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

arXiv:2509.09640v2 Announce Type: replace 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.

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.

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)

arXiv:2509.09640 [quant-ph]

(or arXiv:2509.09640v2 [quant-ph] for this version)

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

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