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arXiv:1509.03890 · Physical Review B

Many-body localization transition in Rokhsar-Kivelson-type wave functions

Xiao Chen, Xiongjie Yu, Gil Young Cho, Bryan K. Clark, Eduardo Fradkin

Published Dec 23, 2015• 20 citations
arXivHTMLPDF

Abstract

We construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the random energy model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with the subsystem size: scaling with the entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a subextensive scaling between these limits. Near the phase transition point, the fluctuations of the R\'enyi entropies are non-Gaussian. We find that R\'enyi entropies with different R\'enyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior.

DOI
10.1103/physrevb.92.214204

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We construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the random energy model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with the subsystem size: scaling with the entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a subextensive scaling between these limits. Near the phase transition point, the fluctuations of the R'enyi entropies are non-Gaussian. We find that R'enyi entropies with different R'enyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior.
Entanglement comes in different forms, some more complex than others. In this paper we study the transitions of entanglement complexity in an exemplary family of states---the Rokhsar-Kivelson-sign wavefunctions---whose degree of entanglement is controlled by a single parameter. This family of states is known to feature a transition between a phase exhibiting volume-law scaling of entanglement entropy and a phase with subextensive scaling of entanglement, reminiscent of the many-body-localization transition of disordered quantum Hamiltonians. We study the singularities of the Rokhsar-Kivelson-sign wavefunctions and their entanglement complexity across the transition using several tools from quantum information theory: fidelity metric, entanglement spectrum statistics, entanglement entropy fluctuations, stabilizer R\'enyi entropy, and the performance of a disentangling algorithm. Across the whole volume-law phase the states feature universal entanglement spectrum statistics. Yet a ``superuniversal'' regime appears for small values of the control parameter in which all metrics become independent of the parameter itself, the entanglement entropy as well as the stabilizer R\'enyi entropy appear to approach their theoretical maximum, the entanglement fluctuations scale to zero as in output states of random universal circuits, and the disentangling algorithm has essentially null efficiency. All these indicators consistently reveal a complex pattern of entanglement. In the sub-volume-law phase, on the other hand, the entanglement spectrum statistics is no longer universal, entanglement fluctuations are larger and exhibiting a nonuniversal scaling, and the efficiency of the disentangling algorithm becomes finite. Our results, based on model wavefunctions, suggest that a similar combination of entanglement scaling properties and of entanglement complexity features may be found in high-energy Hamiltonian eigenstates---a very strong candidate being offered by the many-body localization transition of disordered lattice-spin models.
We analyze the many-body localization- (MBL) to-delocalization transition in the Sherrington-Kirkpatrick (SK) model of Ising spin glass in the presence of a transverse field $\mathrm{\ensuremath{\Gamma}}$. Based on energy-resolved analysis, which is of relevance for a closed quantum system, we show that the quantum SK model has many-body mobility edges separating the MBL phase, which is nonergodic and nonthermal, from the delocalized phase, which is ergodic and thermal. The range of the delocalized regime increases with an increase in the strength of $\mathrm{\ensuremath{\Gamma}}$, and eventually for $\mathrm{\ensuremath{\Gamma}}$ larger than ${\mathrm{\ensuremath{\Gamma}}}_{\text{CP}}$ the entire many-body spectrum is delocalized. We show that the Renyi entropy is almost independent of the system size in the MBL phase while the delocalized phase shows extensive Renyi entropy. We further obtain the spin-glass transition curve in the energy density $\ensuremath{\epsilon}\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}$ plane from the collapse of the eigenstate spin susceptibility. We demonstrate that in most of the parameter regime, the spin-glass transition occurs close to the MBL transition, indicating that the spin-glass phase is nonergodic and nonthermal while the paramagnetic phase is delocalized and thermal.
The transition between ergodic and many-body localization (MBL) phases lies at the heart of understanding quantum thermalization of many-body systems. Here, we predict a many-body critical (MBC) phase with finite-size scaling analysis in the one-dimensional extended Aubry-André-Harper-Hubbard model, which is different from both the ergodic phase and MBL phase, implying that the quantum system hosts three different fundamental phases in the thermodynamic limit. The level statistics in the MBC phase are well characterized by the so-called critical statistics, and the wave functions exhibit deep multifractal behavior only in the critical region. We further study the half-chain entanglement entropy and thermalization properties and show that the former, in the MBC phase, manifest a volume law scaling, while the many-body states violate the eigenstate thermalization hypothesis. The results are confirmed by the state-of-the-art numerical calculations with system size up to L=22. This work unveils a novel many-body phase which is extended but nonthermal.
We introduce a simple, exactly solvable strong-randomness renormalization group (RG) model for the many-body localization (MBL) transition in one dimension. Our approach relies on a family of RG flows parametrized by the asymmetry between thermal and localized phases. We identify the physical MBL transition in the limit of maximal asymmetry, reflecting the instability of MBL against rare thermal inclusions. We find a critical point that is localized with power-law distributed thermal inclusions. The typical size of critical inclusions remains finite at the transition, while the average size is logarithmically diverging. We propose a two-parameter scaling theory for the many-body localization transition that falls into the Kosterlitz-Thouless universality class, with the MBL phase corresponding to a stable line of fixed points with multifractal behavior.
We use exact diagonalization to explore the many-body localization transition in a random-field spin-1/2 chain. We examine the correlations within each many-body eigenstate, looking at all high-energy states and thus effectively working at infinite temperature. For weak random field the eigenstates are thermal, as expected in this nonlocalized, ``ergodic'' phase. For strong random field the eigenstates are localized with only short-range entanglement. We roughly locate the localization transition and examine some of its finite-size scaling, finding that this quantum phase transition at nonzero temperature might be showing infinite-randomness scaling with a dynamic critical exponent $z\ensuremath{\rightarrow}\ensuremath{\infty}$.
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum f(0≤α≤2)=α2f(0 \leq \alpha \leq 2)=\frac{\alpha}{2}f(0≤α≤2)=2α​, well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. As a consequence, the level statistics at criticality is the closest possible to the Poisson statistics of the Localized phase, and the Delocalized phase near the critical point is not ergodic.