We examine applicability of the valence bond basis correlator product state ansatz, equivalent to the restricted Boltzmann machine quantum artificial neural network ansatz, and variational Monte Carlo method for direct optimization of excited energy states to study properties of strongly correlated and frustrated quantum systems. The energy eigenstates are found by stochastic minimization of the variational function for the energy eigenstates which allows direct optimization of particular energy state without knowledge of the lower energy states. This approach combined with numerous tensor network or artificial neural network ansatz wavefunctions then allows further insight into quantum phases and phase transitions in various strongly correlated models by considering properties of these systems beyond the ground state properties. Also, the method is in general applicable to any dimension and has no sign instability. An example that we consider is the square lattice J1-J2 antiferromagnetic Heisenberg model. The model is one of the most studied models in frustrated quantum magnetism since it is closely related to the disappearance of the antiferromagnetic order in the high-Tc superconducting materials and there is still no agreement about the properties of the system in the highly frustrated regime near J2/J1 = 0.5. For J1-J2 model we write the variational ansatz in terms of the two site correlators and in the valence bond basis and calculate lowest energy eigenstates in the highly frustrated regime near J2/J1 = 0.5 where the system has a paramagnetic phase. We find that our results are in good agreement with previously obtained results which confirms applicability of the method to study frustrated spin systems.
Emergence of highly degenerate excited states in frustrated magnet MgCr$_2$O$_4$
基态的高简并导致奇异零能模的产生，其中一个代表性的例子是在受抑磁体中形成分子自旋类液体涨落。在这里，我们提出了单晶非弹性中子散射的结果为阻挫磁体MgCr $_2$ O $_4$，这表明，一组共同的有限能量分子自旋激发模式是持续在液体状顺磁相和磁有序相具有极其复杂的结构。基于这一发现，我们提出了激发态高简并度的概念，它促进了局域共振元激发。这一概念有望对我们理解许多复杂系统中的激发产生影响，这些系统不仅包括自旋，而且包括原子液体、复杂有序系统和无定形系统。
The variational cluster approximation is used to study the frustrated Hubbard model at half filling defined on the two-dimensional square lattice with anisotropic next-nearest-neighbor hopping parameters. We calculate the ground-state phase diagrams of the model in a wide parameter space for a variety of lattice geometries, including square, crossed-square, and triangular lattices. We examine the Mott metal-insulator transition and show that, in the Mott insulating phase, magnetic phases with Néel, collinear, and spiral orders appear in relevant parameter regions, and in an intermediate region…
Photo-excited charge dynamics of interacting charge-frustrated systems are studied using a spinless fermion model on an anisotropic triangular lattice. Real-time evolution of the system after irradiating a pump-photon pulse is analyzed by the exact diagonalization method. We focus on photo-excited states in the two canonical charge-ordered (CO) ground states, i.e., horizontal stripe-type and vertical stripe-type COs, which compete with each other owing to the charge frustration. We find that the photo-induced excited states from the two types of COs are distinct. From the horizontal stripe-typ…
Magnetization, specific heat, and electron spin resonance (ESR) measurements are carried out to clarify the low-energy excitations for the S = 1/2 polyhedral clusters {Mo$_{72}$V$_{30}$} and {W$_{72}$V$_{30}$}. The clusters provide unique model systems of Kagomé network on a quasi-sphere. The linear field variation of magnetization at low temperatures indicates that the ground state is singlet for both clusters. The temperature and the magnetic field dependence of specific heat shows a distinct difference between two clusters with differing structural symmetries. In {W$_{72}$V$_{30}$} with I$_…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
Hybrid quantum-classical algorithms have been proposed as a potentially viable application of quantum computers. A particular example - the variational quantum eigensolver, or VQE - is designed to determine a global minimum in an energy landscape specified by a quantum Hamiltonian, which makes it appealing for the needs of quantum chemistry. Experimental realizations have been reported in recent years and theoretical estimates of its efficiency are a subject of intense effort. Here we consider the performance of the VQE technique for a Hubbard-like model describing a one-dimensional chain of f…
The use of artificial neural networks to represent quantum wave-functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parameterizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper we start addressing this aspect, concen…
Gutzwiller-projected fermionic states can be efficiently implemented within quantum Monte Carlo calculations to define extremely accurate variational wave functions for Heisenberg models on frustrated two-dimensional lattices, not only for the ground state but also for low-energy excitations. The application of few Lanczos steps on top of these states further improves their accuracy, allowing calculations on large clusters. In addition, by computing both the energy and its variance, it is possible to obtain reliable estimations of exact results. Here, we report the cases of the frustrated Heis…
We extend our hybrid linear-method/accelerated-descent variational Monte Carlo optimization approach to excited states and investigate its efficacy in double excitations. In addition to showing a superior statistical efficiency when compared to the linear method, our tests on the carbon dimer and cyclopentadiene show good energetic agreement with benchmark methods and experiment, respectively. We also demonstrate the ability to treat double excitations in systems that are too large for a full treatment by selective configuration interaction methods via an application to 4-aminobenzonitrile. Fi…
The spin-1/2 chain with isotropic Heisenberg exchange $J_1$, $J_2 > 0$ between first and second neighbors is frustrated for either sign of J1. Its quantum phase diagram has critical points at fixed $J_1/J_2$ between gapless phases with nondegenerate ground state (GS) and quasi-long-range order (QLRO) and gapped phases with doubly degenerate GS and spin correlation functions of finite range. In finite chains, exact diagonalization (ED) estimates critical points as level crossing of excited states. GS spin correlations enter in the spin structure factor $S(q)$ that diverges at wave vector $q_m$ …
An equation of motion formalism for excited states in variational Monte Carlo is derived and a pilot implementation for the Jastrow-modified antisymmetric geminal power is tested. In single excitations across a range of small molecules, this combination is shown to be intermediate in accuracy between configuration interaction singles and equation of motion coupled cluster with singles and doubles. For double excitations, energy errors are found to be similar to those for coupled cluster.
We employ a generalized variational principle to improve the stability, reliability, and precision of fully excited-state-specific complete active space self-consistent field theory. Compared to previous approaches that similarly seek to tailor this ansatz's orbitals and configuration interaction expansion for an individual excited state, we find the present approach to be more resistant to root flipping and better at achieving tight convergence to an energy stationary point. Unlike state-averaging, this approach allows orbital shapes to be optimal for individual excited states, which is espec…
Quantum spin ice materials, pyrochlore magnets with competing Ising and transverse exchange interactions, have been widely discussed as candidates for a quantum spin-liquid ground state. Here, motivated by quantum chemical calculations for Pr pyrochlores, we present the results of a study for frustrated transverse exchange. Using a combination of variational calculations, exact diagonalisation, numerical linked-cluster and series expansions, we find that the previously-studied U(1) quantum spin liquid, in its pi-flux phase, transforms into a nematic quantum spin liquid at a high-symmetry, SU(2…
We present a generalization of the variational principle that is compatible with any Hamiltonian eigenstate that can be specified uniquely by a list of properties. This variational principle appears to be compatible with a wide range of electronic structure methods, including mean-field theory, density functional theory, multi-reference theory, and quantum Monte Carlo. Like the standard variational principle, this generalized variational principle amounts to the optimization of a nonlinear function that, in the limit of an arbitrarily flexible wave function, has the desired Hamiltonian eigenst…
We present a variational function that targets excited states directly based on their position in the energy spectrum, along with a Monte Carlo method for its evaluation and minimization whose cost scales polynomially for a wide class of approximate wave functions. Being compatible with both real and Fock space and open and periodic boundary conditions, the method has the potential to impact many areas of chemistry, physics, and materials science. Initial tests on doubly excited states show that using this method, the Hilbert space Jastrow antisymmetric geminal power ansatz can deliver order-o…
The study of quantum frustrated systems remains one of the most challenging subjects of quantum magnetism, as they can hold quantum spin liquids, whose characterization is quite elusive. The presence of gapped quantum spin liquids possessing long range entanglement while being locally indistinguishable often demand highly sophisticated numerical approaches for their description. Here we propose an easy computational method based on exact diagonalization with engineered boundary conditions in very small plaquettes. We apply the method to study the quantum phase diagram of diverse antiferromagne…
We present a method for finding individual excited states' energy stationary points in complete active space self-consistent field theory that is compatible with standard optimization methods and highly effective at overcoming difficulties due to root flipping and near-degeneracies. Inspired by both the maximum overlap method and recent progress in excited state variational principles, our approach combines these ideas in order to track individual excited states throughout the orbital optimization process. In a series of tests involving root flipping, near-degeneracies, charge transfers, and d…
We compute concurrence, a measure of bipartite entanglement, of the first excited state of the $1$-D Heisenberg frustrated $J_1$-$J_2$ spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength $α=J_2/J_1\approx0.241$, where a quantum phase transition from spin-fluid phase to dimer phase has been previously reported. We numerically observe this phenomena for spin-chain with $8$ sites to $16$ sites, and the value of $α$ at which the change in entanglement is observed asymptotically tends to a value $α_c\approx0.24116$. We have calculated the finite-…
Neural quantum states (NQS) attract a lot of attention due to their potential to serve as a very expressive variational ansatz for quantum many-body systems. Here we study the main factors governing the applicability of NQS to frustrated magnets by training neural networks to approximate ground states of several moderately-sized Hamiltonians using the corresponding wavefunction structure on a small subset of the Hilbert space basis as training dataset. We notice that generalization quality, i.e. the ability to learn from a limited number of samples and correctly approximate the target state on…
We introduce a hybrid quantum-classical variational algorithm to simulate ground-state phase diagrams of frustrated quantum spin models in the thermodynamic limit. The method is based on a cluster-Gutzwiller ansatz where the wave function of the cluster is provided by a parameterized quantum circuit whose key ingredient is a two-qubit real XY gate allowing to efficiently generate valence-bonds on nearest-neighbor qubits. Additional tunable single-qubit Z- and two-qubit ZZ-rotation gates allow the description of magnetically ordered and paramagnetic phases while restricting the variational opti…
We investigate the issue of optimization stability in variance-based state-specific variational Monte Carlo, discussing the roles of the objective function, the complexity of wave function ansatz, the amount of sampling effort, and the choice of minimization algorithm. Using a small cyanine dye molecule as a test case, we systematically perform minimizations using variants of the linear method as both a standalone algorithm and in a hybrid combination with accelerated descent. We demonstrate that adaptive step control is crucial for maintaining the linear method's stability when optimizing com…
Variational methods have proven to be excellent tools to approximate ground states of complex many body Hamiltonians. Generic tools like neural networks are extremely powerful, but their parameters are not necessarily physically motivated. Thus, an efficient parametrization of the wave-function can become challenging. In this letter we introduce a neural-network based variational ansatz that retains the flexibility of these generic methods while allowing for a tunability with respect to the relevant correlations governing the physics of the system. We illustrate the success of this approach on…
Highly excited states of quantum many-body systems are central objects in the study of quantum dynamics and thermalization that challenge classical computational methods due to their volume-law entanglement content. In this work, we explore the potential of variational quantum algorithms to approximate such states. We propose an adaptive variational algorithm, adaptive VQE-X, that self-generates a variational ansatz for arbitrary eigenstates of a many-body Hamiltonian $H$ by attempting to minimize the energy variance with respect to $H$. We benchmark the method by applying it to an Ising spin …
Excited state optimization for strongly correlated quantum defects using ensemble variational Monte Carlo
Anthony J. Leggett Institute for Condensed Matter Physics, Department of Physics, Grainger College of Engineering, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格雷格工程学院物理系安东尼·J·雷格特凝聚态物理研究所）
Spin ice magnetic monopoles are fractionalized emergent excitations in a class of frustrated magnets called spin ices. The classical spin ice model has an extensive number of ground state spin configurations, whereas magnetic monopoles can be thought of as the endpoints of string operators applied to these ground states. Introducing quantum fluctuations into the model induces monopoles with dynamics, which would normally lift the degeneracy of the two-monopole energy level at the linear order of the perturbation. Contrary to this expectation, we find that quantum fluctuations in the form of a …
We study the ground state phase diagram of a frustrated spin-1/2 four-leg spin tube in an external magnetic field. We explore the parameter space of this model in the regime of all-antiferromagnetic exchange couplings by means of three different approaches: analysis of low-energy effective Hamiltonian (LEH), a Hartree variational approach (HVA) and density matrix renormalization group (DMRG) for finite clusters. We find that in the limit of weakly interacting plaquettes, low-energy singlet, triplet and quintuplet states play an important role in the formation of fractional magnetization platea…
A programmable quantum device that has a large number of qubits without fault-tolerance has emerged recently. Variational Quantum Eigensolver (VQE) is one of the most promising ways to utilize the computational power of such devices to solve problems in condensed matter physics and quantum chemistry. As the size of the current quantum devices is still not large for rivaling classical computers at solving practical problems, Fujii et al. proposed a method called "Deep VQE" which can provide the ground state of a given quantum system with the smaller number of qubits by combining the VQE and the…
The dynamical spin structure factor is computed within a variational framework to study the one-dimensional $J_1-J_2$ Heisenberg model. Starting from Gutzwiller-projected fermionic wave functions, the low-energy spectrum is constructed from two-spinon excitations. The direct comparison with Lanczos calculations on small clusters demonstrates the excellent description of both gapless and gapped (dimerized) phases, also including incommensurate structures for $J_2/J_1>0.5$. Calculations on large clusters show how the intensity evolves when increasing the frustrating ratio and give an unprecedent…
We develop a technique for calculating three-dimensional classical partition functions using projected entangled-pair states (PEPS). Our method is based on variational PEPS optimization algorithms for two-dimensional quantum spin systems, and allows us to compute free energies directly in the thermodynamic limit. The main focus of this work is classical frustration in three-dimensional many-body systems leading to an extensive ground-state degeneracy. We provide high-accuracy results for the residual entropy of the dimer model on the cubic lattice, water-ice $I_h$ and water-ice $I_c$. In addit…
We explore a variational Ansatz for lattice quantum systems -- named long-range entangled-plaquette state -- in which pairs of clusters of adjacent lattice sites are correlated at any distance. The explicit scale-free structure of correlations built in this wave function makes it fit to reproduce critical states with long-range entanglement. The use of complex weights in the Ansatz allows for an efficient optimization of non positive definite states in a fully variational fashion, namely without any additional bias (arising \emph{e.g.} from pre-imposed sign structures) beyond that imposed by t…
In single-reference coupled-cluster (CC) methods, one has to solve a set of non-linear polynomial equations in order to determine the so-called amplitudes which are then used to compute the energy and other properties. Although it is of common practice to converge to the (lowest-energy) ground-state solution, it is also possible, thanks to tailored algorithms, to access higher-energy roots of these equations which may or may not correspond to genuine excited states. Here, we explore the structure of the energy landscape of variational CC (VCC) and we compare it with its (projected) traditional…
