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arXiv:1509.06258 · Journal of Statistical Mechanics Theory and Experiment

Many-body localization: construction of the emergent local conserved operators via block real-space renormalization

Cécile Monthus

Published Mar 1, 2016• 37 citations
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Abstract

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.

DOI
10.1088/1742-5468/2016/03/033101

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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.
Abstract For the many-body-localized phase of random Majorana models, a general strong disorder real-space renormalization procedure known as RSRG-X (Pekker et al 2014 Phys. Rev . X 4 011052) is described to produce the whole set of excited states, via the iterative construction of the local integrals of motion (LIOMs). The RG rules are then explicitly derived for arbitrary quadratic Hamiltonians (free-fermions models) and for the Kitaev chain with local interactions involving even numbers of consecutive Majorana fermions. The emphasis is put on the advantages of the Majorana language over the usual quantum spin language to formulate unified RSRG-X rules.
We develop a real-space renormalization-group (RSRG) scheme by appropriately inserting the long-range hopping t∼r^{-α} with nearest-neighbor interaction to study the entanglement entropy and maximum block size for the many-body localization (MBL) transition. We show that for α<2 there exists a localization transition with renormalized disorder that depends logarithmically on the finite size of the system. The transition observed for α>2 does not need a rescaling in disorder strength. Most important, we find that even though the MBL transition for α>2 falls in the same universality class as that of the short-range models, the transition for α<2 belongs to a different universality class. Because of the intrinsic nature of the RSRG flow toward delocalization, MBL phase for α>2 might suffer an instability in the thermodynamic limit while the underlying systems support algebraic localization. Moreover, we verify these findings by inserting microscopic details to the RSRG scheme where we additionally find a more appropriate rescaling function for disorder strength; we indeed uncover a power-law scaling with a logarithmic correction and a distinctly different stretched exponential scaling for α<2 and α>2, respectively, by analyzing system with finite size. This finding further suggests that microscopic RSRG scheme is able to give a hint of instability of the MBL phase for α>2 even considering systems of finite size.
For a quantum system to be permanently out-of-equilibrium, some non-trivial mechanism must be at play, to counteract the general tendency of entropy increase and flow toward equilibration. Among the possible ways to protect a system against local thermalization, the phenomenon of localization induced by quenched disorder appears to be one of the most promising. Although the problem of localization was introduced almost sixty years ago, its many-body version is still partly unresolved, despite the recent theoretical effort to tackle it. In this thesis we address a few aspects of the localized phase, mainly focusing on the interacting case. A large part of the thesis is devoted to investigating the underlying “integrable” structure of many-body localized systems, i.e., the existence of non-trivial conservation laws that prevent ergodicity and thermalization. In particular, we show that such conserved operators can be explicitly constructed by dressing perturbatively the non-interacting conserved quantities, in a procedure that converges when scattering processes are weak enough. This is reminiscent of the quasiparticle theory in Fermi liquids, although in the disordered case the construction extends to the full many-body energy spectrum, and it results in operators that are exactly conserved. As an example of how to use the constructive recipe for the conserved quantities, we compute the long-time limit of an order parameter for the MBL phase in antiferromagnetic spin systems. Similar analytical tools as the ones exploited for the construction of the conserved operators are then applied to the problem of the stability of single-particle localization with respect to the coupling to a finite bath. In this context, we identify a quantum-Zeno-type effect, whereby the bath unexpectedly enhances the particle’s localization. In the final part of the thesis, we discuss several mechanisms by which thermal fluctuations may influence the spatial localization of excitations in interacting many-body states.
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.
The presence and character of local integrals of motion---quasilocal operators that commute with the Hamiltonian---encode valuable information about the dynamics of a quantum system. In particular, strongly disordered many-body systems can generically avoid thermalization when there are extensively many such operators. In this work, we explicitly construct local conserved operators in one-dimensional spin chains by directly minimizing their commutator with the Hamiltonian. We demonstrate the existence of an extensively large set of local integrals of motion in the many-body localized phase of the disordered XXZ spin chain. These operators are shown to have exponentially decaying tails, in contrast to the ergodic phase where the decay is (at best) polynomial in the size of the subsystem. We study the algebraic properties of localized operators and confirm that in the many-body localized phase, they are well described by ``dressed'' spin operators.
This paper focuses on developing a new algorithm (En-DMRG) for obtaining highly excited states of disordered and strongly interacting spin systems. This line of research is stimulated by the growing interest in understanding the many-body localized (MBL) state, which emerges in disordered interacting systems at high energy densities through a disorder-driven dynamic phase transition. By studying the one-dimensional Heisenberg spin chain (with up to 72 spins) in a random field, the authors demonstrate the accuracy of the method in obtaining energy eigenstates and the corresponding statistical results of the quantum system in the MBL phase. The method presented in this work has the potential to be applicable in a variety of quantum statistical physics problems, possibly including the MBL phase in higher dimensions.