Many-body Research Articles

Machine learning stochastic differential equations for the evolution of order parameters of classical many-body systems in and out of equilibrium

Abstract We develop a machine learning algorithm to infer the emergent stochastic equation governing the evolution of an order parameter of a many-body system. We train our neural network to independently learn the directed force acting on the order parameter as well as an effective diffusive noise. We illustrate our approach using the classical Ising model endowed with Glauber dynamics, and the contact process as test cases. For both models, which represent paradigmatic equilibrium and nonequilibrium scenarios, the directed force and noise can be efficiently inferred. The directed force term of the Ising model allows us to reconstruct an effective potential for the order parameter which develops the characteristic double-well shape below the critical temperature. Despite its genuine nonequilibrium nature, such an effective potential can also be obtained for the contact process and its shape signals a phase transition into an absorbing state. Also, in contrast to the equilibrium Ising model, the presence of an absorbing state renders the noise term dependent on the value of the order parameter itself.

Phase transitions in quantum many-body scars

We propose a type of phase transition in quantum many-body systems that occurs in highly excited quantum many-body scar states while most of the spectrum is largely unaffected. Such scar-state phase transitions can be realized by embedding a matrix product state, known to undergo a phase transition, as a scar state into the thermal spectrum of a parent Hamiltonian. We find numerically that the mechanism for the scar-state phase transition involves the formation or presence of low-entropy states at energies similar to the scar state in the vicinity of the phase transition point. Published by the American Physical Society 2024

Determinant- and derivative-free quantum Monte Carlo within the stochastic representation of wavefunctions

Abstract Describing the ground states of continuous, real-space quantum many-body systems, like atoms and molecules, is a significant computational challenge with applications throughout the physical sciences. Recent progress was made by variational methods based on machine learning (ML) ansatzes. However, since these approaches are based on energy minimization, ansatzes must be twice differentiable. This (a) precludes the use of many powerful classes of ML models; and (b) makes the enforcement of bosonic, fermionic, and other symmetries costly. Furthermore, (c) the optimization procedure is often unstable unless it is done by imaginary time propagation, which is often impractically expensive in modern ML models with many parameters. The stochastic representation of wavefunctions (SRW), introduced in (Atanasova et al 2023 Nat. Commun. 14 3601), is a recent approach to overcoming (c). SRW enables imaginary time propagation at scale, and makes some headway towards the solution of problem (b), but remains limited by problem (a). Here, we argue that combining SRW with path integral techniques leads to a new formulation that overcomes all three problems simultaneously. As a demonstration, we apply the approach to generalized ‘Hooke’s atoms’: interacting particles in harmonic wells. We benchmark our results against state-of-the-art data where possible, and use it to investigate the crossover between the Fermi liquid and the Wigner molecule within closed-shell systems. Our results shed new light on the competition between interaction-driven symmetry breaking and kinetic-energy-driven delocalization.

Transport and integrability-breaking in non-Hermitian many-body quantum systems

Transport and integrability-breaking in non-Hermitian many-body quantum systems

Distilling the Essential Elements of Nuclear Binding via Neural-Network Quantum States.

To distill the essential elements of nuclear binding, we seek the simplest Hamiltonian capable of modeling atomic nuclei with percent-level accuracy. A critical aspect of this endeavor consists of accurately solving the quantum many-body problem without incurring an exponential computing cost with the number of nucleons. We address this challenge by leveraging a variational MonteCarlo method based on a highly expressive neural-network quantum state ansatz. In addition to computing binding energies and charge radii of nuclei with up to A=20 nucleons, by evaluating their magnetic moments, we demonstrate that neural-network quantum states are able to correctly capture the self-emerging nuclear shell structure. To this end, we introduce a novel computational protocol based on adding an external magnetic field to the nuclear Hamiltonian, which allows the neural network to learn the preferred polarization of the nucleus within the given magnetic field.

Pseudogap Effects in the Strongly Correlated Regime of the Two-Dimensional Fermi Gas.

The two-species Fermi gas with attractive short-range interactions in two spatial dimensions provides a paradigmatic system for the understanding of strongly correlated Fermi superfluids in two dimensions. It is known to exhibit a BEC to BCS crossover as a function of ln(k_{F}a), where a is the scattering length, and to undergo a Berezinskii-Kosterlitz-Thouless superfluid transition below a critical temperature T_{c}. However, the extent of a pseudogap regime in the strongly correlated regime of ln(k_{F}a)∼1, in which pairing correlations persist above T_{c}, remains largely unexplored with controlled theoretical methods. Here, we use finite-temperature auxiliary-field quantum MonteCarlo methods on discrete lattices in the canonical ensemble formalism to calculate thermodynamical observables in the strongly correlated regime. We extrapolate to continuous time and the continuum limit to eliminate systematic errors and present results for particle numbers ranging from N=42 to N=162. We estimate T_{c} by a finite-size scaling analysis, and observe clear pseudogap signatures above T_{c} and below a temperature T^{*} in both the spin susceptibility and free-energy gap. We also present results for the contact, a fundamental thermodynamic property of quantum many-body systems with short-range interactions.

Symmetry Restoration and Quantum Mpemba Effect in Symmetric Random Circuits.

Entanglement asymmetry, which serves as a diagnostic tool for symmetry breaking and a proxy for thermalization, has recently been proposed and studied in the context of symmetry restoration for quantum many-body systems undergoing a quench. In this Letter, we investigate symmetry restoration in various symmetric random quantum circuits, particularly focusing on the U(1) symmetry case. In contrast to nonsymmetric random circuits where the U(1) symmetry of a small subsystem can always be restored at late times, we reveal that symmetry restoration can fail in U(1)-symmetric circuits for certain weak symmetry-broken initial states in finite-size systems. In the early-time dynamics, we observe an intriguing quantum Mpemba effect implying that symmetry is restored faster when the initial state is more asymmetric. Furthermore, we also investigate the entanglement asymmetry dynamics for SU(2) and Z_{2} symmetric circuits and identify the presence and absence of the quantum Mpemba effect for the corresponding symmetries, respectively. A unified understanding of these results is provided through the lens of quantum thermalization with conserved charges.

Exact dynamics of the Tomonaga-Luttinger liquid and shortcuts to adiabaticity

Controlling many-body quantum systems is a highly challenging task required to advance quantum technologies. Here, we report progress in controlling gapless many-body quantum systems described by the Tomonaga-Luttinger liquid (TLL). To do so, we investigate the exact dynamics of the TLL induced by an arbitrary interaction quench, making use of the SU(1,1) dynamical symmetry group and the Schrödinger picture. First, we demonstrate that this approach is useful to perform a shortcut to adiabaticity, which cancels the final nonadiabatic residual energy of the driven TLL and is experimentally implementable in the semiclassical limit of the sine-Gordon model. Second, we apply this framework to analyze various driving schemes in finite time, including linear ramps and smooth protocols. Published by the American Physical Society 2024

Quantum turbulence, superfluidity, non-Markovian dynamics, and wave function thermalization

While quantum turbulence has been addressed both experimentally (predominantly for superfluid He4 and He3) and theoretically, the dynamics of various ensembles of quantized vortices has been followed in time only until the vortices have decayed into phonons. How this “thermalization” is achieved is still an unaddressed and thus an unelucidated question. The unitary Fermi gas (UFG) is a unique quantum system, which has no classical counterpart and is of relevance to neutron stars, cold atoms, condensed-matter and nuclear many-body systems. The non-Markovian evolution of an isolated UFG is put in evidence and its entire nonequilibrium evolution can be studied theoretically within a unified theoretical framework. The initial lattice of quantum vortices and antivortices evolves through a couple of vortex tangles and excitation of Kelvin waves, where vortices cross and reconnect, until very slowly thermalization sets in. Published by the American Physical Society 2024

Revealing symmetries in quantum computing for many-body systems

Abstract We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan–Wigner form in which they can act on multi-qubit states. Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian. They are, however, masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits. To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations. They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices. We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces effort in quantum computations for many-body systems.

Emergence of unitary symmetry of microcanonically truncated operators in chaotic quantum systems.

We study statistical properties of matrix elements of observables written in the energy eigenbasis and truncated to small microcanonical windows. We present numerical evidence indicating that for all few-body operators in chaotic many-body systems, truncated below acertain energy scale, collective statistical properties of matrix elements exhibit emergent unitary symmetry. Namely, we show that below acertain scale the spectra of the truncated operators exhibit universal behavior, matching our analytic predictions, which are numerically testable for system sizes beyond exact diagonalization. We discuss operator and system-size dependence of the energy scale of emergent unitary symmetry and put our findings inthe context of previous works exploring theemergence of random-matrix behavior at small energy scales.

Commutative families in DIM algebra, integrable many-body systems and q, t matrix models

We extend our consideration of commutative subalgebras (rays) in different representations of the W1+∞ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra Uq,tgl̂̂1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {U}_{q,t}\\left({\\hat{\\hat{\\mathfrak{gl}}}}_1\\right) $$\\end{document}). Its advantage is that it possesses the Miki automorphism, which makes all commutative rays equivalent. Integrable systems associated with these rays become finite-difference and, apart from the trigonometric Ruijsenaars system not too much familiar. We concentrate on the simplest many-body and Fock representations, and derive explicit formulas for all generators of the elliptic Hall algebra en,m. In the one-body representation, they differ just by normalization from znqmD̂\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {z}^n{q}^{m\\hat{D}} $$\\end{document} of the W1+∞ Lie algebra, and, in the N -body case, they are non-trivially generalized to monomials of the Cherednik operators with action restricted to symmetric polynomials. In the Fock representation, the resulting operators are expressed through auxiliary polynomials of n variables, which define weights in the residues formulas. We also discuss q, t-deformation of matrix models associated with constructed commutative subalgebras.

Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries.

We study quantum many-body systems in the presence of an exotic antiunitary translation or inversion symmetry involving time reversal. Based on a symmetry-twisting method and spectrum robustness, we propose that a half-integer spin chain that respects any of these two antiunitary crystalline symmetries in addition to the discrete Z_{2}×Z_{2} global spin-rotation symmetry must either be gapless or possess degenerate ground states. This explains the gaplessness of a class of chiral spin models not indicated by the Lieb-Schultz-Mattis theorem and its known extensions. Moreover, we present symmetry classes with minimal sets of generators that give nontrivial Lieb-Schultz-Mattis-type constraints, argued by the bulk-boundary correspondence in 2D symmetry-protected topological phases as well as lattice homotopy. Our results for detecting the ingappability of 1D quantum magnets from the interplay between spin-rotation symmetries and magnetic space groups are applicable to systems with a broader class of spin interactions, including Dzyaloshinskii-Moriya and triple-product interactions.

Study of quantum nonlocality by CHSH function and its extension in disordered fermions

Quantum nonlocality is an important concept in quantum physics. In this work, we study the quantum nonlocality in a fermion many-body system under quasi-periodic disorders. The Clauser-Horne-Shimony-Holt (CHSH) inequality is systematically investigated, which quantifies quantum nonlocality between two sites. We find particular behaviors of the quantifiers of quantum nonlocality around the extended and critical phase transitions in the system and further that the CHSH inequality is not broken in the globally averaged picture of the maximum value of the quantum nonlocality, but the violation probability of the CHSH inequality for two site pairs in the system becomes sufficiently finite in the critical phase and on a critical phase boundary. Further, we investigate an extension of the CHSH inequality, Mermin-Klyshko-Svetlichny (MKS) polynomials, which can characterize multipartite quantum nonlocality. We also find a similar behavior to the case of CHSH inequality. In particular, in the critical regime and on a transition point, the adjacent three-qubit MKS polynomial in a portion of the system exhibits a quantum nonlocal violation regime with a finite probability in the critical phase.

General Criterion for Non-Hermitian Skin Effects and Application: Fock Space Skin Effects in Many-Body Systems.

Non-Hermiticity enables macroscopic accumulation of bulk states, named non-Hermitian skin effects. The non-Hermitian skin effects are well established for single-particle systems, but their proper characterization for general systems is elusive. Here, we propose a general criterion of non-Hermitian skin effects, which works for any finite-dimensional system evolved by a linear operator. The applicable systems include many-body systems and network systems. A system meeting the criterion exhibits enhanced non-normality of the evolution operator, accompanied by exceptional characteristics intrinsic to non-Hermitian systems. Applying the criterion, we discover a new type of non-Hermitian skin effect in many-body systems, which we dub the Fock space skin effect. We also discuss the Fock space skin effect-induced slow dynamics, which gives an experimental signal for the Fock space skin effect.

Realization of quantum secure direct communication by Kitaev Abelian anyons

The quantum secure direct communication (QSDC) is provably secure and has a high capacity. The QSDC protocols are usually experimented in photon systems. Here we propose four schemes to realize the QSDC in the Kitaev Abelian anyon model: 1) the two-step scheme, 2) 2⊗2 dimensional superdense coding scheme, 3) the hyperentanglement-assisted one-step scheme and 4) one-time pad scheme. Our work provides a possibility to implement the quantum secure direct communication in many-body systems and gives a new research perspective of the quantum information on the unitary evolution of quantum field systems.

Continuous Space-Time Crystal State Driven by Nonreciprocal Optical Forces.

We show that the continuous time crystal state can arise in an ensemble of linear oscillators from nonconservative coupling via optical radiation pressure forces. This new mechanism comprehensively explains observations of the time crystal state in an array of nanowires illuminated with light [T. Liu et al., Nat. Phys. 19, 986 (2023).NPAHAX1745-247310.1038/s41567-023-02023-5]. Being fundamentally different from regimes of nonlinear synchronization, it has relevance to a wide range of interacting many-body systems, including in the realms of chemistry, biology, weather, and nanoscale matter.

A Tutorial on the Use of Physics-Informed Neural Networks to Compute the Spectrum of Quantum Systems

Quantum many-body systems are of great interest for many research areas, including physics, biology, and chemistry. However, their simulation is extremely challenging, due to the exponential growth of the Hilbert space with system size, making it exceedingly difficult to parameterize the wave functions of large systems by using exact methods. Neural networks and machine learning, in general, are a way to face this challenge. For instance, methods like tensor networks and neural quantum states are being investigated as promising tools to obtain the wave function of a quantum mechanical system. In this tutorial, we focus on a particularly promising class of deep learning algorithms. We explain how to construct a Physics-Informed Neural Network (PINN) able to solve the Schrödinger equation for a given potential, by finding its eigenvalues and eigenfunctions. This technique is unsupervised, and utilizes a novel computational method in a manner that is barely explored. PINNs are a deep learning method that exploit automatic differentiation to solve integro-differential equations in a mesh-free way. We show how to find both the ground and the excited states. The method discovers the states progressively by starting from the ground state. We explain how to introduce inductive biases in the loss to exploit further knowledge of the physical system. Such additional constraints allow for a faster and more accurate convergence. This technique can then be enhanced by a smart choice of collocation points in order to take advantage of the mesh-free nature of the PINN. The methods are made explicit by applying them to the infinite potential well and the particle in a ring, a challenging problem to be learned by an artificial intelligence agent due to the presence of complex-valued eigenfunctions and degenerate states

Different effective Hamiltonians in a periodic-driven Bose-Josephson junction.

The bosonic Josephson junction, one of the maximally simple models for periodic-driven many-body systems, has been intensively studied in the past two decades. Here, we revisit this problem with five different methods, all of which have solid theoretical reasoning. We find that to the order of ω^{-2} (ω is the modulating frequency), these approaches will yield slightly different effective Hamiltonians. In particular, the parameters in the effective Hamiltonians may be unchanged, increased, or decreased, depending on the approximations used. Especially, some of the methods generate new interactions, which still preserve the total number of particles, and the others do not. The validity of these five effective models is verified using dynamics of population imbalance and self-trapping phase transition, and we find thatthe method based on the rotating frame by a unitary transformation has the highest accuracy. The difference between these methods will become significant when the driving amplitude or interaction strengths are comparable with the driving frequency. These differences can also be manifested from their long-time dynamics. We demonstrate this physics using a Bose-Josephson junction, and it is to be hoped that the validity of these methods and their tiny differences put forward in this paper can be verified in realistic experiments in the future using quantum simulating platforms, including but not limited to ultracold atoms.

Feedback-based quantum algorithms for ground state preparation

The ground state properties of quantum many-body systems are a subject of interest across chemistry, materials science, and physics. Thus, algorithms for finding ground states can have broad impacts. Variational quantum algorithms are one class of ground state algorithms that has received significant attention in recent years. These algorithms utilize a hybrid quantum-classical computing framework to prepare ground states on quantum computers. However, this requires solving a classical optimization problem that can become prohibitively expensive in high dimensions. Here, we develop formulations of feedback-based quantum algorithms for ground state preparation that can be used to address this challenge for two broad classes of Hamiltonians: Fermi-Hubbard Hamiltonians, and molecular Hamiltonians represented in second quantization. Feedback-based quantum algorithms are optimization-free; in place of classical optimization, quantum circuit parameters are set according to a deterministic feedback law derived from quantum Lyapunov control principles. This feedback law guarantees a monotonic improvement in solution quality with respect to the depth of the quantum circuit. A variety of numerical illustrations are provided that analyze the convergence and robustness of feedback-based quantum algorithms for these problem classes. Published by the American Physical Society 2024