured, and a discrete, predictable relaxation law offers a sharp signature to distinguish integrable from chaotic behavior in device characterization. The paper reports no device-level experiment, dataset, or industry partnership.

Future Directions

• Analytical derivation. The authors state that their results rest primarily on numerical fitting and that a rigorous analytical derivation of the discrete exponents remains an important direction.

• The m_A(φ) ≠ 0 case with divergent m_S′. The classification is incomplete here; the antisymmetric contribution was set to zero because its dependence is "overly involved."

• The m_S(0) = m_S(π) = 0 branch of the divergent-derivative flowchart. This branch is explicitly left unspecified in the paper.

• Beyond the Ising chain. The authors propose examining whether discrete subsystem-distance decay persists in other one-dimensional integrable models and in higher-dimensional systems, and deriving a general criterion for the subsystem-size exponent α, for which no simple rule was identified.

Target Audience

Specialists in quantum many-body theory and non-equilibrium statistical mechanics—particularly researchers working on quantum quenches, generalized Gibbs ensembles, and equilibration in integrable systems—will get the most from this paper. It is also valuable to readers interested in quantum information measures of mixed states, since the Bures distance and its numerical evaluation for Gaussian states are central. Graduate students with a background in second quantization and free-fermion techniques could follow the methodology, though the classification flowcharts and analytical case distinctions require careful reading. Experimentalists in cold-atom, trapped-ion, or superconducting-qubit groups may find the discrete exponents useful as reference predictions.

Authors’ abstract

We present a numerical study of subsystem distance decay following a global quantum quench in the infinite one-dimensional transverse-field Ising chain, using the mathematically rigorous Bures distance $B_A(t)$ to quantify the deviation of the time-evolved reduced density matrix from its stationary generalized Gibbs ensemble state. We show that the late-time decay follows a discrete power law $B_A(t) \sim t^{-λ}$, with the exponent $λ$ confined to discrete values: $1$, $5/4$, $3/2$, $7/4$, $2$, $5/2$, and potentially further values. The specific exponent is jointly determined by the pre- and post-quench transverse fields, as well as by properties of the symmetric excitation-fraction function $m_S(\varphi)$, defined on $\varphi\in[0,π]$ to characterize the pre-quench Hamiltonian eigenstates, including continuity, boundary values, and first-derivative boundary values, among others. The previously established $t^{-3/2}$ decay for the initial ground state of the pre-quench Hamiltonian is naturally recovered as a special case of this general classification. Our results reveal a universal discrete structure governing local equilibration dynamics in integrable quantum systems.