Forward approximation as a mean-field approximation for the Anderson and many-body localization transitions

Pietracaprina, Francesca; Ros, Valentina; Scardicchio, Antonello

doi:10.1103/physrevb.93.054201

Cited by 80 publications

(95 citation statements)

References 59 publications

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“…The eigenstate phases of H p do as well, yet the eigenstates are never localized at any finite p. They are instead non-ergodic, in a manner that comes to resemble localization as p increases. We show this by studying the eigenstates within perturbation theory and the forward-scattering approximation [42].Before we turn to detailed analysis, it is useful to consider the p-spin models in terms of Anderson localization on the N -dimensional hypercube defined by the σ z configuration space. H C p is then a random potential and H Q causes hops along the edges of the hypercube.…”

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Clustering of Nonergodic Eigenstates in Quantum Spin Glasses

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The two primary categories for eigenstate phases of matter at finite temperature are manybody localization (MBL) and the eigenstate thermalization hypothesis (ETH). We show that in the paradigmatic quantum p-spin models of spin-glass theory, eigenstates violate ETH yet are not MBL either. A mobility edge, which we locate using the forward-scattering approximation and replica techniques, separates the non-ergodic phase at small transverse field from an ergodic phase at large transverse field. The non-ergodic phase is also bounded from above in temperature, by a transition in configuration-space statistics reminiscent of the clustering transition in spin-glass theory. We show that the non-ergodic eigenstates are organized in clusters which exhibit distinct magnetization patterns, as characterized by an eigenstate variant of the Edwards-Anderson order parameter.Many systems under experimental investigation as platforms for many-body localization (MBL) [1][2][3][4][5][6][7][8][9] have long-range interactions that mediate the direct transport of excitations. This includes disordered electronic materials [10,11], ion traps [12], interacting NV centers in diamond [13,14], and superconducting qubit devices developed for adiabatic quantum computing [15][16][17]. In sufficiently long-ranged systems, the proliferation of longdistance resonances precludes quantum mechanical localization [18][19][20][21][22], an intuitive result strongly supported by analytic work over the last half century. Nevertheless, the quantum Random Energy Model (QREM), an infiniterange spin glass, was recently shown to exhibit a phase with localized eigenstates at finite energy density [23,24]. The QREM provides an analytically tractable framework for studying mobility edges and configuration-space localization. This raises the obvious question of how localization survives despite the infinite-range interactions and what role it plays in more realistic long-range systems.Some insight comes from considering the distribution of local fields -i.e., the energy required to flip one of the system's N spins relative to a given configuration. In the QREM, flipping a spin typically changes the energy by O(N ). Thus the quantum fluctuations which lead to the proliferation of resonances are strongly suppressed. However, short-range models have O(1) local fields, and in fact, so do power-law and infinite-range systems with general p-body interactions. This suggests that the eigenstate-localized phase of the QREM is an exceptional case among long-range models: strict configuration-space localization cannot exist in any model with O(1) local fields, since the introduction of quantum dynamics causes resonant fluctuations.In this paper, we study the eigenstate properties of the quantum p-spin models [25][26][27][28]. Over the past four decades, these models have become paradigms for the mean-field theory of spin glasses [29][30][31][32] Sherrington-Kirkpatrick model (p = 2) [33,34]. The QREM corresponds to the p → ∞ limit in many senses, but the local-field distrib...

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Clustering of Nonergodic Eigenstates in Quantum Spin Glasses

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“…For the Heisenberg model with random fields the transition value has been estimated through other numerical evidence 38,53,54 to be equal to h H c = 3.7(2) at the center of the band for the parameters that we used, although its actual value could be larger (h H c ≥ 4.5 according to 55 ); an equivalent highquality numerical result is not available for the Aubry-André model, although experimental works find the localization transition at similar values 49 . Interestingly, as soon as interactions are introduced, the Aubry-André model acquires almost identical features to the Heisenberg with random fields model first studied by total correlations in 39 .…”

Section: B Many-body Localizationmentioning

confidence: 99%

Total correlations of the diagonal ensemble as a generic indicator for ergodicity breaking in quantum systems

2017

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The diagonal ensemble is the infinite time average of a quantum state following unitary dynamics in systems without degeneracies. In analogy to the time average of a classical phase space dynamics, it is intimately related to the ergodic properties of the quantum system giving information on the spreading of the initial state in the eigenstates of the Hamiltonian. In this work we apply a concept from quantum information, known as total correlations, to the diagonal ensemble. Forming an upper-bound on the multipartite entanglement, it quantifies the combination of both classical and quantum correlations in a mixed state. We generalize the total correlations of the diagonal ensemble to more general α-Renyi entropies and focus on the the cases α = 1 and α = 2 with further numerical extensions in mind. Here we show that the total correlations of the diagonal ensemble is a generic indicator of ergodicity breaking, displaying a sub-extensive behaviour when the system is ergodic. We demonstrate this by investigating its scaling in a range of spin chain models focusing not only on the cases of integrability breaking but also emphasize its role in understanding the transition from an ergodic to a many-body localized phase in systems with disorder or quasi-periodicity.

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“…Since the simple formula (6) is derived under the sole assumption that the conserved operators have spectrum ±1, it could be applied to the conserved pseudo-spins constructed numerically in Refs. [46][47][48] for non-perturbative interactions by means of renormalization procedures or diagonalizing flows.It would be interesting to extend this calculation beyond the lowest orders, exploiting for example the expansion for the conserved quantities in the forward approximation [30,49], to discuss the behavior of the remanent magnetization when approaching the delocalization threshold. An interesting question is whether at the delocalization transition, i.e.…”

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“…It would be interesting to extend this calculation beyond the lowest orders, exploiting for example the expansion for the conserved quantities in the forward approximation [30,49], to discuss the behavior of the remanent magnetization when approaching the delocalization threshold. An interesting question is whether at the delocalization transition, i.e.…”

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Remanent Magnetization: Signature of Many-Body Localization in Quantum Antiferromagnets

Müller

2017

Phys. Rev. Lett.

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We study the remanent magnetization in antiferromagnetic, many-body localized quantum spin chains, initialized in a fully magnetized state. Its long time limit is an order parameter for the localization transition, which is readily accessible by standard experimental probes in magnets. We analytically calculate its value in the strong-disorder regime exploiting the explicit construction of quasi-local conserved quantities of the localized phase. We discuss analogies in cold atomic systems.Introduction. The non-equilibrium dynamics in disordered, isolated quantum systems have been subject to theoretical investigations ever since the notion of localization was introduced in [1]. Spin systems in random fields are prototypical models to analyze the disorderinduced breakdown of thermalization: a large number of studies on disordered spin chains [2][3][4][5][6][7][8][9][10][11][12][13] has provided evidence for a dynamical phase transition between a weakdisorder phase which thermalizes, and a Many-Body Localized (MBL) phase in which excitations do not diffuse, ergodicity is broken and local memory of the initial conditions persists for infinite time [14][15][16].Signatures of MBL are found in the properties of individual many-body eigenstates. Even highly excited eigenstates exhibit area-law scaling of the bipartite entanglement entropy [4,7,8,17] and Poissonian level statistics [2,18,19], both being incompatible with thermalization [20][21][22]. Novel dynamical properties such as the logarithmic spreading of entanglement have been observed in direct simulations of the time evolution [3,23]. The non-equilibrium physics of MBL systems has been probed experimentally in artificial quantum systems made of cold atomic gases [24,25] and trapped ion systems [26], while an indirect signature in the from of strongly suppressed absorption of radiation was found in electron-glasses [27]. However, direct observations of MBL in solid-state materials are still lacking.It has been argued [28][29][30] that the properties of MBL systems are related to the existence of extensively many quasi-local conserved operators that strongly constrain the quantum dynamics, preventing both transport and thermalization. Their existence also follows as a corollary from Imbrie's rigorous arguments in favor of MBL [31,32].In this work, we propose a experimentally readily observable consequence of MBL in quantum magnets: the out-of-equilibrium remanent magnetization that persists after ferromagnetically polarizing an antiferromagnet whose total magnetization is not a conserved. The remanence implies non-ergodicity, since ergodic dynamics would relax the magnetization completely (cf. Fig. 1 for a schematic sketch of the protocol). As an example, we consider an antiferromagnetic, anisotropic Heisenberg spin-1/2 chain(1) subject to random fields h k along the Ising axis. We assume J z < 0, as well as J x = J y to ensure the nonconservation of the total magnetization. Such Hamiltonians can be realized, e.g., in Ising compounds with both exchange and dipola...

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Forward approximation as a mean-field approximation for the Anderson and many-body localization transitions

Cited by 80 publications

References 59 publications

Clustering of Nonergodic Eigenstates in Quantum Spin Glasses

Clustering of Nonergodic Eigenstates in Quantum Spin Glasses

Total correlations of the diagonal ensemble as a generic indicator for ergodicity breaking in quantum systems

Remanent Magnetization: Signature of Many-Body Localization in Quantum Antiferromagnets

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