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Register a new userEmergence of many-body quantum chaos via spontaneous breaking of unitarity
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It is suggested that many-body quantum chaos appears as spontaneous symmetry breaking of unitarity in interacting quantum many-body systems. It has been shown that many-body level statistics, probed by the spectral form factor (SFF) defined as $K(beta,t)=langle|{rm Tr}, exp(-beta H + itH)|^2rangle$, is dominated by a diffusion-type mode in a field theory analysis. The key finding of this paper is that the unitary $beta=0$ case is different from the $beta to 0^+$ limit, with the latter leading to a finite mass of these modes due to interactions. This mass suppresses a rapid exponential ramp in the SFF, which is responsible for the fast emergence of Poisson statistics in the non-interacting case, and gives rise to a non-trivial random matrix structure of many-body levels. The interaction-induced mass in the SFF shares similarities with the dephasing rate in the theory of weak localization and the Lyapunov exponent of the out-of-time-ordered correlators.
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We consider a non-interacting many-fermion system populating levels of a unitary random matrix ensemble (equivalent to the q=2 complex Sachdev-Ye-Kitaev model) - a generic model of single-particle quantum chaos. We study the corresponding many-particle level statistics by calculating the spectral form factor analytically using algebraic methods of random matrix theory, and match it with an exact numerical simulation. Despite the integrability of the theory, the many-body spectral rigidity is found to have a surprisingly rich landscape. In particular, we find a residual repulsion of distant many-body levels stemming from single-particle chaos, together with islands of level attraction. These results are encoded in an exponential ramp in the spectral form-factor, which we show to be a universal feature of non-ergodic many-fermion systems embedded in a chaotic medium.
We propose a new type of locally interacting quantum circuits which are generated by unitary interactions round-a-face (IRF). Specifically, we discuss a set (or manifold) of dual-unitary IRFs with local Hilbert space dimension $d$ (DUIRF$(d)$) which generate unitary evolutions both in space and time directions of an extended 1+1 dimensional lattice. We show how arbitrary dynamical correlation functions of local observables can be evaluated in terms of finite dimensional completely positive trace preserving unital maps, in complete analogy to recently studied circuits made of dual unitary brick gates (DUBG). In fact, we show that the simplest non-trivial (non-vanishing) local correlation functions in dual-unitary IRF circuits involve observables non-trivially supported on at least two sites. We completely characterise the 10-dimensional manifold of DUIRF$(2)$ for qubits ($d=2$) and provide, for $d=3,4,5,6,7$, empirical estimates of its dimensionality based on numerically determined dimensions of tangent spaces at an ensemble of random instances of dual-unitary IRF gates. In parallel, we apply the same algorithm to determine ${rm dim},{rm DUBG}(d)$ and show that they are of similar order though systematically larger than ${rm dim},{rm DUIRF}(d)$ for $d=2,3,4,5,6,7$. It is remarkable that both sets have rather complex topology for $dge 3$ in the sense that the dimension of the tangent space varies among different randomly generated points of the set. Finally, we provide additional data on dimensionality of the chiral extension of DUBG circuits with distinct local Hilbert spaces of dimensions $d eq d$ residing at even/odd lattice sites.
We investigate spectral statistics in spatially extended, chaotic many-body quantum systems with a conserved charge. We compute the spectral form factor $K(t)$ analytically for a minimal Floquet circuit model that has a $U(1)$ symmetry encoded via auxiliary spin-$1/2$ degrees of freedom. Averaging over an ensemble of realizations, we relate $K(t)$ to a partition function for the spins, given by a Trotterization of the spin-$1/2$ Heisenberg ferromagnet. Using Bethe Ansatz techniques, we extract the Thouless time $t^{vphantom{*}}_{rm Th}$ demarcating the extent of random matrix behavior, and find scaling behavior governed by diffusion for $K(t)$ at $tlesssim t^{vphantom{*}}_{rm Th}$. We also report numerical results for $K(t)$ in a generic Floquet spin model, which are consistent with these analytic predictions.
We present a framework in which the transition between a many-body localised (MBL) phase and an ergodic one is symmetry breaking. We consider random Floquet spin chains, expressing their averaged spectral form factor (SFF) as a function of time in terms of a transfer matrix that acts in the space direction. The SFF is determined by the leading eigenvalues of this transfer matrix. In the MBL phase the leading eigenvalue is unique, as in a symmetry-unbroken phase, while in the ergodic phase and at late times the leading eigenvalues are asymptotically degenerate, as in a system with degenerate symmetry-breaking phases. We identify the broken symmetry of the transfer matrix, introduce a local order parameter for the transition, and show that the associated correlation functions are long-ranged only in the ergodic phase.
We study the consequences of having translational invariance in space and in time in many-body quantum chaotic systems. We consider an ensemble of random quantum circuits, composed of single-site random unitaries and nearest neighbour couplings, as a minimal model of translational invariant many-body quantum chaotic systems. We evaluate the spectral form factor (SFF) as a sum over many-body Feynman diagrams, which simplifies in the limit of large local Hilbert space dimension $q$. At sufficiently large $t$, diagrams corresponding to rigid translations dominate, reproducing the random matrix theory (RMT) prediction. At finite $t$, we show that translational invariance introduces an additional mechanism which delays the emergence of RMT. Specifically, we identify two universality classes characterising the approach to RMT: in $d=1$, corrections to RMT are generated by different translations applied to extended domains, known as the crossed diagrams; in $d>1$, corrections are the consequence of deranged defects diagrams, whose defects are dilute and localized due to confinement. We introduce a scaling limit of SFF where these universality classes reduce to simple scaling functions. Lastly, we demonstrate universality of the scaling forms with numerical simulations of two circuit models and discuss the validity of the large $q$ limit in the different cases.
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