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Eigenstate Thermalization Scaling in Majorana Clusters: from Integrable to Chaotic SYK Models

Masudul Haque, Paul McClarty

Published Nov 7, 2017• 9 citations
OpenAlex

Abstract

The eigenstate thermalization hypothesis (ETH) is a conjecture on the nature of isolated quantum systems that guarantees the thermal behavior of subsystems when it is satisfied. ETH has been tested in various forms on a number of local many-body interacting systems. Here we examine the validity of ETH in a class of nonlocal disordered many-body interacting systems --- the Sachdev-Ye-Kitaev Majorana (SYK) models --- that may be tuned from chaotic behavior to integrability. Our analysis shows that SYK$_4$ (with quartic couplings) which is maximally chaotic in the large system size limit satisfies the standard ETH scaling while SYK$_2$ (with quadratic couplings), which is integrable, does not. We show that the low-energy and high-energy properties are affected drastically differently when the two Hamiltonians are mixed.

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The eigenstate thermalization hypothesis (ETH) is a conjecture on the nature of isolated quantum systems that guarantees the thermal behavior of subsystems when it is satisfied. ETH has been tested in various forms on a number of local many-body interacting systems. Here we examine the validity of ETH in a class of nonlocal disordered many-body interacting systems --- the Sachdev-Ye-Kitaev Majorana (SYK) models --- that may be tuned from chaotic behavior to integrability. Our analysis shows that SYK4_44​ (with quartic couplings) which is maximally chaotic in the large system size limit satisfies the standard ETH scaling while SYK2_22​ (with quadratic couplings), which is integrable, does not. We show that the low-energy and high-energy properties are affected drastically differently when the two Hamiltonians are mixed.
The eigenstate thermalization hypothesis is a compelling conjecture which strives to explain the apparent thermal behavior of generic observables in closed quantum systems. Although we are far from a complete analytic understanding, quantum chaos is often seen as a strong indication that the ansatz holds true. In this paper, we address the thermalization of energy eigenstates in the Sachdev-Ye-Kitaev model, a maximally chaotic model of strongly-interacting Majorana fermions. We numerically investigate eigenstate thermalization for specific few-body operators in the original SYK model as well as its N \mathcal{N} N = 1 supersymmetric extension and find evidence that these models satisfy ETH. We discuss the implications of ETH for a gravitational dual and the quantum information-theoretic properties of SYK it suggests.
The eigenstate thermalization hypothesis is a compelling conjecture which strives to explain the apparent thermal behavior of generic observables in closed quantum systems. Although we are far from a complete analytic understanding, quantum chaos is often seen as a strong indication that the ansatz holds true. In this paper, we address the thermalization of energy eigenstates in the Sachdev-Ye-Kitaev model, a maximally chaotic model of strongly-interacting Majorana fermions. We numerically investigate eigenstate thermalization for specific few-body operators in the original SYK model as well as its N = 1 supersymmetric extension and find evidence that these models satisfy ETH. In conclusion, we discuss the implications of ETH for a gravitational dual and the quantum information-theoretic properties of SYK it suggests.
The Eigenstate Thermalization Hypothesis (ETH) is a framework for discussing thermal behavior originating from chaotic dynamics in isolated many-body quantum systems. The PXP model, where certain states do not thermalize, has been compared with the Sachdev-Ye Kitaev (SYK) model, which is believed to be fully thermalizing. A gate-based quantum circuit approach is utilized to simulate time evolution and compute the Out-of-Time-Ordered Correlator (OTOC), a measure of the extent of chaos. Considering restrictions on implementing SYK on gate-based hardware, a simplified model called Spin-XY4 (SXY4), which has a thermal behavior similar to SYK, is tested. An alternate method, which optimizes control on an analog quantum device with the GRAPE (GRadient Ascent Pulse Engineering) algorithm, is also utilized to simulate the SYK model.
A study of Rényi entanglement entropy in the Sachdev-Ye-Kitaev (SYK) chain of Majorana fermions suggested that the model does not rapidly thermalize, despite being maximally chaotic. In this work, I examine the eigenstate thermalization hypothesis (ETH) for both the SYK chain and the two-site SYK models using exact diagonalization. I focus on two physically motivated few-body observables: the one-body particle number operator and the two-body hopping operator. These operators provide insight into local occupation and, in certain contexts, may reflect aspects of transport. Both operators have been employed in previous studies of the ETH in related SYK models. I show that single realizations of both models approximately satisfy ETH conditions for these operators, while ensemble averages strictly satisfy the ETH. Therefore, I conclude that the finite-size SYK chain and two-site SYK models can rapidly thermalize with respect to these observables through the ETH mechanism. This suggests that the subthermal behavior observed in previous studies of Rényi entanglement entropy does not manifest in finite-size systems and that these systems can thermalize rapidly via their light modes. It also indicates that the proposed gravitational dual may undergo rapid thermalization.
In an isolated quantum many-body system undergoing unitary evolution, we study the thermalization of a subsystem, treating the rest of the system as a bath. In this setting, the eigenstate thermalization hypothesis (ETH) was proposed to explain thermalization. Consider a nearly integrable Sachdev-Ye-Kitaev model obtained by adding random all-to-all 4-body interactions as a perturbation to a random free-fermion model. When the subsystem size is larger than the square root of but is still a vanishing fraction of the system size, we prove thermalization if the system is initialized in a random product state, while almost all eigenstates violate the ETH. In this sense, the ETH is not a necessary condition for thermalization.