Hamiltonian deformations in quantum mechanics, $T\bar T$, and SYK

TL;DR

<3-5 sentence high-level summary> The paper develops a universal framework for exactly solvable deformations of quantum-mechanical systems, defined by H -> f(H), which map energies to f(E) while preserving eigenvectors. It derives integral-kernel and momentum-space transforms that express all finite-temperature correlators of the deformed theory in terms of undeformed ones, and applies the machinery to AdS/CFT, the SYK model, and Schwarzian theory. Key results include explicit kernels for notable deformations (e.g., one-dimensional T\bar T), a holographic interpretation as mixed boundary conditions with finite-cutoff JT gravity, and the finding that the maximal Lyapunov exponent in the Schwarzian/SYK sectors is preserved under general f(H) deformations. The work also discusses worldline gravity viewpoints and potential extensions to multi-system and higher-dimensional setups.</p>

Abstract

Motivated by $T\bar T$, 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 AdS/CFT, SYK, 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.

Figures (3)

• Figure 1: The leading order diagrams that contribute to $\Sigma (\tau , \tau')$ when $\lambda \sim 1/N$. Unlabelled vertices are integrated over. Chains formed of links are made of $q$ vertices. Each additional link in the chain adds a factor of $\lambda N$. The aqua blue dashed line indicates the disorder average.
• Figure 2: Numerically computed kernel $K(\beta,\beta')$ for the deformation $f(H) = H + \lambda (H^2 + H^4)$ for $\beta = 1$ and $\lambda = 0.5$.
• Figure 3: Solid orange line: The undeformed partition function for the simple harmonic oscillator ($\omega = 1$). Solid aqua blue line: Truncated sum (to 40 terms) of the partition function for the deformed simple harmonic oscillator for the deformation $f(H) = H + \lambda (H^2 + H^4)$ with $\lambda = 0.5$. Dots: Numerically computed deformed partition function through the integral transform.