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arXiv:1903.10499 · Physical review. B./Physical review. B

Slow scrambling and hidden integrability in a random rotor model

Dan Mao, Debanjan Chowdhury, T. Senthil

Published Sep 17, 2020• 6 citations
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Abstract

We analyze the out-of-time-order correlation functions of a solvable model of a large number $N$ of $M$-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. We focus on the growth of commutators of operators at a temperature $T$ above the zero temperature quantum critical point separating the spin-glass and paramagnetic phases. In the large $N,\phantom{\rule{0.16em}{0ex}}M$ limit, the squared commutators of the rotor fields do not display any exponential growth of commutators, in spite of the absence of any sharp quasiparticlelike excitations in the disorder-averaged theory. We show that in this limit, the problem is integrable and points out interesting connections to random-matrix theory. At leading order in $1/M$, there are no modifications to the critical behavior but an irrelevant term in the fixed-point action leads to a small exponential growth of the squared commutator. We also introduce and comment on a generalized model involving $p$-pair rotor interactions.

DOI
10.1103/physrevb.102.094306

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We study scrambling in a model consisting of a number NNN of MMM-component quantum rotors coupled by random infinite-range interactions. This model is known to have both a paramagnetic phase and a spin glass phase separated by second order phase transition. We calculate in perturbation theory the squared commutator of rotor fields at different sites in the paramagnetic phase, to leading non-trivial order at large NNN and large MMM. This quantity diagnoses the onset of quantum chaos in this system, and we show that the squared commutator grows exponentially with time, with a Lyapunov exponent proportional to 1M\frac{1}{M}M1​. At high temperature, the Lyapunov exponent limits to a value set by the microscopic couplings, while at low temperature, the exponent exhibits a T4T^4T4 dependence on temperature TTT.
We examine a model of M-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. A complete solution is obtained at M=\ensuremath{\infty} in the spin-glass and quantum-disordered phases. The quantum phase transition separating them is found to possess logarithmic violations of scaling, with no further modifications to the leading critical behavior at any order in 1/M; this suggests that the critical properties of the transverse-field Ising model (believed to be identical to the M\ensuremath{\rightarrow}1 limit) are the same as those of the M=\ensuremath{\infty} quantum rotors.
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
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We study the dynamics of excitations in a system of O(N)O(N)O(N) quantum rotors in the presence of random fields and random anisotropies. Below the lower critical dimension dlc=4{d}_{\mathrm{lc}}=4dlc​=4 the system exhibits a quasi-long-range order with a power-law decay of correlations. At zero temperature the spin waves are localized at the length scale Lloc{L}_{\mathrm{loc}}Lloc​ beyond which the quantum tunneling is exponentially suppressed, $c\ensuremath{\sim}{e}^{\ensuremath{-}{(L/{L}_{\mathrm{loc}})}^{2(\ensuremath{\theta}+1)}}$. At finite temperature TTT the spin waves propagate by thermal activation over energy barriers that scale as ${L}^{\ensuremath{\theta}}$. Above dlc{d}_{\mathrm{lc}}dlc​ the system undergoes an order-disorder phase transition with activated dynamics such that the relaxation time grows with the correlation length $\ensuremath{\xi}$ as $\ensuremath{\tau}\ensuremath{\sim}{e}^{C{\ensuremath{\xi}}^{\ensuremath{\theta}}/T}$ at finite temperature and as $\ensuremath{\tau}\ensuremath{\sim}{e}^{{C}^{\ensuremath{'}}{\ensuremath{\xi}}^{2(\ensuremath{\theta}+1)}/{\ensuremath{\hbar}}^{2}}$ in the vicinity of the quantum critical point.
The growth of commutators of initially commuting local operators diagnoses the onset of chaos in quantum many-body systems. We compute such commutators of local field operators with NNN components in the (2+12+12+1)-dimensional O(N)O(N)O(N) nonlinear sigma model to leading order in 1/N1/N1/N. The system is taken to be in thermal equilibrium at a temperature TTT above the zero temperature quantum critical point separating the symmetry broken and unbroken phases. The commutator grows exponentially in time with a rate denoted ${\ensuremath{\lambda}}_{L}$. At large NNN the growth of chaos as measured by ${\ensuremath{\lambda}}_{L}$ is slow because the model is weakly interacting, and we find ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}3.2T/N$. The scaling with temperature is dictated by conformal invariance of the underlying quantum critical point. We also show that operators grow ballistically in space with a ``butterfly velocity'' given by ${v}_{B}/c\ensuremath{\approx}1$ where ccc is the Lorentz-invariant speed of particle excitations in the system. We briefly comment on the behavior of ${\ensuremath{\lambda}}_{L}$ and vB{v}_{B}vB​ in the neighboring symmetry broken and unbroken phases.