Leading and beyond leading-order spectral form factor in chaotic quantum many-body systems across all Dyson symmetry classes

• URL: http://arxiv.org/abs/2502.04152v1
• Date: Thu, 06 Feb 2025 15:37:18 GMT
• Title: Leading and beyond leading-order spectral form factor in chaotic quantum many-body systems across all Dyson symmetry classes
• Authors: Vijay Kumar, Tomaž Prosen, Dibyendu Roy,
• Abstract summary: We show the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic periodically kicked interacting many-body systems. We analytically calculating spectral form factor (SFF), $K(t)$, up to two leading orders in time, $t$. Our derivation only assumes random phase approximation to enable ensemble average.
• Score: 8.105213101498085
• License:
• Abstract: We show the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic periodically kicked interacting quantum many-body systems by analytically calculating spectral form factor (SFF), $K(t)$, up to two leading orders in time, $t$. We explicitly consider the presence or absence of time reversal ($\mathcal{T}$) symmetry to investigate all three Dyson's symmetry classes. Our derivation only assumes random phase approximation to enable ensemble average. For $\mathcal{T}$-invariant systems with $\mathcal{T}^2=1$, we show that beyond the Thouless time $t^*$, the SFF takes the form $K(t)\simeq 2t-2t^2/\mathcal{N}$ up to second order in time, where $\mathcal{N}$ is the Hilbert space dimension. This is identical to the result from circular orthogonal ensemble of RMT. In the absence of $\mathcal{T}$-symmetry, we show that $K(t)\simeq t$ beyond $t^*$, and there is no universal term in the second order, unlike the $\mathcal{T}^2=1$ case, in agreement with the result of circular unitary ensemble. For $\mathcal{T}$-invariant systems with $\mathcal{T}^2=-1$, we show that $K(t)\simeq 2t+2t^2/\mathcal{N}$ up to two orders in time beyond $t^*$, in agreement with the result of circular symplectic ensemble. In all three cases, the system-size, $L$, scaling of $t^*$ is determined by eigenvalues of a doubly stochastic matrix $\mathcal{M}$. For strongly interacting fermionic chains, $\mathcal{M}$ is $SU(2)$ invariant in all three cases, leading to $t^*\propto L^2$ in the presence of $U(1)$ symmetry. In the absence of $U(1)$ symmetry, we find $t^*\propto L^0$, due to gapped non-degenerate second-largest eigenvalue of $\mathcal{M}$ or $t^*\propto \ln(L)$ due to gapped second-largest eigenvalue with degeneracy $\propto L^\zeta$. Our calculation of SFF is plausible in higher space dimensions as well, where similar system-size scalings of $t^*$ can be obtained.

Related papers

• Characteristic time operators as quantum clocks [0.0]
  We show that $mathsfT$ satisfies the canonical relation $[mathsfT,mathsfH]|psirangle=ihbar|psirangle$ in a dense subspace of the Hilbert space. One can construct a quantum clock that tells the time in the neighborhood of $mathscrT$ by measuring a compatible observable.
  arXiv  Detail & Related papers  (2024-09-05T09:10:38Z)
• Scaling of symmetry-restricted quantum circuits [42.803917477133346]
  In this work, we investigate the properties of $mathcalMSU(2N)$, $mathcalM$-invariant subspaces of the special unitary Lie group $SU(2N)$.
  arXiv  Detail & Related papers  (2024-06-14T12:12:15Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Monogamy of entanglement between cones [68.8204255655161]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones. Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Beyond the Berry Phase: Extrinsic Geometry of Quantum States [77.34726150561087]
  We show how all properties of a quantum manifold of states are fully described by a gauge-invariant Bargmann. We show how our results have immediate applications to the modern theory of polarization.
  arXiv  Detail & Related papers  (2022-05-30T18:01:34Z)
• Matrix concentration inequalities and efficiency of random universal sets of quantum gates [0.0]
  For a random set $mathcalS subset U(d)$ of quantum gates we provide bounds on the probability that $mathcalS$ forms a $delta$-approximate $t$-design. We show that for $mathcalS$ drawn from an exact $t$-design the probability that it forms a $delta$-approximate $t$-design satisfies the inequality $mathbbPleft(delta geq x right)leq 2D_t
  arXiv  Detail & Related papers  (2022-02-10T23:44:09Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Topological entanglement and hyperbolic volume [1.1909611351044664]
  Chern-Simons theory provides setting to visualise the $m$-moment of reduced density matrix as a three-manifold invariant $Z(M_mathcalK_m)$. For SU(2) group, we show that $Z(M_mathcalK_m)$ can grow at mostly in $k$. We conjecture that $ln Z(M_mathcalK_m)$ is the hyperbolic volume of the knot complement $S3backslash mathcalK_m
  arXiv  Detail & Related papers  (2021-06-07T07:51:03Z)
• $\mathcal{PT}$ symmetry of a square-wave modulated two-level system [23.303857456199328]
  We study a non-Hermitian two-level system with square-wave modulated dissipation and coupling. Based on the Floquet theory, we achieve an effective Hamiltonian from which the boundaries of the $mathcalPT$ phase diagram are captured.
  arXiv  Detail & Related papers  (2020-08-17T03:18:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.