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

Miguel Tierz

doi:10.22331/q-2026-10-22-2241

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

Miguel Tierz

Shanghai Institute for Mathematics and Interdisciplinary Sciences, Block A, International Innovation Plaza, No. 657 Songhu Road, Yangpu District, Shanghai, China

Published:	2026-10-22, volume 10, page 2241
Editor:	Michele Campisi
Eprint:	arXiv:2509.09640v2
Doi:	https://doi.org/10.22331/q-2026-10-22-2241
Citation:	Quantum 10, 2241 (2026).

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Abstract

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 $W=\sum_{r\le m} a_r \Re\operatorname{Tr} U^r$ (and in models with pairing terms -superconducting pairing- additional $b_r \Im\operatorname{Tr} U^r$ terms appear). By invoking multivariate central limit theorems for vectors of traces of unitaries, we obtain a Gaussian distribution for $P(W)$ with variance $\mathrm{Var}(W)=\frac{1}{2}\sum_r r (a_r^2+b_r^2)$ 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.

Featured image: Sudden quantum quenches and work fluctuations. The inset depicts an abrupt change of the chain’s couplings at t = 0. For the quenches studied, work statistics map to weighted sums of traces of random unitary matrices, represented here by their eigenvalues on the unit circle. This connection explains the emergence of Gaussian distributions and helps identify departures from Gaussianity. Conceptual illustration.

Popular summary

How much work is done when a quantum system is changed abruptly? Repeat the same abrupt change — a quench — many times, measure the work each time, and the answers scatter. The pattern of these outcomes, the work distribution, is what this paper is about.

We study quenches in periodic quantum chains (rings) described by noninteracting fermions, a textbook setting that includes the XY spin chain. For the quenches considered, the work distribution coincides with that of a classic object of random matrix theory: a weighted sum of traces of powers of a random unitary matrix. Results about such matrices then explain when a Gaussian distribution (the familiar bell curve) emerges in large systems, and give its width directly from the couplings between sites.

The same dictionary shows where the Gaussian description stops being the whole story. Couplings that decay sufficiently slowly with distance can make the width grow with system size. Singularities tied to certain critical or boundary settings change the fluctuations, and finite systems show corrections to the limiting distribution. A growing width does not by itself spoil the Gaussian shape, which can survive once the width is scaled out. Examples in the XY chain and numerical tests illustrate these mechanisms. The upshot is a direct link between the shape and size of work fluctuations and the structure of the underlying model.

► BibTeX data

@article{Tierz2026workstatisticsof,
  doi = {10.22331/q-2026-10-22-2241},
  url = {https://doi.org/10.22331/q-2026-10-22-2241},
  title = {Work statistics of sudden quantum quenches: {A} random matrix theory perspective on {G}aussianity and its deviations},
  author = {Tierz, Miguel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2241},
  month = oct,
  year = {2026}
}

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