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arXiv:1811.00442 · Physical review. B./Physical review. B

Approximating observables on eigenstates of large many-body localized systems

Abishek K. Kulshreshtha, Arijeet Pal, Thorsten B. Wahl, Steven H. Simon

Published Mar 11, 2019• 14 citations
arXivHTMLPDF

Abstract

Eigenstates of fully many-body localized (FMBL) systems can be organized into spin algebras based on quasilocal operators called $l$ bits. These spin algebras define quasilocal $l$-bit measurement (${\ensuremath{\tau}}_{i}^{z}$) and $l$-bit flip (${\ensuremath{\tau}}_{i}^{x}$) operators. For a disordered Heisenberg spin chain in the MBL regime we approximate $l$-bit flip operators by first calculating them exactly on small windows of systems using an algorithm called operator localization optimization. We then extend the $l$-bit operators onto the whole system by exploiting their quasilocal nature. We subsequently use these operators to represent approximate eigenstates of the Hamiltonian. Finally, we describe a method to calculate products of local observables on these eigenstates for systems of size $L$ in $O({L}^{2})$ time. This method is used to calculate the variance of the energy of the approximate eigenstates, yielding an estimate of the error of the approximation.

DOI
10.1103/physrevb.99.104201

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Eigenstates of fully many-body localized (FMBL) systems can be organized into spin algebras based on quasilocal operators called lll bits. These spin algebras define quasilocal lll-bit measurement (\ensuremathτiz{\ensuremath{\tau}}_{i}^{z}\ensuremathτiz​) and lll-bit flip (\ensuremathτix{\ensuremath{\tau}}_{i}^{x}\ensuremathτix​) operators. For a disordered Heisenberg spin chain in the MBL regime we approximate lll-bit flip operators by first calculating them exactly on small windows of systems using an algorithm called operator localization optimization. We then extend the lll-bit operators onto the whole system by exploiting their quasilocal nature. We subsequently use these operators to represent approximate eigenstates of the Hamiltonian. Finally, we describe a method to calculate products of local observables on these eigenstates for systems of size LLL in O(L2)O({L}^{2})O(L2) time. This method is used to calculate the variance of the energy of the approximate eigenstates, yielding an estimate of the error of the approximation.
Eigenstates of fully many-body localized (FMBL) systems are described by quasilocal operators ${\ensuremath{\tau}}_{i}^{z}$ (l-bits), which are conserved exactly under Hamiltonian time evolution. The algebra of the operators ${\ensuremath{\tau}}_{i}^{z}$ and ${\ensuremath{\tau}}_{i}^{x}$ associated with l-bits $({\mathbf{\ensuremath{\tau}}}_{i})$ completely defines the eigenstates and the matrix elements of local operators between eigenstates at all energies. We develop a nonperturbative construction of the full set of l-bit algebras in the many-body localized phase for the canonical model of MBL. Our algorithm to construct the Pauli algebra of l-bits combines exact diagonalization and a tensor network algorithm developed for efficient diagonalization of large FMBL Hamiltonians. The distribution of localization lengths of the l-bits is evaluated in the MBL phase and used to characterize the MBL-to-thermal transition.
Many-body localization (MBL) addresses the absence of thermalization in interacting quantum systems, with nonergodic high-energy eigenstates behaving as ground states, only area-law entangled. However, computing highly excited many-body eigenstates using exact methods is very challenging. Instead, we show that one can address high-energy MBL physics using ground-state methods, which are much more amenable to many efficient algorithms. We find that a localized many-body ground state of a given interacting disordered Hamiltonian H0{\mathcal{H}}_{0}H0​ is a very good approximation for a high-energy eigenstate of a parent Hamiltonian, close to H0{\mathcal{H}}_{0}H0​ but more disordered. This construction relies on computing the covariance matrix, easily achieved using density-matrix renormalization group for disordered Heisenberg chains up to L=256L=256L=256 sites.
We provide a pedagogical review on the calculation of highly excited eigenstates of disordered interacting quantum systems which can undergo a many-body localization (MBL) transition, using shift-invert exact diagonalization. We also provide an example code at https://bitbucket.org/dluitz/sinvert_mbl. Through a detailed analysis of the simulational parameters of the random field Heisenberg spin chain, we provide a practical guide on how to perform efficient computations. We present data for mid-spectrum eigenstates of spin chains of sizes up to L=26 L = 26 . This work is also geared towards readers with interest in efficiency of parallel sparse linear algebra techniques that will find a challenging application in the MBL problem.
We provide a pedagogical review on the calculation of highly excited eigenstates of disordered interacting quantum systems which can undergo a many-body localization (MBL) transition, using shift-invert exact diagonalization. We also provide an example code at https://bitbucket.org/dluitz/sinvert_mbl. Through a detailed analysis of the simulational parameters of the random field Heisenberg spin chain, we provide a practical guide on how to perform efficient computations. We present data for mid-spectrum eigenstates of spin chains of sizes up to L = 26. This work is also geared towards readers with interest in efficiency of parallel sparse linear algebra techniques that will find a challenging application in the MBL problem.
Disorder and interactions can lead to the breakdown of statistical mechanics in certain quantum systems, a phenomenon known as many-body localization (MBL). Much of the phenomenology of MBL emerges from the existence of ℓ bits, a set of conserved quantities that are quasilocal and binary (i.e., possess only ±1 eigenvalues). While MBL and ℓ bits are known to exist in one-dimensional systems, their existence in dimensions greater than one is a key open question. To tackle this question, we develop an algorithm that can find approximate binary ℓ bits in arbitrary dimensions by adaptively generating a basis of operators in which to represent the ℓ bit. We use the algorithm to study four models: the one-, two-, and three-dimensional disordered Heisenberg models and the two-dimensional disordered hard-core Bose-Hubbard model. For all four of the models studied, our algorithm finds high-quality ℓ bits at large disorder strength and rapid qualitative changes in the distributions of ℓ bits in particular ranges of disorder strengths, suggesting the existence of MBL transitions. These transitions in the one-dimensional Heisenberg model and two-dimensional Bose-Hubbard model coincide well with past estimates of the critical disorder strengths in these models, which further validates the evidence of MBL phenomenology in the other two- and three-dimensional models we examine. In addition to finding MBL behavior in higher dimensions, our algorithm can be used to probe MBL in various geometries and dimensionality.
Abstract A fully many-body localized (FMBL) quantum disordered system is characterized by the emergence of an extensive number of local conserved operators that prevents the relaxation towards thermal equilibrium. These local conserved operators can be seen as the building blocks of the whole set of eigenstates. In this paper, we propose to construct them explicitly via block real-space renormalization. The principle is that each renormalization group step diagonalizes the smallest remaining blocks and produces a conserved operator for each block. The final output for a chain of N spins is a hierarchical organization of the N conserved operators with layers. The system size nature of the conserved operators of the top layers is necessary to describe the possible long-range order of the excited eigenstates and the possible critical points between different FMBL phases. We discuss the similarities and the differences with the strong disorder RSRG-X method that generates the whole set of the 2 N eigenstates via a binary tree of N layers. The approach is applied to the long-range quantum spin-glass Ising model, where the constructed excited eigenstates are found to be exactly like ground states in another disorder realization, so that they can be either in the paramagnetic phase, in the spin-glass phase or critical.