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Register a new userProbing Criticality in Quantum Spin Chains with Neural Networks
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The numerical emulation of quantum systems often requires an exponential number of degrees of freedom which translates to a computational bottleneck. Methods of machine learning have been used in adjacent fields for effective feature extraction and dimensionality reduction of high-dimensional datasets. Recent studies have revealed that neural networks are further suitable for the determination of macroscopic phases of matter and associated phase transitions as well as efficient quantum state representation. In this work, we address quantum phase transitions in quantum spin chains, namely the transverse field Ising chain and the anisotropic XY chain, and show that even neural networks with no hidden layers can be effectively trained to distinguish between magnetically ordered and disordered phases. Our neural network acts to predict the corresponding crossovers finite-size systems undergo. Our results extend to a wide class of interacting quantum many-body systems and illustrate the wide applicability of neural networks to many-body quantum physics.
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Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems as well as disordered ones. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization group transformation. These discrete sequences are therefore fixed points of a emph{functional} renormalization group. This observation allows for an asymptotically exact treatment of the critical points. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains, as well as a phenomenological model for the quasiperiodic many-body localization transition.
We study the infinite-temperature properties of an infinite sequence of random quantum spin chains using a real-space renormalization group approach, and demonstrate that they exhibit non-ergodic behavior at strong disorder. The analysis is conveniently implemented in terms of SU(2)$_k$ anyon chains that include the Ising and Potts chains as notable examples. Highly excited eigenstates of these systems exhibit properties usually associated with quantum critical ground states, leading us to dub them quantum critical glasses. We argue that random-bond Heisenberg chains self-thermalize and that the excited-state entanglement crosses over from volume-law to logarithmic scaling at a length scale that diverges in the Heisenberg limit $krightarrowinfty$. The excited state fixed points are generically distinct from their ground state counterparts, and represent novel non-equilibrium critical phases of matter.
We propose a new framework to understand how quantum effects may impact on the dynamics of neural networks. We implement the dynamics of neural networks in terms of Markovian open quantum systems, which allows us to treat thermal and quantum coherent effects on the same footing. In particular, we propose an open quantum generalisation of the celebrated Hopfield neural network, the simplest toy model of associative memory. We determine its phase diagram and show that quantum fluctuations give rise to a qualitatively new non-equilibrium phase. This novel phase is characterised by limit cycles corresponding to high-dimensional stationary manifolds that may be regarded as a generalisation of storage patterns to the quantum domain.
The classical simulation of quantum systems typically requires exponential resources. Recently, the introduction of a machine learning-based wavefunction ansatz has led to the ability to solve the quantum many-body problem in regimes that had previously been intractable for existing exact numerical methods. Here, we demonstrate the utility of the variational representation of quantum states based on artificial neural networks for performing quantum optimization. We show empirically that this methodology achieves high approximation ratio solutions with polynomial classical computing resources for a range of instances of the Maximum Cut (MaxCut) problem whose solutions have been encoded into the ground state of quantum many-body systems up to and including 256 qubits.
Using strong-disorder renormalization group, numerical exact diagonalization, and quantum Monte Carlo methods, we revisit the random antiferromagnetic XXZ spin-1/2 chain focusing on the long-length and ground-state behavior of the average time-independent spin-spin correlation function C(l)=upsilon l^{-eta}. In addition to the well-known universal (disorder-independent) power-law exponent eta=2, we find interesting universal features displayed by the prefactor upsilon=upsilon_o/3, if l is odd, and upsilon=upsilon_e/3, otherwise. Although upsilon_o and upsilon_e are nonuniversal (disorder dependent) and distinct in magnitude, the combination upsilon_o + upsilon_e = -1/4 is universal if C is computed along the symmetric (longitudinal) axis. The origin of the nonuniversalities of the prefactors is discussed in the renormalization-group framework where a solvable toy model is considered. Moreover, we relate the average correlation function with the average entanglement entropy, whose amplitude has been recently shown to be universal. The nonuniversalities of the prefactors are shown to contribute only to surface terms of the entropy. Finally, we discuss the experimental relevance of our results by computing the structure factor whose scaling properties, interestingly, depend on the correlation prefactors.
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