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Register a new userMany-body systems with random spatially local interactions
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Added by
Siddhardh Morampudi
Publication date
2018
fields
Physics
and research's language is
English
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We extend random matrix theory to consider randomly interacting spin systems with spatial locality. We develop several methods by which arbitrary correlators may be systematically evaluated in a limit where the local Hilbert space dimension $N$ is large. First, the correlators are given by sums over stacked planar diagrams which are completely determined by the spectra of the individual interactions and a dependency graph encoding the locality in the system. We then introduce heap freeness as a generalization of free independence, leading to a second practical method to evaluate the correlators. Finally, we generalize the cumulant expansion to a sum over dependency partitions, providing the third and most succinct of our methods. Our results provide tools to study dynamics and correlations within extended quantum many-body systems which conserve energy. We further apply the formalism to show that quantum satisfiability at large-$N$ is determined by the evaluation of the independence polynomial on a wide class of graphs.
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We consider the spectral statistics of the Floquet operator for disordered, periodically driven spin chains in their quantum chaotic and many-body localized phases (MBL). The spectral statistics are characterized by the traces of powers $t$ of the Floquet operator, and our approach hinges on the fact that, for integer $t$ in systems with local interactions, these traces can be re-expressed in terms of products of dual transfer matrices, each representing a spatial slice of the system. We focus on properties of the dual transfer matrix products as represented by a spectrum of Lyapunov exponents, which we call textit{spectral Lyapunov exponents}. In particular, we examine the features of this spectrum that distinguish chaotic and MBL phases. The transfer matrices can be block-diagonalized using time-translation symmetry, and so the spectral Lyapunov exponents are classified according to a momentum in the time direction. For large $t$ we argue that the leading Lyapunov exponents in each momentum sector tend to zero in the chaotic phase, while they remain finite in the MBL phase. These conclusions are based on results from three complementary types of calculation. We find exact results for the chaotic phase by considering a Floquet random quantum circuit with on-site Hilbert space dimension $q$ in the large-$q$ limit. In the MBL phase, we show that the spectral Lyapunov exponents remain finite by systematically analyzing models of non-interacting systems, weakly coupled systems, and local integrals of motion. Numerically, we compute the Lyapunov exponents for a Floquet random quantum circuit and for the kicked Ising model in the two phases. As an additional result, we calculate exactly the higher point spectral form factors (hpSFF) in the large-$q$ limit, and show that the generalized Thouless time scales logarithmically in system size for all hpSFF in the large-$q$ chaotic 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.
We study the spectral statistics of spatially-extended many-body quantum systems with on-site Abelian symmetries or local constraints, focusing primarily on those with conserved dipole and higher moments. In the limit of large local Hilbert space dimension, we find that the spectral form factor $K(t)$ of Floquet random circuits can be mapped exactly to a classical Markov circuit, and, at late times, is related to the partition function of a frustration-free Rokhsar-Kivelson (RK) type Hamiltonian. Through this mapping, we show that the inverse of the spectral gap of the RK-Hamiltonian lower bounds the Thouless time $t_{mathrm{Th}}$ of the underlying circuit. For systems with conserved higher moments, we derive a field theory for the corresponding RK-Hamiltonian by proposing a generalized height field representation for the Hilbert space of the effective spin chain. Using the field theory formulation, we obtain the dispersion of the low-lying excitations of the RK-Hamiltonian in the continuum limit, which allows us to extract $t_{mathrm{Th}}$. In particular, we analytically argue that in a system of length $L$ that conserves the $m^{th}$ multipole moment, $t_{mathrm{Th}}$ scales subdiffusively as $L^{2(m+1)}$. We also show that our formalism directly generalizes to higher dimensional circuits, and that in systems that conserve any component of the $m^{th}$ multipole moment, $t_{mathrm{Th}}$ has the same scaling with the linear size of the system. Our work therefore provides a general approach for studying spectral statistics in constrained many-body chaotic systems.
We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of integrability of Hamiltonians; TMI can be negative or positive for both integrable and non-integrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by non-integrability. Furthermore, we calculate TMI in disordered systems such as many-body localized (MBL) systems and the Sachdev-Ye-Kitaev (SYK) model. We find that scrambling occurs but is slow in a MBL phase, while disorder in the SYK model does not make scrambling slower but makes it smoother.
While there are well established methods to study delocalization transitions of single particles in random systems, it remains a challenging problem how to characterize many body delocalization transitions. Here, we use a generalized real-space renormalization group technique to study the anisotropic Heisenberg model with long-range interactions, decaying with a power $alpha$, which are generated by placing spins at random positions along the chain. This method permits a large-scale finite-size scaling analysis. We examine the full distribution function of the excitation energy gap from the ground state and observe a crossover with decreasing $alpha$. At $alpha_c$ the full distribution coincides with a critical function. Thereby, we find strong evidence for the existence of a many body localization transition in disordered antiferromagnetic spin chains with long range interactions.
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