A
bstract
The fine grained energy spectrum of quantum chaotic systems is widely believed to be described by random matrix statistics. A basic scale in such a system is the energy range over which ...this behavior persists. We define the corresponding time scale by the time at which the linearly growing ramp region in the spectral form factor begins. We call this time
t
ramp
. The purpose of this paper is to study this scale in many-body quantum systems that display strong chaos, sometimes called scrambling systems. We focus on randomly coupled qubit systems, both local and
k
-local (all-to-all interactions) and the Sachdev-Ye-Kitaev (SYK) model. Using numerical results, analytic estimates for random quantum circuits, and a heuristic analysis of Hamiltonian systems we find the following results. For geometrically local systems with a conservation law we find
t
ramp
is determined by the
diffusion time
across the system, order
N
2
for a 1D chain of
N
qubits. This is analogous to the behavior found for local one-body chaotic systems. For a
k
-local system like SYK the time is order log
N
but with a different prefactor and a different mechanism than the scrambling time. In the absence of any conservation laws, as in a generic random quantum circuit, we find
t
ramp
∼ log
N
, independent of connectivity.
Quantum Lyapunov spectrum Gharibyan, Hrant; Hanada, Masanori; Swingle, Brian ...
The journal of high energy physics,
04/2019, Letnik:
2019, Številka:
4
Journal Article
Recenzirano
Odprti dostop
A
bstract
We introduce a simple quantum generalization of the spectrum of classical Lyapunov exponents. We apply it to the SYK and XXZ models, and study the Lyapunov growth and entropy production. ...Our numerical results suggest that a black hole is not just the fastest scrambler, but also the fastest entropy generator. We also study the statistical features of the quantum Lyapunov spectrum and find universal random matrix behavior, which resembles the recently-found universality in classical chaos. The random matrix behavior is lost when the system is deformed away from chaos, towards integrability or a many-body localized phase. We propose that quantum systems holographically dual to gravity satisfy this universality in a strong form. We further argue that the quantum Lyapunov spectrum contains important additional information beyond the largest Lyapunov exponent and hence provides us with a better characterization of chaos in quantum systems.
Black holes and random matrices Cotler, Jordan S.; Gur-Ari, Guy; Hanada, Masanori ...
The journal of high energy physics,
05/2017, Letnik:
2017, Številka:
5
Journal Article
Recenzirano
Odprti dostop
A
bstract
We argue that the late time behavior of horizon fluctuations in large anti-de Sitter (AdS) black holes is governed by the random matrix dynamics characteristic of quantum chaotic systems. ...Our main tool is the Sachdev-Ye-Kitaev (SYK) model, which we use as a simple model of a black hole. We use an analytically continued partition function |
Z
(
β
+
it
)|
2
as well as correlation functions as diagnostics. Using numerical techniques we establish random matrix behavior at late times. We determine the early time behavior exactly in a double scaling limit, giving us a plausible estimate for the crossover time to random matrix behavior. We use these ideas to formulate a conjecture about general large AdS black holes, like those dual to 4D super-Yang-Mills theory, giving a provisional estimate of the crossover time. We make some preliminary comments about challenges to understanding the late time dynamics from a bulk point of view.
We generalize Page's result on the entanglement entropy of random pure states to the many-body eigenstates of realistic disordered many-body systems subject to long-range interactions. This extension ...leads to two principal conclusions: first, for increasing disorder the "shells" of constant energy supporting a system's eigenstates fill only a fraction of its full Fock space and are subject to intrinsic correlations absent in synthetic high-dimensional random lattice systems. Second, in all regimes preceding the many-body localization transition individual eigenstates are thermally distributed over these shells. These results, corroborated by comparison to exact diagonalization for an SYK model, are at variance with the concept of "nonergodic extended states" in many-body systems discussed in the recent literature.
A model of randomly-coupled Pauli spins Hanada, Masanori; Jevicki, Antal; Liu, Xianlong ...
The journal of high energy physics,
05/2024, Letnik:
2024, Številka:
5
Journal Article
Recenzirano
Odprti dostop
A
bstract
We construct a model of Pauli spin operators with all-to-all 4-local interactions by replacing Majorana fermions in the SYK model with spin operators. Equivalently, we replace fermions with ...hard-core bosons. We study this model numerically and compare the properties with those of the SYK model. We observe a striking quantitative coincidence between the spin model and the SYK model, which suggests that this spin model is strongly chaotic and, perhaps, can play some role in holography. We also discuss the path-integral approach with multi-local fields and the possibility of quantum simulations. This model may be an interesting target for quantum simulations because Pauli spins are easier to implement than fermions on qubit-based quantum devices.
We model backreaction in AdS2 JT gravity via a proposed boundary dual Sachdev-Ye-Kitaev quantum dot coupled to Dirac fermion matter and study it from the perspective of quantum entanglement and ...chaos. The boundary effective action accounts for the backreaction through a linear coupling of the Dirac fermions to the Gaussian-random two-body Majorana interaction term in the low-energy limit. We calculate the time evolution of the entanglement entropy between graviton and Dirac fermion fields for a separable initial state and find that it initially increases and then saturates to a finite value. Moreover, in the limit of a large number of fermions, we find a maximally entangled state between the Majorana and Dirac fields in the saturation region, implying a transition of the von Neumann algebra of observables from type I to type II. This transition in turn indicates a loss of information in the holographically dual emergent spacetime. We corroborate these observations with a detailed numerical computation of the averaged nearest-neighbor gap ratio of the boundary spectrum and provide a useful complement to quantum entanglement studies of holography.
We present a fully analytical description of a many-body localization (MBL) transition in a microscopically defined model. Its Hamiltonian is the sum of one- and two-body operators, where both ...contributions obey a maximum-entropy principle and have no symmetries except Hermiticity (not even particle number conservation). These two criteria paraphrase that our system is a variant of the Sachdev-Ye-Kitaev model. We will demonstrate how this simple zero-dimensional system displays numerous features seen in more complex realizations of MBL. Specifically, it shows a transition between an ergodic and a localized phase, and nontrivial wave-function statistics indicating the presence of nonergodic extended states. We check our analytical description of these phenomena by a parameter-free comparison to high performance numerics for systems of up to N=15 fermions. In this way, our study becomes a test bed for concepts of high-dimensional quantum localization, previously applied to synthetic systems such as Cayley trees or random regular graphs. The minimal model describes a many-body system for which an effective theory is derived and solved from first principles. The hope is that the analytical concepts developed in this study may become a stepping stone for the description of MBL in more complex systems.
Randomness and chaos in qubit models Lau, Pak Hang Chris; Ma, Chen-Te; Murugan, Jeff ...
Physics letters. B,
08/2019, Letnik:
795
Journal Article
Recenzirano
Odprti dostop
We introduce randomness into a class of integrable models and study the spectral form factor as a diagnostic to distinguish between randomness and chaos. Spectral form factors exhibit a ...characteristic dip-ramp-plateau behavior in the N>2 SYK2 model at high temperatures that is absent in the N=2 SYK2 model. Our results suggest that this dip-ramp-plateau behavior implies the existence of random eigenvectors in a quantum many-body system. To further support this observation, we examine the Gaussian random transverse Ising model and obtain consistent results without suffering from small N issues. Finally, we demonstrate numerically that expectation values of observables computed in a random quantum state at late times are equivalent to the expectation values computed in the thermal ensemble in a Gaussian random one-qubit model.
Abstract
Low-dimensional systems of interacting fermions in a synthetic gauge field have been experimentally realized using two-component ultra-cold Fermi gases in optical lattices. Using a two-leg ...ladder model that is relevant to these experiments, we have studied the signatures of topological Lifshitz transitions and the effects of the inter-species interaction
U
on the gauge-invariant orbital current in the regime of large intra-leg hopping Ω. Focusing on non-insulating regimes, we have carried out numerically exact density-matrix renormalization-group (DMRG) calculations to compute the orbital current at fixed particle number as a function of the interaction strength and the synthetic gauge flux per plaquette. Signatures of topological Lifshitz transitions where the number Fermi points changes are found to persist even in the presence of very strong repulsive interactions. This numerical observation suggests that the orbital current can be computed from an appropriately renormalized mean-field band structure, which is also described here. Quantitative agreement between the mean-field and the DMRG results in the intermediate interaction regime where
U
≲ Ω is demonstrated. We also have observed that interactions can change the sign of the current susceptibility at zero field and induce Lifshitz transitions between two metallic phases, which is also captured by the mean-field theory. Correlation effects beyond mean-field theory in the oscillations of the local inter-leg current are also reported. We argue that the observed robustness against interactions makes the orbital current a good indicator of the topological Lifshitz transitions.
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