Work statistics of sudden quantum quenches: A random matrix theory perspective on Gaussianity and its deviations

The gist: The work distribution for sudden quantum quenches in quadratic fermionic chains reduces to Gaussian statistics governed by random matrix theory, with non-Gaussian tails arising from specific structural features like many interaction terms or Fisher–Hartwig singularities.

Work Definition and Mapping

Work is defined through a protocol where external controls are varied during a time window t ∈ [0, τ], leading to the unitary operator Uτ = T exp − iZ τ 0 H(λ(t)) dt <ref:2509.09640#pg2> The work random variable w is calculated as w = E(f) m − E(i) n <ref:2509.09640#pg4>. In a sudden quench, this reduces to the Loschmidt amplitude G(−u) where G(t):= ⟨ψ0e − iHf t ψ0⟩ <ref:2509.09640#pg7>. For quadratic fermion models, the problem is mapped to a unitary matrix integral. The work characteristic function χW (u) has the standard operator form χW (u) = Tr e − iuHf Uτ e −iuHi ρ0 U † τ <ref:2509.09640#pg6>.

Random Matrix Theory Connection

The key structural observation is that for sudden quenches in quadratic models with periodic boundary conditions, the Loschmidt amplitude admits a Toeplitz determinant representation whose symbol is e i u ε(k) <ref:2509.09640#pg9>. This means the characteristic function can be written as an average over the unitary group <ref:2509.09640#pg10>. The logarithm of the characteristic function is the cumulant generating functional of a linear statistic of traces of powers of a Haar-distributed unitary <ref:2509.09640#pg10>. For fixed m as N → ∞, these variables converge jointly to independent complex Gaussians with E[TrU m] = 0, Var(TrU m) = m <ref:2509.09640#pg11>.

Gaussian Core and Variance

To leading order, the work distribution P(w) is Gaussian with zero mean shift determined by a0 and a variance Var[w] = 2 Xm≥1 m αm2 <ref:2509.09640#pg12>. The Gaussian core is rigorously guaranteed under the strong Szeg˝o condition P r≥1 r(a2r + b2r) < ∞, which ensures that W converges in distribution to N (0, 1/2 Pr(a2r +b2r)) as N → ∞. The variance is given by Var(W) = 1/2 Xm≥1 r a2r = 2Xm≥1 r J 2r. This result holds even when m(N) grows with N as long as m(N) = o(N2/3), and the total-variation error is super-exponentially small if m ≪ √N.

Sources of Non-Gaussian Tails

Non-Gaussian tails arise from several mechanisms, including:

1. Many interaction terms or their slow decay <ref:2509.09640#pg12>.

2. The appearance of Fisher–Hartwig singularities. These are nonanalytic jumps or zeros in the symbol of the Toeplitz matrix.

3. Critical quenches (DQPTs) where Fisher zeros of G(t) approach the real axis, which translates into χW (u) acquiring singularities for real u.

4. Finite-N mesoscopic windows where the third and fourth cumulants are visible due to slow decay of Fourier coefficients or borderline 1/r behavior.

Experimental Signatures

The paper suggests that the Gaussian ‘core’ of the work distribution, with model-specific non-Gaussian tails, implies clear experimental signatures. Specifically, the prevalence of Gaussian statistics in sudden quenches is expected unless the system is tuned to special conditions like long-range interactions or critical points that produce visible deviations. The next step suggested is driving the system across quantum critical manifolds or considering open boundaries to study how these tails reorganize.

XY Chain Example

For the XY chain, the work variable is written as W = P r≤m ar R TrU r + br I TrU r. The analysis shows that if the quench does not cross a critical line, the Fourier coefficients α˜m decay exponentially, resulting in a Gaussian core with O(1) width. If it crosses a critical line, Toeplitz FH asymptotics yield deterministic power-law/oscillatory corrections on top of the CLT-driven Gaussian core for finite harmonics.

Numerical Diagnostics

Numerical diagnostics confirm the analytical results by showing that histograms and Q–Q plots display the Gaussian core and isolate any tail curvature. Scatter plots of trace components with theory ellipses show that the data agree with the random-matrix CLT for traces, confirming a Gaussian bulk with variance set by the quadratic form P r≥1 r(a2r + b2r). The ratio Tm/S3/2m tracks the relative size of cubic cumulants, which is small when Fourier weights decay fast and grows in borderline regimes <ref:2509.09640#pg11>.

Summary of Key Findings

The work distribution for sudden quantum quenches reduces to Gaussian statistics governed by random matrix theory, with non-Gaussian tails arising from specific structural features like many interaction terms or Fisher–Hartwig singularities. The Gaussian core is determined by the Fourier components of the post-quench dispersion, and deviations are controlled by Szeg˝o regularity and the presence of nonanalytic features in the symbol.

How it works

The work characteristic function χW (u) is identified as e − iuE0 E[e i u SN (U)]. This identifies P(w) as the distribution of the random variable SN (U) − E0 when U is Haar-random. The Fourier inversion yields PN = E δ(· − (SN (U) − E0)) as distributions.

Gaussian Core and Variance

The Gaussian core for W is determined by the multivariate CLT for traces, yielding W ⇒ N 0, 1/2 Xm≥1 r(a2r + b2r). This intrinsic variance of order m rather than vanishing with N is a direct physical consequence of the random-matrix type central limit behavior.

Non-Gaussian Features

Non-Gaussian features are identified when the conditions of the CLT are weakened, such as multiple slow-decaying modes or symbol singularities. These deviations include persistent non-Gaussian edges and oscillatory factors dictated by Fisher–Hartwig structure in the symbol.

Conclusion

The paper maps work statistics to random matrix problems, predicting a Gaussian core with model-specific non-Gaussian tails controlled by the Fourier components of the dispersion relation. The next step involves studying how these tails reorganize when crossing critical manifolds or considering open boundaries.

How it works

The work characteristic function is representation-invariant and factorizes over k, admitting a compact determinant/Pfaffian form. This leads to the work variable being the linear statistic W = Xr≥1 ar R TrU r + br I TrU r. The variance is Var(W) = 1/2 P r≥1 r(a2r + b2r).

1. Better characterization of work distribution tails via model selection: The AI system can distinguish between Gaussian cores and non-Gaussian tails by analyzing the ratio Tm/S3/2m, which tracks the relative size of the cubic cumulant and grows in borderline/FH regimes.

2. Enhanced prediction of non-Gaussian behavior: The system can predict when deviations occur based on conditions such as (A) Too many modes / slow decay or (B) Fisher–Hartwig singularities, which generate subleading cumulants, allowing for a more precise assessment of tail curvature.

3. Robust error control for large systems: When analyzing work statistics, the system can utilize quantitative multivariate central limit theorem bounds to ensure accuracy when the number of active harmonics, m(N), grows with N, specifically maintaining super-exponentially small total-variation error if m ≪ √N.

4. Experimental signature prediction: The improved system can identify experimental observables based on the theoretical predictions, such as anticipating that when G(t) develops Fisher zeros that approach the real time axis, the strip of analyticity of χW(u) narrows, revealing structured, non-Gaussian tails.

5. Universality mapping: The AI can connect quench thermodynamics to established physics by tuning parameters like the Fourier components (ar, br), which allows for probing the large-deviation regime of the matrix model, connecting results to unitary-matrix universality classes.

We show that, for sudden quenches, the work distribution reduces to the statistics of the traces of powers of Haar unitaries, which are random unitary matrices drawn uniformly from the unitary group. For translation-invariant quadratic fermionic chains with interactions extending to m neighbors and periodic boundary conditions, the Loschmidt amplitude admits a unitary matrix model / Toeplitz representation, which yields a work variable of the form W= sum r m a r Tr U r (and in models with pairing terms-superconducting pairing- additional b r Tr U r terms appear). By invoking multivariate central limit theorems for vectors of traces of unitaries, we obtain a Gaussian distribution for P(W) with variance Var(W)= 1 over 2 sum r r (a r 2+b r 2) and asymptotic independence across different powers. We also characterise the conditions under which non-Gaussian tails arise, for example from many interaction terms or their slow decay, as well as the appearance of Fisher-Hartwig singularities. We illustrate these mechanisms in the XY chain. Various numerical diagnostics support the analytical results.