Criterion for Many-Body Localization-Delocalization Phase Transition

Serbyn, Maksym; Papić, Zlatko; Abanin, Dmitry A.

doi:10.1103/physrevx.5.041047

Cited by 292 publications

(317 citation statements)

References 63 publications

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“…1) that indicates the location of the ETH-MBL transition as a function of energy density and disorder strength. It is in good agreement with results obtained using conventional methods [18,36,43,49]. This paper is organized as follows.…”

Section: Introductionsupporting

confidence: 86%

“…(8)], which is designed to favor networks which confidently classify transition region states, makes physical sense and is essential to make contact with established methods. We note that by refining and combining conventional methods [18,36,43,49] the same or better classification success can be achieved. However, here we want to point out that there is no equally simple method, assuming as little prior knowledge, that performs equally well as the machine learning based approach.…”

Section: B Single Disorder Realizationmentioning

confidence: 89%

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Probing many-body localization with neural networks

2017

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We show that a simple artificial neural network trained on entanglement spectra of individual states of a many-body quantum system can be used to determine the transition between a many-body localized and a thermalizing regime. Specifically, we study the Heisenberg spin-1/2 chain in a random external field. We employ a multilayer perceptron with a single hidden layer, which is trained on labeled entanglement spectra pertaining to the fully localized and fully thermal regimes. We then apply this network to classify spectra belonging to states in the transition region. For training, we use a cost function that contains, in addition to the usual error and regularization parts, a term that favors a confident classification of the transition region states. The resulting phase diagram is in good agreement with the one obtained by more conventional methods and can be computed for small systems. In particular, the neural network outperforms conventional methods in classifying individual eigenstates pertaining to a single disorder realization. It allows us to map out the structure of these eigenstates across the transition with spatial resolution. Furthermore, we analyze the network operation using the dreaming technique to show that the neural network correctly learns by itself the power-law structure of the entanglement spectra in the many-body localized regime.

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Section: Introductionsupporting

confidence: 86%

Section: B Single Disorder Realizationmentioning

confidence: 89%

Probing many-body localization with neural networks

2017

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“…In the present case, the variable w n introduced in Eq. 21 involving the matrix element of the local operator I between two consecutive eigenstates has been studied in detail in Ref [26] in terms of the notation…”

Section: Scaling With the System Size L Of The Amplitudes Of The Hmentioning

confidence: 99%

Many-body-localization transition: sensitivity to twisted boundary conditions

Monthus¹

2017

J. Phys. A: Math. Theor.

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For disordered interacting quantum systems, the sensitivity of the spectrum to twisted boundary conditions depending on an infinitesimal angle φ can be used to analyze the Many-Body-Localization Transition. The sensitivity of the energy levels En(φ) is measured by the level curvature Kn = E ′′ n (0), or more precisely by the Thouless dimensionless curvature kn = Kn/∆n, where ∆n is the level spacing that decays exponentially with the size L of the system. For instance ∆n ∝ 2 −L in the middle of the spectrum of quantum spin chains of L spins, while the Drude weight Dn = LKn studied recently by M. Filippone, P.W. Brouwer, J. Eisert and F. von Oppen [arXiv:1606.07291v1] involves a different rescaling. The sensitivity of the eigenstates |ψn(φ) > is characterized by the susceptibility χn = −F ′′ n (0) of the fidelity Fn = | < ψn(0)|ψn(φ) > |. Both observables are distributed with probability distributions displaying power-law tails2 , where β is the level repulsion index taking the values β GOE = 1 in the ergodic phase and β loc = 0 in the localized phase. The amplitudes A β and B β of these two heavy tails are given by some moments of the off-diagonal matrix element of the local current operator between two nearby energy levels, whose probability distribution has been proposed as a criterion for the Many-Body-Localization transition by M. Serbyn, Z. Papic and D.A. Abanin [Phys. Rev. X 5, 041047 (2015)].

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“…This form holds for matrix elements of local operators in a finite-size, ETH system [9,45]. The renormalization of intercluster couplings is different from those of [37,38] Approximating Γ ij by the limiting MBL and ETH forms becomes self-consistently justified since the width of the distribution of resonance parameters g ij = Γ ij /∆E ij [37,38,47] increases with each RG step. In an infinite critical system, the width of the distribution of g increases without bound along the RG flow so that one asymptotically encounters only the cases g 1 (MBL) or g 1 (ETH) and almost never faces marginal cases where g ≈ 1.…”

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confidence: 99%

Scaling Theory of Entanglement at the Many-Body Localization Transition

2017

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We study the universal properties of eigenstate entanglement entropy across the transition between many-body localized (MBL) and thermal phases. We develop an improved real space renormalization group approach that enables numerical simulation of large system sizes and systematic extrapolation to the infinite system size limit. For systems smaller than the correlation length, the average entanglement follows a sub-thermal volume law, whose coefficient is a universal scaling function. The full distribution of entanglement follows a universal scaling form, and exhibits a bimodal structure that produces universal subleading power-law corrections to the leading volume-law. For systems larger than the correlation length, the short interval entanglement exhibits a discontinuous jump at the transition from fully thermal volume-law on the thermal side, to pure area-law on the MBL side.Recent experimental advances in synthesizing isolated quantum many-body systems, such as cold-atoms [1][2][3][4], trapped ions [5,6], or impurity spins in solids [7,8], have raised fundamental questions about the nature of statistical mechanics. Even when decoupled from external sources of dissipation, large interacting quantum systems tend to act as their own heat-baths and reach thermal equilibrium. This behavior is formalized in the eigenstate thermalization hypothesis (ETH) [9,10]. Generic excited eigenstates of such thermal systems are highly entangled, with the entanglement of a subregion scaling as the volume of that region ("volume law"). This results in incoherent, classical dynamics at long times. In contrast, strong disorder can dramatically alter this picture by pinning excitations that would otherwise propagate heat and entanglement [11][12][13][14][15][16][17]. In such many-body localized (MBL) systems [18][19][20], generic eigenstates have properties akin to those of ground states. They exhibit short-range entanglement that scales like the perimeter of the subregion [17] ("area law"), and have quantum coherent dynamics up to arbitrarily long time scales [21][22][23][24][25][26][27], even at high energy densities [17,26,[28][29][30][31].A transition between MBL and thermal regimes requires a singular rearrangement of eigenstates from area-law to volume-law entanglement. This many-body (de)localization transition (MBLT) represents an entirely new class of critical phenomena, outside the conventional framework of equilibrium thermal or quantum phase transitions. Developing a systematic theory of this transition promises not only to expand our understanding of possible critical phenomena, but also to yield universal insights into the nature of the proximate MBL and thermal phases.The eigenstate entanglement entropy can be viewed as a non-equilibrium analog of the thermodynamic free energy for a conventional thermal phase transition, and plays a central role in our conceptual understanding of the MBL and ETH phases. Describing the entanglement across the MBLT requires addressing the challenging combination of disorder, interactio...

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Criterion for Many-Body Localization-Delocalization Phase Transition

Cited by 292 publications

References 63 publications

Probing many-body localization with neural networks

Probing many-body localization with neural networks

Many-body-localization transition: sensitivity to twisted boundary conditions

Scaling Theory of Entanglement at the Many-Body Localization Transition

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