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arXiv:1912.06132 · Physical review. D/Physical review. D.

Hamiltonian deformations in quantum mechanics, TT¯ , and the SYK model

David J. Gross, Jorrit Kruthoff, Andrew Rolph, Edgar Shaghoulian

Published Aug 31, 2020• 92 citations
arXivHTMLPDF

Abstract

Motivated by $T\overline{T}$, we introduce and study a wide class of solvable deformations of quantum-mechanical theories. These deformations map the Hamiltonian to a function of itself. We solve these theories by computing all finite-temperature correlation functions of the deformed theory in terms of the correlators of the undeformed theory. Applications to $\mathrm{AdS}/\mathrm{CFT}$, Sachdev-Ye-Kitaev, and the Schwarzian theory are considered. We write down the deformed Schwarzian action for an arbitrary Hamiltonian deformation and find that the maximal Lyapunov exponent is unchanged.

DOI
10.1103/physrevd.102.046019

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Motivated by TT‾T\overline{T}TT, we introduce and study a wide class of solvable deformations of quantum-mechanical theories. These deformations map the Hamiltonian to a function of itself. We solve these theories by computing all finite-temperature correlation functions of the deformed theory in terms of the correlators of the undeformed theory. Applications to AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT, Sachdev-Ye-Kitaev, and the Schwarzian theory are considered. We write down the deformed Schwarzian action for an arbitrary Hamiltonian deformation and find that the maximal Lyapunov exponent is unchanged.
A bstract The construction of exactly-solvable models has recently been advanced by considering integrable TT‾ T\overline{T} TT T T ¯ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise in the description of energy diffusion that emerges in quantum systems from time-keeping errors in a real clock used to track time evolution. We show that the latter can be related to an inverse TT‾ T\overline{T} TT T T ¯ deformation with a purely imaginary deformation parameter. In this case, the integral transforms take a particularly simple form when the initial state is a coherent Gibbs state or a thermofield double state, as we illustrate by characterizing the purity, Rényi entropies, logarithmic negativity, and the spectral form factor. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we further study the spectral properties of the Liouvillians, i.e., the dynamical generators associated with the deformed theories. As an application, we discuss the interplay between decoherence and quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and the Sachdev-Ye-Kitaev model.
Abstract We explore computationally tractable deformations of the SYK model. The deformed theories are described by the sum of two SYK Hamiltonians with differing numbers, q and q∼ \overset{\sim }{q} q∼​ q ~ , of interacting fermions. In the large N limit, employing analytic and numerical tools, we compute finite temperature correlation functions and thermodynamic quantities. We identify a novel analytically solvable model in the large q limit. We find that, under certain circumstances, the thermal RG flow in the strongly coupled infrared phase exhibits two regions of linear-in-temperature entropy, which we interpret in terms of Schwarzian actions. Using conformal perturbation theory we compute the leading relevant correction away from the intermediate near-conformal fixed point. Holographic spacetimes in two spacetime dimensions that reproduce the thermodynamics of the microphysical theory are discussed. These are flow geometries that interpolate between two Euclidean near-AdS2 spacetimes with different radii. The Schwarzian soft mode corresponding to the AdS2 region in the deep interior resides entirely within the geometric regime.
Abstract We review recent progress regarding the double scaled Sachdev–Ye–Kitaev model and other p -local quantum mechanical random Hamiltonians. These models exhibit an expansion using chord diagrams, which can be solved by combinatorial methods. We describe exact results in these models, including their spectrum, correlation functions, and Lyapunov exponent. In a certain limit, these techniques manifest the relation to the Schwarzian quantum mechanics, a theory of quantum gravity in AdS 2 . More generally, the theory is controlled by a rigid algebraic structure of a quantum group, suggesting a theory of quantum gravity on non-commutative q -deformed AdS 2 . We conclude with discussion of related universality classes, and survey some of the current research directions.
We study the TT‾T\overline{T}TT deformation on multiquantum mechanical systems. By introducing dynamical coordinate transformation, we reformulate the one-dimensional TT‾T\overline{T}TT deformation of generic quantum mechanical systems, which is consistent with the previous proposal in the literature. We further study the thermo-field-double state under the TT‾T\overline{T}TT deformation on these systems, which include the conformal quantum mechanical system, the Sachdev-Ye-Kitaev model, and the model satisfying the eigenstate thermalization hypothesis. We find common regenesis phenomena in which the signal injected into one local system can regenerate from the other local system. From the AdS2/CFT1{\mathrm{AdS}}_{2}/{\mathrm{CFT}}_{1}AdS2​/CFT1​ perspective, we study the deformation of Jackiw-Teitelboim gravity governed by Schwarzian action and find that these regenesis phenomena are realized by exchanging boundaries graviton via the nonlocal TT‾T\overline{T}TT coupling.
Abstract We compute the exact, all energy scale, 4-point function of the large N doublescaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL(2), that generalizes the Schwarzian to the complete range of energies.
Abstract The Sachdev–Ye–Kitaev (SYK) model is a strongly coupled, quantum many-body system that is chaotic, nearly conformally invariant, and exactly solvable. This remarkable and, to date, unique combination of properties have driven the intense activity surrounding the SYK model and its applications within both high energy and condensed matter physics. In this review we give an introduction to the SYK model and recent developments connected to it. We discuss: SYK and tensor models as a new class of large N quantum field theories, the near-conformal invariance in the infrared, the computation of correlation functions, generalizations of the SYK model, and applications to AdS/CFT and strange metals.