Neural network setups for a precise detection of the many-body localization transition: finite-size scaling and limitations

Determining phase diagrams and phase transitions semi-automatically using machine learning has received a lot of attention recently, with results in good agreement with more conventional approaches in most cases. When it comes to more quantitative predictions, such as the identification of universality class or precise determination of critical points, the task is more challenging. As an exacting test-bed, we study the Heisenberg spin-1/2 chain in a random external field that is known to display a transition from a many-body localized to a thermalizing regime, which nature is not entirely characterized. We introduce different neural network structures and dataset setups to achieve a finite-size scaling analysis with the least possible physical bias (no assumed knowledge on the phase transition and directly inputing wave-function coefficients), using state-of-the-art input data simulating chains of sizes up to L=24. In particular, we use domain adversarial techniques to ensure that the network learns scale-invariant features. We find a variability of the output results with respect to network and training parameters, resulting in relatively large uncertainties on final estimates of critical point and correlation length exponent which tend to be larger than the values obtained from conventional approaches. We put the emphasis on interpretability throughout the paper and discuss what the network appears to learn for the various used architectures. Our findings show that a it quantitative analysis of phase transitions of unknown nature remains a difficult task with neural networks when using the minimally engineered physical input.Comment: v2: published versio

Many-body localization: an introduction and selected topics

What happens in an isolated quantum system when both disorder and interactions are present? Over the recent years, the picture of a non-thermalizing phase of matter, the many-localized phase, has emerged as a stable solution. We present a basic introduction to the topic of many-body localization, using the simple example of a quantum spin chain which allows us to illustrate several of the properties of this phase. We then briefly review the current experimental research efforts probing this physics. The largest part of this review is a selection of more specialized questions, some of which are currently under active investigation. We conclude by summarizing the connections between many-body localization and quantum simulations.Comment: Review article. 28 pages, 8 figures, Comptes Rendus Physique (2018

The semiflexible fully-packed loop model and interacting rhombus tilings

Motivated by a recent adsorption experiment [M.O. Blunt et al., Science 322, 1077 (2008)], we study tilings of the plane with three different types of rhombi. An interaction disfavors pairs of adjacent rhombi of the same type. This is shown to be a special case of a model of fully-packed loops with interactions between monomers at distance two along a loop. We solve the latter model using Coulomb gas techniques and show that its critical exponents vary continuously with the interaction strenght. At low temperature it undergoes a Kosterlitz-Thouless transition to an ordered phase, which is predicted from numerics to occur at a temperature T \sim 110K in the experiments.Comment: 4 pages, 4 figures, v2: corrected typo, v3: minor modifications, published versio

Critical Correlations for Short-Range Valence-Bond Wave Functions on the Square Lattice

We investigate the arguably simplest $SU(2)$-invariant wave functions capable of accounting for spin-liquid behavior, expressed in terms of nearest-neighbor valence-bond states on the square lattice and characterized by different topological invariants. While such wave-functions are known to exhibit short-range spin correlations, we perform Monte Carlo simulations and show that four-point correlations decay algebraically with an exponent $1.16(4)$. This is reminiscent of the {\it classical} dimer problem, albeit with a slower decay. Furthermore, these correlators are found to be spatially modulated according to a wave-vector related to the topological invariants. We conclude that a recently proposed spin Hamiltonian that stabilizes the here considered wave-function(s) as its (degenerate) ground-state(s) should exhibit gapped spin and gapless non-magnetic excitations.Comment: 4 pages, 5 figures. Updated versio

Out-of-time-ordered measurements as a probe of quantum dynamics

Probing the out-of-equilibrium dynamics of quantum matter has gained renewed interest owing to immense experimental progress in artifcial quantum systems. Dynamical quantum measures such as the growth of entanglement entropy (EE) and out-of-time ordered correlators (OTOCs) have been shown, theoretically, to provide great insight by exposing subtle quantum features invisible to traditional measures such as mass transport. However, measuring them in experiments requires either identical copies of the system, an ancilla qubit coupled to the whole system, or many measurements on a single copy, thereby making scalability extremely complex and hence, severely limiting their potential. Here, we introduce an alternate quantity $-$ the out-of-time-ordered measurement (OTOM) $-$ which involves measuring a single observable on a single copy of the system, while retaining the distinctive features of the OTOCs. We show, theoretically, that OTOMs are closely related to OTOCs in a doubled system with the same quantum statistical properties as the original system. Using exact diagonalization, we numerically simulate classical mass transport, as well as quantum dynamics through computations of the OTOC, the OTOM, and the EE in quantum spin chain models in various interesting regimes (including chaotic and many-body localized systems). Our results demonstrate that an OTOM can successfully reveal subtle aspects of quantum dynamics hidden to classical measures, and crucially, provide experimental access to them.Comment: 7 pages, 4 figure

N\'eel to valence-bond solid transition on the honeycomb lattice: Evidence for deconfined criticality

We study a spin-1/2 SU(2) model on the honeycomb lattice with nearest-neighbor antiferromagnetic exchange $J$ that favors N\'eel order, and competing 6-spin interactions $Q$ which favor a valence bond solid (VBS) state in which the bond-energies order at the "columnar" wavevector ${\mathbf K} = (2\pi/3,-2\pi/3)$. We present quantum Monte-Carlo evidence for a direct continuous quantum phase transition between N\'eel and VBS states, with exponents and logarithmic violations of scaling consistent with those at analogous deconfined critical points on the square lattice. Although this strongly suggests a description in terms of deconfined criticality, the measured three-fold anisotropy of the phase of the VBS order parameter shows unusual near-marginal behaviour at the critical point.Comment: published version with extensive T > 0 data; author list rearranged to reflect these new result

Many-body localization in a quasiperiodic Fibonacci chain

We study the many-body localization (MBL) properties of a chain of interacting fermions subject to a quasiperiodic potential such that the non-interacting chain is always delocalized and displays multifractality. Contrary to naive expectations, adding interactions in this systems does not enhance delocalization, and a MBL transition is observed. Due to the local properties of the quasiperiodic potential, the MBL phase presents specific features, such as additional peaks in the density distribution. We furthermore investigate the fate of multifractality in the ergodic phase for low potential values. Our analysis is based on exact numerical studies of eigenstates and dynamical properties after a quench

Generalized Directed Loop Method for Quantum Monte Carlo Simulations

Efficient quantum Monte Carlo update schemes called directed loops have recently been proposed, which improve the efficiency of simulations of quantum lattice models. We propose to generalize the detailed balance equations at the local level during the loop construction by accounting for the matrix elements of the operators associated with open world-line segments. Using linear programming techniques to solve the generalized equations, we look for optimal construction schemes for directed loops. This also allows for an extension of the directed loop scheme to general lattice models, such as high-spin or bosonic models. The resulting algorithms are bounce-free in larger regions of parameter space than the original directed loop algorithm. The generalized directed loop method is applied to the magnetization process of spin chains in order to compare its efficiency to that of previous directed loop schemes. In contrast to general expectations, we find that minimizing bounces alone does not always lead to more efficient algorithms in terms of autocorrelations of physical observables, because of the non-uniqueness of the bounce-free solutions. We therefore propose different general strategies to further minimize autocorrelations, which can be used as supplementary requirements in any directed loop scheme. We show by calculating autocorrelation times for different observables that such strategies indeed lead to improved efficiency; however we find that the optimal strategy depends not only on the model parameters but also on the observable of interest.Comment: 17 pages, 16 figures; v2 : Modified introduction and section 2, Changed title; v3 : Added section on supplementary strategies; published versio

Valence bond distribution and correlation in bipartite Heisenberg antiferromagnets

Every singlet state of a quantum spin 1/2 system can be decomposed into a linear combination of valence bond basis states. The range of valence bonds within this linear combination as well as the correlations between them can reveal the nature of the singlet state, and are key ingredients in variational calculations. In this work, we study the bipartite valence bond distributions and their correlations within the ground state of the Heisenberg antiferromagnet on bipartite lattices. In terms of field theory, this problem can be mapped to correlation functions near a boundary. In dimension d >= 2, a non-linear sigma model analysis reveals that at long distances the probability distribution P(r) of valence bond lengths decays as |r|^(-d-1) and that valence bonds are uncorrelated. By a bosonization analysis, we also obtain P(r) proportional to |r|^(-d-1) in d=1 despite the different mechanism. On the other hand, we find that correlations between valence bonds are important even at large distances in d=1, in stark contrast to d >= 2. The analytical results are confirmed by high-precision quantum Monte Carlo simulations in d=1, 2, and 3. We develop a single-projection loop variant of the valence bond projection algorithm, which is well-designed to compute valence bond probabilities and for which we provide algorithmic details.Comment: 15 pages, 11 figures. Final version after minor revision

Sign-problem-free Monte Carlo simulation of certain frustrated quantum magnets

We introduce a Quantum Monte Carlo (QMC) method which efficiently simulates in a sign-problem-free way a broad class of frustrated $S=1/2$ models with competing antiferromagnetic interactions. Our scheme uses the basis of total spin eigenstates of clusters of spins to avoid the severe sign problem faced by other QMC methods. We also flag important limitations of the new method, and comment on possibilities for further progress.Comment: 6 pages + appendix with supplemental informatio