Lattice Quantum Research Articles

From short-range to mean-field models in quantum lattices

From short-range to mean-field models in quantum lattices

On the notion of strong correlation in electronic structure theory.

Strong correlation has been said to have many faces, and appears to have many synonyms of questionable suitability. In this work we aim not to define the term once and for all, but to highlight one possibility that is both rigorously defined and physically transparent, and remains so in reference to molecules and quantum lattice models. We survey both molecular examples - hydrogen systems (Hn, n = 2, 4, 6), Be2, H-He-H, and benzene - and the half-filled Hubbard model over a range of correlation regimes. Various quantities are examined including the extent of spin symmetry breaking in correlated single-reference wave functions, energetic ratios inspired by the Hubbard model and the Virial theorem, and metrics derived from the one- and two-electron reduced density matrices (RDMs). The trace and the square norm of the cumulant of the two-electron reduced density matrix capture what may well be defined as strong correlation. Accordingly, strong correlation is understood as a statistical dependence between two electrons, and is distinct from the concepts of "correlation energy" and more general than entanglement quantities that require a partitioning of a quantum system into distinguishable subspaces. This work enables us to build a bridge between a rigorous and quantifiable regime of strong electron correlation and more familiar chemical concepts such as anti-aromaticity in the context of Baird's rule.

Coherent Spatial Control of Wave Packet Dynamics on Quantum Lattices.

Quantum lattices are pivotal in the burgeoning fields of quantum materials and information science. Novel experimental techniques allow the preparation and monitoring of wave packet dynamics on quantum lattices with high spatiotemporal resolution. We present an analytical study of wave packet diffusivity and diffusion length on tight-binding quantum lattices subject to stochastic noise. Our analysis reveals the crucial role of spatial coherence and predicts a set of novel phenomena: (1) noise can enhance the transient diffusivity and diffusion length of spatially extended initial states; (2) standing or traveling initial states, with large momentum, spread faster than a localized initial state and exhibit a noise-induced peak in the transient diffusivity; (3) the differences in the diffusivity or diffusion length of extended and localized initial states have a universal dependence on initial width. These predictions suggest the possibility of controlling the wave packet dynamics by spatial manipulations, which will have implications for materials science and quantum technologies.

Quantum lattice models that preserve continuous translation symmetry

Quantum lattice models that preserve continuous translation symmetry

Correction: Large Deviations in Quantum Lattice Systems: One-Phase Region

Correction: Large Deviations in Quantum Lattice Systems: One-Phase Region

Correspondence between the Hamiltonian cycle problem and the quantum lattice gauge theory

We propose the correspondence between the Hamiltonian cycle (HC) problem in graph theory and the quantum lattice gauge theory (QZ2LGT) defined on the lattice dual to that graph. For the QZ2LGT, when the coupling parameter g is less than the critical value g c , the ground state is a superposition of all configurations with closed strings of same spins, which can be obtained by using an adiabatic quantum algorithm. A subsequent search for a HC among those closed strings solves the original HC problem. The method is demonstrated for random samples of small graphs.

Almost Everywhere Ergodicity in Quantum Lattice Models

We rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the C^* algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on C^* algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.

Quantum lattice dynamics and their importance in ternary superhydride clathrates

The quantum nature of the hydrogen lattice in superconducting hydrides can have crucial effects on the material’s properties. Taking a detailed look at the dynamic stability of the recently predicted BaSiH8 phase, we find that the inclusion of anharmonic quantum ionic effects leads to an increase in the critical dynamical pressure to 20 GPa as compared to 5 GPa within the harmonic approximation. We identify the change in the crystal structure due to quantum ionic effects to be the main driving force for this increase and demonstrate that this can already be understood at the harmonic level by considering zero-point energy corrections to the total electronic energy. In fact, the previously determined critical pressure of kinetic stability pkin = 30 GPa still poses a stricter bound for the synthesizability of BaSiH8 and similar hydride materials than the dynamical stability and therefore constitutes a more rigorous and accurate estimate for the experimental realizability of these structures.

Projective cluster-additive transformation for quantum lattice models

We construct a projection-based cluster-additive transformation that block-diagonalizes wide classes of lattice Hamiltonians \mathcal{H}=\mathcal{H}_0 +Vℋ=ℋ0+V. Its cluster additivity is an essential ingredient to set up perturbative or non-perturbative linked-cluster expansions for degenerate excitation subspaces of \mathcal{H}_0ℋ0. Our transformation generalizes the minimal transformation known amongst others under the names Takahashi’s transformation, Schrieffer-Wolff transformation, des Cloiseaux effective Hamiltonian, canonical van Vleck effective Hamiltonian or two-block orthogonalization method. The effective cluster-additive Hamiltonian and the transformation for a given subspace of \mathcal{H}ℋ, that is adiabatically connected to the eigenspace of \mathcal{H}_0ℋ0 with eigenvalue e_0^ne0n, solely depends on the eigenspaces of \mathcal{H}ℋ connected to e_0^me0m with e_0^m\leq e_0^ne0m≤e0n. In contrast, other cluster-additive transformations like the multi-block orthogonalization method or perturbative continuous unitary transformations need a larger basis. This can be exploited to implement the transformation efficiently both perturbatively and non-perturbatively. As a benchmark, we perform perturbative and non-perturbative linked-cluster expansions in the low-field ordered phase of the transverse-field Ising model on the square lattice for single spin-flips and two spin-flip bound-states.

Structural, optical and temperature dependent electric modulus property of few layer MoS2 nanosheets

Few layered molybdenum di-sulphide (MoS2) nanosheets were synthesised using low temperature hydrothermal method. Hexagonal structure of MoS2 nanosheets was confirmed by the X-ray diffraction (XRD) and Raman spectroscopy techniques. High resolution transmission electron microscopy (HRTEM) studies revealed the formation of 5–6 layered nanosheets with thickness of 3–4 nm. Crystalline size and micro strain in MoS2 were calculated by various model including Williamson-Hall plot, Size-strain plot, Halder-Wagner and Scherrer formula. UV–Vis studies reveal sharp absorption cut off at 216 nm and large band gap of 2.67 eV caused by quantum confinement effect and lattice strain induced shift of orbital overlapping. Difference between two Raman peaks of 24 cm−1 confirmed the formation of 5–6 layers. Very high dielectric constant of 2600 was obtained at room temperature. Temperature dependent complex electric-modulus and dipolar relaxation of MoS2 nanosheets was investigated to understand the relaxation process and transport of charge carriers for the first time.

Dissipative quantum many-body dynamics in (1+1)D quantum cellular automata and quantum neural networks

Classical artificial neural networks, built from elementary units, possess enormous expressive power. Here we investigate a quantum neural network (QNN) architecture, which follows a similar paradigm. It is structurally equivalent to so-called (1+1)D quantum cellular automata, which are two-dimensional quantum lattice systems on which dynamics takes place in discrete time. Information transfer between consecutive time slices—or adjacent network layers—is governed by local quantum gates, which can be regarded as the quantum counterpart of the classical elementary units. Along the time-direction an effective dissipative evolution emerges on the level of the reduced state, and the nature of this dynamics is dictated by the structure of the elementary gates. We show how to construct the local unitary gates to yield a desired many-body dynamics, which in certain parameter regimes is governed by a Lindblad master equation. We study this for small system sizes through numerical simulations and demonstrate how collective effects within the quantum cellular automaton can be controlled parametrically. Our study constitutes a step towards the utilization of large-scale emergent phenomena in large QNNs for machine learning purposes.

Highly Resolved Spectral Functions of Two-Dimensional Systems with Neural Quantum States.

Spectral functions are central to link experimental probes to theoretical models in condensed matter physics. However, performing exact numerical calculations for interacting quantum matter has remained a key challenge especially beyond one spatial dimension. In this work, we develop a versatile approach using neural quantum states to obtain spectral properties based on simulations of the dynamics of excitations initially localized in real or momentum space. We apply this approach to compute the dynamical structure factor in the vicinity of quantum critical points (QCPs) of different two-dimensional quantum Ising models, including one that describes the complex density wave orders of Rydberg atom arrays. When combined with deep network architectures we find that our method reliably describes dynamical structure factors of arrays with up to 24×24 spins, including the diverging timescales at critical points. Our approach is broadly applicable to interacting quantum lattice models in two dimensions and consequently opens up a route to compute spectral properties of correlated quantum matter in yet inaccessible regimes.

Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners

Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners

Cubic ferromagnet and emergent U(1) symmetry on its phase boundary

We study the simplest quantum lattice spin model for the two-dimensional (2D) cubic ferromagnet by means of mean-field analysis and tensor network calculation. While both methods give rise to similar results in detecting related phases, the 2D infinite projected entangled-pair state (iPEPS) calculation provides more accurate values of transition points. Near the phase boundary, moreover, our iPEPS results indicate that it is more difficult to pin down the orientation of magnetic easy axes, and we interpret it as the easy-axis softening. This phenomenon implies an emergence of continuous $U(1)$ symmetry, which is indicated by the low-energy effective model and has been analytically shown by the field theory. Our model and study provide a concrete example for utilizing iPEPS near the critical region, showing that the emergent phenomenon living on the critical points can already be captured by iPEPS with a rather small bond dimension.

Long-Time Dynamics in Quantum Spin Lattices: Ergodicity and Hydrodynamic Projections at All Frequencies and Wavelengths

AbstractObtaining rigorous and general results about the non-equilibrium dynamics of extended many-body systems is a difficult task. In quantum lattice models with short-range interactions, the Lieb–Robinson bound tells us that the spatial extent of operators grows at most linearly in time. But what happens within this light-cone? We discuss rigorous results on ergodicity and the emergence of the Euler hydrodynamic scale in correlation functions, which establish fundamental principles at the root of non-equilibrium physics. One key idea of the present work is that general structures of Euler hydrodynamics, obtained under ballistic scaling, follow independently from the details of the microscopic dynamics, and in particular do not necessitate chaos; they are consequences of “extensivity”. Another crucial observation is that these apply at arbitrary frequencies and wavelengths. That is, long-time, persistent oscillations of correlation functions over ballistic regions of spacetime, which may be of microscopic frequencies and wavelengths, are predicted by a general Euler-hydrodynamic theory that takes the same form as that for smoothed-out correlation functions. This involves a natural extension of notions of conserved quantities and hydrodynamic projection and shows that the Euler hydrodynamic paradigm covers the full frequency-wavelength plane.

Efficient calculation of three-dimensional tensor networks

We have proposed an efficient algorithm to calculate physical quantities in the translational invariant three-dimensional tensor networks, which is particularly relevant to the study of the three-dimensional classical statistical models and the ($2+1$)-dimensional quantum lattice models. In the context of a classical model, we determine the partition function by solving the dominant eigenvalue problem of the transfer matrix, whose left and right dominant eigenvectors are represented by two projected entangled simplex states. These two projected entangled simplex states are not Hermitian conjugate to each other but are appropriately arranged so that their inner product can be computed much more efficiently than in the usual prescription. For the three-dimensional Ising model, the calculated internal energy and spontaneous magnetization agree with the published results in the literature. The possible improvement and extension to other models are also discussed.

Preparation for quantum simulation of the (1+1) -dimensional O(3) nonlinear σ model using cold atoms

The 1+1D O(3) non-linear {\sigma}-model is a model system for future quantum lattice simulations of other asymptotically-free theories, such as non-Abelian gauge theories. We find that utilizing dimensional reduction can make efficient use of two-dimensional layouts presently available on cold atom quantum simulators. A new definition of the renormalized coupling is introduced, which is applicable to systems with open boundary conditions and can be measured using analog quantum simulators. Monte Carlo and tensor network calculations are performed to determine the quantum resources required to reproduce perturbative short-distance observables. In particular, we show that a rectangular array of 48 Rydberg atoms with existing quantum hardware capabilities should be able to adiabatically prepare low-energy states of the perturbatively-matched theory. These states can then be used to simulate non-perturbative observables in the continuum limit that lie beyond the reach of classical computers.

Exact solution of the q-deformed D3(1) vertex model with open boundaries

In this paper, we study the exact solution of the $q$-deformed $D^{(1)}_3$ quantum lattice model with non-diagonal open boundary condition. We demonstrate the crossing symmetry of the transfer matrix and obtain the quantum determinant. We construct the independent transfer matrix fusion identities and show that the fusion processes can be closed. Based on the fusion hierarchies and polynomial analysis, we obtain the inhomogeneous $T-Q$ relations, exact energy spectrum and Bethe ansatz equations of the system.

Gauge-invariant and anyonic-symmetric autoregressive neural network for quantum lattice models

Symmetries such as gauge invariance and anyonic symmetry play a crucial role in quantum many-body physics. We develop a general approach to constructing gauge-invariant or anyonic-symmetric autoregressive neural networks, including a wide range of architectures such as transformer and recurrent neural network, for quantum lattice models. These networks can be efficiently sampled and explicitly obey gauge symmetries or anyonic constraint. We prove that our methods can provide exact representation for the ground and excited states of the two- and three-dimensional toric codes, and the X-cube fracton model. We variationally optimize our symmetry-incorporated autoregressive neural networks for ground states as well as real-time dynamics for a variety of models. We simulate the dynamics and the ground states of the quantum link model of U(1) lattice gauge theory, obtain the phase diagram for the two-dimensional Z2 gauge theory, determine the phase transition and the central charge of the SU(2)3 anyonic chain, and also compute the ground-state energy of the SU(2) invariant Heisenberg spin chain. Our approach provides powerful tools for exploring condensed-matter physics, high-energy physics, and quantum information science.26 MoreReceived 3 December 2021Revised 26 May 2022Accepted 15 December 2022DOI:https://doi.org/10.1103/PhysRevResearch.5.013216Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasGauge theoriesMachine learningTopological phases of matterTechniquesGauge symmetriesLattice models in condensed matterVariational approachQuantum InformationCondensed Matter, Materials & Applied PhysicsParticles & Fields

Ternary Unitary Quantum Lattice Models and Circuits in 2+1 Dimensions.

We extend the concept of dual unitary quantum gates introduced in Phys. Rev. Lett. 123, 210601 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.210601 to quantum lattice models in 2+1 dimensions, by introducing and studying ternary unitary four-particle gates, which are unitary in time and both spatial dimensions. When used as building blocks of lattice models with periodic boundary conditions in time and space (corresponding to infinite temperature states), dynamical correlation functions exhibit a light ray structure. We also generalize solvable matrix product states introduced in Phys. Rev. B 101, 094304 (2020)PRBMDO2469-995010.1103/PhysRevB.101.094304 to two spatial dimensions with cylindrical boundary conditions, by showing that the analogous solvable projected entangled pair states can be identified with matrix product unitaries. In the resulting tensor network for evaluating equal-time correlation functions, the bulk ternary unitary gates cancel out. We delineate and implement a numerical algorithm for computing such correlations by contracting the remaining tensors.