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arXiv:2106.09036 · Physical review. B./Physical review. B

Fock-space anatomy of eigenstates across the many-body localization transition

Sthitadhi Roy, David E. Logan

Published Nov 1, 2021• 46 citations
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Abstract

We explore the Fock-space structure of eigenstates across the many-body localization (MBL) transition in a disordered, interacting quantum spin-$\frac{1}{2}$ chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition, and is intimately connected to the discontinuous jump in the multifractal exponents characterizing the Fock-space wave functions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the length scale also implies discontinuous behavior of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume scale in the ergodic phase and a length scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless--type scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.

DOI
10.1103/physrevb.104.174201

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We explore the Fock-space structure of eigenstates across the many-body localization (MBL) transition in a disordered, interacting quantum spin-12\frac{1}{2}21​ chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition, and is intimately connected to the discontinuous jump in the multifractal exponents characterizing the Fock-space wave functions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the length scale also implies discontinuous behavior of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume scale in the ergodic phase and a length scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless--type scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.
We explore the Fock-space landscape of eigenstates across the many-body localisation (MBL) transition in a disordered, interacting quantum spin-1/2 chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition; and is intimately connected to the discontinuous jump in the multifractal exponents characterising the Fock-space wavefunctions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the lengthscale also implies discontinuous behaviour of the local observables across the transition. The scaling behaviour of this lengthscale, which is closely connected to that of the inverse participation ratios, shows that the nature of the MBL transition is consistent with it being Kosterlitz-Thouless like, as predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.
This article reviews recent progress in understanding the physics of many-body localisation (MBL) in disordered and interacting quantum many-body systems, from the perspective of ergodicity breaking on the associated Fock space. This approach to MBL is underpinned by mapping the dynamics of the many-body system onto that of a fictitious single particle on the high-dimensional, correlated and disordered Fock-space graph; yet, as we elaborate, the problem is fundamentally different from that of conventional Anderson localisation on high-dimensional or hierarchical graphs. We discuss in detail the nature of eigenstate correlations on the Fock space, both static and dynamic, and in the ergodic and many-body localised phases as well as in the vicinity of the MBL transition. The latter in turn sheds light on the nature of the transition, and motivates a scaling theory for it in terms of Fock-space based quantities. We also illustrate how these quantities can be concretely connected to real-space observables. An overview is given of several analytical and numerical techniques which have proven important in developing a comprehensive picture. Finally, we comment on some open questions in the field of MBL where the Fock-space approach is likely to prove insightful.
We implement a recursive Green's function method to extract the Fock space (FS) propagator and associated self-energy across the many-body localization (MBL) transition, for one-dimensional interacting fermions in a random on-site potential. We show that the typical value of the imaginary part of the local FS self-energy, ${\mathrm{\ensuremath{\Delta}}}_{t}$, related to the decay rate of an initially localized state, acts as a probabilistic order parameter for the thermal to MBL phase transition and can be used to characterize critical properties of the transition as well as the multifractal nature of MBL states as a function of disorder strength WWW. In particular, we show that a fractal dimension Ds{D}_{s}Ds​ extracted from ${\mathrm{\ensuremath{\Delta}}}_{t}$ jumps discontinuously across the transition, from Ds<1{D}_{s}<1Ds​<1 in the MBL phase to Ds=1{D}_{s}=1Ds​=1 in the thermal phase. Moreover, ${\mathrm{\ensuremath{\Delta}}}_{t}$ follows an asymmetrical finite-size scaling form across the thermal-MBL transition, where a nonergodic volume in the thermal phase diverges with a Kosterlitz-Thouless--like essential singularity at the critical point Wc{W}_{c}Wc​ and controls the continuous vanishing of ${\mathrm{\ensuremath{\Delta}}}_{t}$ as Wc{W}_{c}Wc​ is approached. In contrast, a correlation length ($\ensuremath{\xi}$) extracted from ${\mathrm{\ensuremath{\Delta}}}_{t}$ exhibits a power-law divergence on approaching Wc{W}_{c}Wc​ from the MBL phase.
Many-body localised (MBL) phases of disordered, interacting quantum systems allow for exotic localisation protected quantum order in eigenstates at arbitrarily high energy densities. In this work, we analyse the manifestation of such order on the Hilbert-space anatomy of eigenstates. Quantified in terms of non-local Hilbert-spatial correlations of eigenstate amplitudes, we find that the spread of the eigenstates on the Hilbert-space graph is directly related to the order parameters which characterise the localisation protected order, and hence these correlations, in turn, characterise the order or lack thereof. Higher-point eigenstate correlations also characterise the different entanglement structures in the many-body localised phases, with and without order, as well as in the ergodic phase. The results pave the way for characterising the transitions between MBL phases and the ergodic phase in terms of scaling of emergent correlation lengthscales on the Hilbert-space graph.
Many-body localization (MBL) is a novel prototype of ergodicity breaking due to the emergence of local integrals of motion (LIOMs) in a disordered interacting quantum system. To better understand the role played by the existence of such LIOMs, we explore and study some of their structural properties across the MBL transition. We first consider a one-dimensional XXZ spin chain in a disordered magnetic field and introduce and implement a nonperturbative, fast, and accurate method of constructing LIOMs. In contrast to already existing methods, our scheme allows obtaining LIOMs not only in the deep MBL phase but, rather, near the transition point too. Then, we take the matrix representation of LIOM operators as an adjacency matrix of a directed graph whose elements describe the connectivity of ordered eigenbasis in the Hilbert space. Our cluster-size analysis for this graph shows that the MBL transition coincides with a percolation transition in the Hilbert space. By performing finite-size scaling, we compare the critical disorder and correlation exponent $\ensuremath{\nu}$ both in the presence and absence of interactions. Finally, we also discuss how the distribution of diagonal elements of LIOM operators in a typical cluster signals the transition.
In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in Hilbert space, the support of the eigenstates. In this Letter, using exact diagonalization techniques we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases for up to L=24 spin-1/2 particles (i.e., Hilbert space of size N≃2.7×10^{6}). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions.