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arXiv:2104.03852 · Physical review. D/Physical review. D.

TT¯ deformation on multiquantum mechanics and regenesis

Song He, Zhuo-Yu Xian

Published Aug 10, 2022• 14 citations
arXivHTMLPDF

Abstract

We study the $T\overline{T}$ deformation on multiquantum mechanical systems. By introducing dynamical coordinate transformation, we reformulate the one-dimensional $T\overline{T}$ 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 $T\overline{T}$ 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 ${\mathrm{AdS}}_{2}/{\mathrm{CFT}}_{1}$ 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 $T\overline{T}$ coupling.

DOI
10.1103/physrevd.106.046002

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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.
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 We study the dynamics of the geometric entanglement entropy of a 2D CFT in the presence of a boundary. We show that this dynamics is governed by local equations of motion, that take the same form as 2D Jackiw-Teitelboim gravity coupled to the CFT. If we assume that the boundary has a small thickness ϵ and constant boundary entropy, we derive that its location satisfies the equations of motion of Schwarzian quantum mechanics with coupling constant C = c ϵ/ 12 π . We rederive this result via energy-momentum conservation.
On the flow of states under TT‾T\overline{T}TT
We study the TT deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the TT operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, “dressed” operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the TT deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.
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.
Generalized 1 + 0-dimensional Liouvillean dynamics describing deformations of the Sachdev–Ye–Kitaev (SYK) model, as well as the various 1 + 1-dimensional dilaton and Horava–Lifshitz gravity theories, can all be mapped onto the single-particle quantum mechanics of a non-relativistic charge propagating in a (generally, curved) 2d space and subject to a (generally, non-uniform) magnetic field. The latter description sets a stage for the phenomenon of quantum Hall effect (QHE), thereby elucidating the intrinsically topological nature of the pertinent gravity theories and demystifying their (pseudo)holographic connection to a broad class of the SYK-like models.
We derive a large-N wormhole saddle in the Sachdev-Ye-Kitaev (SYK) model for the variance of the thermal one-point function of a heavy operator across realizations of the microscopic couplings. We propose that this saddle describes a double trumpet geometry. In the double-scaling limit, we express the wormhole as a transition amplitude in an emergent bulk Hilbert space of chords connecting the two boundaries. In the Schwarzian limit, the chords proliferate and the wormhole defines a quantum transition amplitude in the open-universe Hilbert space of Jackiw-Teitelboim (JT) gravity. Comparing the SYK wormhole amplitude with its counterpart in JT gravity coupled to matter provides a direct test of the heavy-operator randomness hypothesis in SYK. The microscopic SYK couplings specify the α-parameters of the wormhole. For a fixed realization of the couplings, we compute the observable in the corresponding α-state. We provide the bulk wavefunction of the associated half-wormhole, which exhibits erratic correlations in the length basis that do not self-average. We extend the bulk description to the Hilbert space of a closed universe. We point out that the correlation chords from which the wormhole geometry emerges structurally resemble local observer worldlines.