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

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The gist

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

In short

The work distribution from sudden quantum quenches in quadratic fermionic chains is shown to follow Gaussian statistics governed by random matrix theory. The work variable is mapped to traces of powers of Haar-distributed unitaries, leading to a predicted variance dependent on the interaction structure. Non-Gaussian tails arise from specific structural features like many interaction terms or Fisher–Hartwig singularities.

Key concepts

Work Variable (W)
Work is a measurable quantity derived from varying external controls during a time window in the quantum system. In this context, it is mathematically represented as a linear combination of the real and imaginary parts of traces of powers of unitary matrices, which are drawn from random matrix statistics.
Random Matrix Theory Connection
The Loschmidt amplitude for sudden quenches in quadratic models can be represented by a Toeplitz determinant whose symbol relates to the eigenvalues. This allows the characteristic function to be expressed as an average over the unitary group, linking quantum dynamics directly to random matrix theory.
Gaussian Core and Variance
The leading-order distribution of work is Gaussian, governed by multivariate central limit theorems applied to traces of unitaries. The variance of this core is determined by a specific quadratic form involving the coefficients ($a_r, b_r$) that define the interaction terms in the model.
Non-Gaussian Tails
Deviations from Gaussianity occur when conditions for the central limit theorem are weakened. These non-Gaussian features are caused by factors like having many interaction terms, slow decay of these terms, or specific nonanalytic jumps known as Fisher–Hartwig singularities in the underlying mathematical symbol.

Terminology used across episodes

This episode discusses

The paper

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

Miguel Tierz

Shanghai Institute for Mathematics and Interdisciplinary Sciences

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.

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: Today's paper: "Work statistics of sudden quantum quenches".

Mira: The gist: The work distribution for sudden quantum quenches in quadratic fermionic chains reduces to Gaussian statistics governed by random matrix theory,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, we just touched on how this paper connects the work distribution to random matrix theory, focusing on quadratic fermionic chains with periodic boundary conditions. The thesis is that these systems map onto a unitary matrix model.

Mira: They establish that for sudden quenches in quadratic models with periodic boundary conditions, the Loschmidt amplitude has a Toeplitz determinant representation whose symbol is e iu epsilon(k), where epsilon(k) is the single-particle dispersion of the post-quench Hamiltonian after rotation.

Lev: That means you can write that Loschmidt amplitude as

T N(phi): for some function, and the Fourier coefficients are elements of a structured matrix called T N(phi), which has constant values along diagonals.

Kai: Expanding that dispersion relation epsilon(k) in Fourier modes gives you epsilon(k) = a zero + sum m one (a m mk + b m mk), and then using Heine–Szeg˝o identities, the characteristic function can be written as an average over the unitary group <ref:2509.09640#pg3>.

Mira: That average is chi(u) = E U(N)

(iu X m one alpha m Tr U m + alpha* m Tr U-m): , where the coefficients alpha m are determined by the quench and microscopic couplings.

Lev: So, what's interesting is that the logarithm of this characteristic function becomes the cumulant generating functional of a linear statistic of traces of powers of a Haar-distributed unitary.

Kai: This leads to them obtaining a Gaussian distribution for P(W) using multivariate central limit theorems for these vectors of traces. The key result there is that the variance is Var(W) = one over two sum r r (a r 2+b r two), and they also get asymptotic independence across different powers <ref:2509.09640#pg1,asymptotic independence across different powers>.

Mira: They also rigorously guarantee this Gaussian core under a strong Szeg˝o condition, which requires P r one r(a 2r + b 2r) < infinity, ensuring that W converges in distribution to a normal distribution as N goes to infinity <ref:2509.09640#pg1>.

Lev: But the caveat is that this convergence holds even when the number of terms m(N) grows with N as long as it's less than or equal to two-thirds times N, and the total-variation error stays super-exponentially small if m is much smaller than square root of N.

Kai: So, even though the Gaussian core is strong, they spend time characterizing when things go wrong with non-Gaussian tails, which happen from many interaction terms or slow decay.

Conclusion: Mira: Thinking about this paper, "Work statistics of sudden quantum quenches: A random matrix theory perspective on Gaussianity and its deviations," the authors are essentially showing how work statistics in these systems reduce to random matrix theory.

Kai: And what that means for us is that the work distribution is fundamentally a Gaussian core, but it's not just a simple bell curve; it has specific non-Gaussian tails controlled by the structure of the post-quench dispersion.

Mira: So, if you're listening only to this show, you should understand that these deviations come from things like having too many interaction terms or those singularities in the symbol of the Toeplitz matrix.

Lev: For anyone building hardware, it means that while the Gaussian part is predictable, you need to watch for those tails when scaling up because they dictate where your error correction overhead really lands.

Kai: The paper suggests that studying how these tails reorganize when you cross critical manifolds or look at open boundaries could give us clues about the behavior of these systems in more complex regimes.

Mira: It boils down to understanding that the work is governed by random matrix statistics, but the fine details are controlled by the Fourier components of the dispersion relation.

Lev: So, to sum up for anyone listening, you have a Gaussian core determined by those Fourier components, with non-Gaussian tails arising from slow decay or symbol singularities.

Kai: That's it for this paper on "Work statistics of sudden quantum quenches: A random matrix theory perspective on Gaussianity and its deviations."

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