Exact Eigenstates Research Articles

CFTD from TQFTD+1 via Holographic Tensor Network, and Precision Discretization of CFT2

We show that the path integral of conformal field theories in D dimensions (CFTD) can be constructed by solving for eigenstates of a renormalization group (RG) operator following from the Turaev-Viro formulation of a topological field theory (topological quantum field theory) (TQFT) in D+1 dimensions (TQFTD+1), explicitly realizing the holographic sandwich relation between a symmetric theory and a TQFT. Generically, exact eigenstates corresponding to symmetric TQFTD follow from Frobenius algebra in TQFTD+1. For D=2, we construct eigenstates that produce 2D rational CFT path integrals exactly, which curiously connect a continuous-field theoretic path integral with the Turaev-Viro state sum. We also devise and illustrate numerical methods for D=2, 3 to search for CFTD as phase transition points between symmetric TQFTD. Finally, since the RG operator is in fact an exact analytic holographic tensor network, we compute “bulk-boundary” correlators and compare them with the AdS/CFT dictionary at D=2. Promisingly, they are numerically compatible given our accuracy, although further works will be needed to explore the precise connection to the AdS/CFT correspondence. Published by the American Physical Society 2024

Tangent Space Generators of Matrix Product States and Exact Floquet Quantum Scars

The advancement of quantum simulators motivates the development of a theoretical framework to assist with efficient state preparation in quantum many-body systems. Generally, preparing a target entangled state via unitary evolution with time-dependent couplings is a challenging task and very little is known about the existence of solutions and their properties. In this work we develop a constructive approach for preparing matrix product states (MPS) via continuous unitary evolution. We provide an explicit construction of the operator that exactly implements the evolution of a given MPS along a specified direction in its tangent space. This operator can be written as a sum of local terms of finite range, yet it is in general non-Hermitian. Relying on the explicit construction of the non-Hermitian generator of the dynamics, we demonstrate the existence of a Hermitian sequence of operators that implements the desired MPS evolution with an error that decreases exponentially with the operator range. The construction is benchmarked on an explicit periodic trajectory in a translationally invariant MPS manifold. We demonstrate that the Floquet unitary generating the dynamics over one period of the trajectory features an approximate MPS-like eigenstate embedded among a sea of thermalizing eigenstates. These results show that our construction is not only useful for state preparation and control of many-body systems, but also provides a generic route towards Floquet scars—periodically driven models with quasilocal generators of dynamics that have exact MPS eigenstates in their spectrum. Published by the American Physical Society 2024

Electronic correlations in parallel-coupled double quantum dot system: An exact analytical approach

Exact eigenstates of the parallel coupled double quantum dots attached to the non-interacting leads taken in the zero-bandwidth limit (effectively quadruple quantum dots) are analytically obtained in each particle and spin subspace. The ground state of the half-filled system is determined within a four-dimensional subspace of the twenty-dimensional Hilbert space. The effects of tunable parameters, such as quantum dot energy levels, interdot tunneling, ondot and interdot Coulomb interactions, on spin-spin correlation and dot occupancies are analyzed. In the parameter space defined by interdot tunneling and ondot Coulomb interaction, the tunnel-coupled dots exhibit both ferromagnetic and antiferromagnetic correlations, suggesting suitability for singlet-triplet qubits in quantum computing. A critical dependency of interdot tunneling on ondot Coulomb interaction leads to a transition from ferromagnetic to antiferromagnetic correlation as interdot tunneling increases. These correlations persist even without interdot tunneling via indirect exchange through the leads, with the interdot Coulomb interaction significantly influencing this dependency.

Estimating Bethe roots with VQE

Abstract Bethe equations, whose solutions determine exact eigenvalues and eigenstates of corresponding integrable Hamiltonians, are generally hard to solve. We implement a Variational Quantum Eigensolver approach to estimating Bethe roots of the spin-1/2 XXZ quantum spin chain, by using Bethe states as trial states, and treating Bethe roots as variational parameters. In numerical simulations of systems of size up to 6, we obtain estimates for Bethe roots corresponding to both ground states and excited states with up to 5 down-spins, for both the closed and open XXZ chains. This approach is not limited to real Bethe roots.

Approximate Functionals for Multistate Density Functional Theory.

Recently Lu and Gao [J. Phys. Chem. Lett. 2022, 13 (33), 7762-7769] published a new, rigorous density functional theory for excited states and proved that the projection of the kinetic and electron-repulsion operators into the subspace of the lowest electronic states are a universal functional of the matrix density D(r). This is the first attempt to find an approximation to the multistate universal functional . It is shown that (i) does not explicitly depend on the number of electronic states and (ii) is an analytic matrix functional. The Thomas-Fermi-Dirac-von Weizsäcker model and the correlation energy of the homogeneous electron gas are turned into matrix functionals guided by two principles: that each matrix functional should transform properly under basis set transformations and that the ground state functional should be recovered for a single electronic state. Lieb-Oxford-like bounds on the average kinetic and electron-repulsion energies in the subspace are given. When evaluated on the numerically exact matrix density of LiF, this simple approximation reproduces the matrix elements of the electron-repulsion operator in the basis of the exact eigenstates accurately for all bond lengths. In particular the off-diagonal elements of the effective Hamiltonian that come from the interactions of different electronic states can be calculated with the same or better accuracy than the diagonal elements. Unsurprisingly, the largest error comes from the kinetic energy functional. More exact conditions that constrain the functional form of are needed to go beyond the local density approximation.

Analytical Solutions of Symmetric Isotropic Spin Clusters Using Spin and Point Group Projectors

Spin models like the Heisenberg Hamiltonian effectively describe the interactions of open-shell transition-metal ions on a lattice and can account for various properties of magnetic solids and molecules. Numerical methods are usually required to find exact or approximate eigenstates, but for small clusters with spatial symmetry, analytical solutions exist, and a few Heisenberg systems have been solved in closed form. This paper presents a simple, generally applicable approach to analytically solve isotropic spin clusters, based on adapting the basis to both total spin and point group symmetry to factor the Hamiltonian matrix into sufficiently small blocks. We demonstrate applications to small rings and polyhedra, some of which are straightforward to solve by successive spin-coupling for Heisenberg terms only; additional interactions, such as biquadratic exchange or multi-center terms necessitate symmetry adaptation.

Quantum Electrodynamics in 2+1 Dimensions as the Organizing Principle of a Triangular Lattice Antiferromagnet

Quantum electrodynamics in 2+1 dimensions (QED3) has been proposed as a critical field theory describing the low-energy effective theory of a putative algebraic Dirac spin liquid or of quantum phase transitions in two-dimensional frustrated magnets. We provide compelling evidence that the intricate spectrum of excitations of the elementary but strongly frustrated J1−J2 Heisenberg model on the triangular lattice is in one-to-one correspondence to a zoo of excitations from QED3, in the quantum spin liquid regime. This evidence includes a large manifold of explicitly constructed monopole and bilinear excitations of QED3, which is thus shown to serve as an organizing principle of phases of matter in triangular lattice antiferromagnets and their low-lying excitations. Moreover, we observe signatures of emergent valence-bond solid (VBS) correlations, which can be interpreted either as evidence of critical VBS fluctuations of an emergent Dirac spin liquid or as a transition from the 120° Néel order to a VBS whose quantum critical point is described by QED3. Our results are obtained by comparing ansatz wave functions from a parton construction to exact eigenstates obtained using large-scale exact diagonalization up to N=48 sites. Published by the American Physical Society 2024

Ensemble variational Monte Carlo for optimization of correlated excited state wave functions

Variational Monte Carlo methods have recently been applied to the calculation of excited states; however, it is still an open question what objective function is most effective. A promising approach is to optimize excited states using a penalty to minimize overlap with lower eigenstates, which has the drawback that states must be computed one at a time. We derive a general framework for constructing objective functions with minima at the the lowest N eigenstates of a many-body Hamiltonian. The objective function uses a weighted average of the energies and an overlap penalty, which must satisfy several conditions. We show this objective function has a minimum at the exact eigenstates for a finite penalty, and provide a few strategies to minimize the objective function. The method is demonstrated using ab initio variational Monte Carlo to calculate the degenerate first excited state of a CO molecule.

Thermal states of two identical two-level atoms interacting with the single-mode fields of the Arik–Coon and Biedenharn–Macfarlane q-oscillators

We consider the Jaynes–Cummings models (JCMs) of two identical two-level atoms interacting with a single mode of the Arik–Coon and Biedenharn–Macfarlane [Formula: see text]-boson fields in a cavity. It is shown that there is a constant of motion, so-called the total number of excitation, which leads to decomposition of the total Hilbert space of [Formula: see text]-deformed system into three invariant subspaces. So, we have the block diagonal structure with three blocks as the Hamiltonian matrices. The exact energy spectrum and eigenstates of the [Formula: see text]-deformed models are obtained by diagonalizing each block. Assuming that the cavity fields and atoms are in thermal equilibrium, the partition function and thermal density matrix of the models are calculated. Finally, the influence of deformation parameters of the Arik–Coon and Biedenharn–Macfarlane [Formula: see text]-oscillator light fields at low-temperatures on the entropy and internal energy of the Jaynes–Cummings models as well as on the atom–atom entanglement via the concurrence measure is analyzed.

Integrability and quench dynamics in the spin-1 central spin XX model

Central spin models provide an idealized description of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. We show that the family of models with a spin-1 at the center and XX interactions of arbitrary strength with surrounding spins is integrable. Specifically, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz. As in the homogenous limit, the states divide into two exponentially large classes: bright states, in which the spin-1 is entangled with its surroundings, and dark states, in which it is not. On resonance, the bright states further break up into two classes depending on their weight on states with central spin polarization zero. These classes are probed in quench dynamics wherein they prevent the central spin from reaching thermal equilibrium. In the single spin-flip sector we explicitly construct the bright states and show that the central spin exhibits oscillatory dynamics as a consequence of the semilocalization of these eigenstates. We relate the integrability to the closely related class of integrable Richardson-Gaudin models, and conjecture that the spin-s central spin XX model is integrable for any s.

Even Shorter Quantum Circuit for Phase Estimation on Early Fault-Tolerant Quantum Computers with Applications to Ground-State Energy Estimation

We develop a phase estimation method with a distinct feature: its maximal runtime (which determines the circuit depth) is $\delta/\epsilon$, where $\epsilon$ is the target precision, and the preconstant $\delta$ can be arbitrarily close to $0$ as the initial state approaches the target eigenstate. The total cost of the algorithm satisfies the Heisenberg-limited scaling $\widetilde{\mathcal{O}}(\epsilon^{-1})$. This is different from all previous proposals, where $\delta \gtrsim \pi$ is required even if the initial state is an exact eigenstate. As a result, our algorithm may significantly reduce the circuit depth for performing phase estimation tasks on early fault-tolerant quantum computers. The key technique is a simple subroutine called quantum complex exponential least squares (QCELS). Our algorithm can be readily applied to reduce the circuit depth for estimating the ground-state energy of a quantum Hamiltonian, when the overlap between the initial state and the ground state is large. If this initial overlap is small, we can combine our method with the Fourier filtering method developed in [Lin, Tong, PRX Quantum 3, 010318, 2022], and the resulting algorithm provably reduces the circuit depth in the presence of a large relative overlap compared to $\epsilon$. The relative overlap condition is similar to a spectral gap assumption, but it is aware of the information in the initial state and is therefore applicable to certain Hamiltonians with small spectral gaps. We observe that the circuit depth can be reduced by around two orders of magnitude in numerical experiments under various settings.

Spectral features of polaronic excitations in a superconducting analog simulator

We investigate spectral properties of polaronic excitations within the framework of an analog quantum simulator based on inductively coupled superconducting transmon qubits and microwave resonators. This system emulates a lattice model that describes a nonlocal coupling of an itinerant spinless-fermion excitation to dispersionless (Einstein-type) phonons through the Peierls and breathing-mode interaction mechanisms. The model is characterized by a sharp, level-crossing transition at a critical value of the effective excitation-phonon coupling strength; above the transition point, the ground state of this model corresponds to a heavily dressed (small-polaron) excitation. Using the kernel-polynomial method, we evaluate the momentum-frequency resolved spectral function of this system for a broad range of parameters. In particular, we underscore the ramifications of the fact that the zero-quasimomentum Bloch state of a bare excitation represents the exact eigenstate of the Hamiltonian of this system for an arbitrary excitation-phonon coupling strength. We also show that -- based on the numerically evaluated spectral function and its well-known relation with the survival probability of the initial, bare-excitation Bloch state (the Loschmidt echo) -- one can make predictions about the system dynamics following an excitation-phonon interaction quench. To make contact with anticipated experimental realizations, we utilize a previously proposed method for extracting dynamical-response functions in systems with local (single-qubit) addressability using the multiqubit (many-body) version of the Ramsey interference protocol.

The frustration-free fully packed loop model

We consider a quantum fully packed loop model on the square lattice with a frustration-free projector Hamiltonian and ring-exchange interactions acting on plaquettes. A boundary Hamiltonian is added to favor domain-wall boundary conditions and link ground state properties to the combinatorics and six-vertex model literature. We discuss how the boundary term fractures the Hilbert space into Krylov subspaces, and we prove that the Hamiltonian is ergodic within each subspace, leading to a series of energy-equidistant exact eigenstates in the lower end of the spectrum. Among them we systematically classify both finitely entangled eigenstates and product eigenstates. Using a recursion relation for enumerating half-plane configurations, we compute numerically the exact entanglement entropy of the ground state, confirming area law scaling. Finally, the spectrum is shown to be gapless in the thermodynamic limit with a trial state constructed by adding a twist to the ground state superposition.

Relaxation of Phonons in the Lieb-Liniger Gas by Dynamical Refermionization.

Motivated by recent experiments, we investigate the Lieb-Liniger gas initially prepared in an out-of-equilibrium state that is Gaussian in terms of the phonons, namely whose density matrix is the exponential of an operator quadratic in terms of phonon creation and annihilation operators. Because the phonons are not exact eigenstates of the Hamiltonian, the gas relaxes to a stationary state at very long times whose phonon population is apriori different from the initial one. Thanks to integrability, that stationary state needs not be a thermal state. Using the Bethe-ansatz mapping between the exact eigenstates of the Lieb-Liniger Hamiltonian and those of a noninteracting Fermi gas and bosonization techniques we completely characterize the stationary state of the gas after relaxation and compute its phonon population distribution. We apply our results to the case where the initial state is an excited coherent state for a single phonon mode, and we compare them to exact results obtained in the hard-core limit.

Excited states of zero seniority based on a pair condensate

We study the excited states of zero seniority for various like-particle systems interacting by pairing forces and by general two-body interactions. We consider two types of excitations, generated from a ground state described by a pair condensate. One type is obtained by breaking a pair from the ground state condensate and replacing it by "excited" collective pairs built on time-reversed single-particle orbits. The second type of zero seniority excited states is described by a condensate of identical excited pairs. The structure of these excited states is analysed for the picked fence model and for the valence neutrons of $^{108}$Sn. For a state-depending pairing interaction, the first type of excited states agree well with the J=0 states which are known in $^{108}$Sn. At the same time, these states can be also associated unambiguously with those exact states which are the closest in energy to the experimental levels. The states corresponding to the excited pair condensate appear at low energies, around the energy of the second excited state of the first type, and they do not have a simple correspondence with exact eigenstates. However, at a much higher excitation energy there is an exact state which is similar in structure to an EPC state. It is shown that this EPC state has the features of a giant pairing vibration.

Confined states in the tight-binding model on the hexagonal golden-mean tiling

We study the tight-binding model with two distinct hoppings (tL, tS ) on the two-dimensional hexagonal golden-mean tiling and examine the confined states with E = 0, where E is the eigenenergy. Some confined states found in the case tL = tS are exact eigenstates even for the system with tL ≠ t S, where their amplitudes are smoothly changed. By contrast, the other states are no longer eigenstates of the system with tL ≠ t S. This may imply the existence of macroscopically degenerate states which are characteristic of the system with tL = t S, and that a discontinuity appears in the number of the confined states in the thermodynamic limit.

Finiteness of entanglement entropy in collective field theory

We explore the question of finiteness of the entanglement entropy in gravitational theories whose emergent space is the target space of a holographic dual. In the well studied duality of two-dimensional non-critical string theory and c = 1 matrix model, this question has been studied earlier using fermionic many-body theory in the space of eigenvalues. The entanglement entropy of a subregion of the eigenvalue space, which is the target space entanglement in the matrix model, is finite, with the scale being provided by the local Fermi momentum. The Fermi momentum is, however, a position dependent string coupling, as is clear in the collective field theory formulation. This suggests that the finiteness is a non-perturbative effect. We provide evidence for this expectation by an explicit calculation in the collective field theory of matrix quantum mechanics with vanishing potential. The leading term in the cumulant expansion of the entanglement entropy is calculated using exact eigenstates and eigenvalues of the collective Hamiltonian, yielding a finite result, in precise agreement with the fermion answer. Treating the theory perturbatively, we show that each term in the perturbation expansion is UV divergent. However the series can be resummed, yielding the exact finite result. Our results indicate that the finiteness of the entanglement entropy for higher dimensional string theories is non-perturbative as well, with the scale provided by Newton’s constant.

Escaping many-body localization in an exact eigenstate

Isolated quantum systems typically follow the eigenstate thermalization hypothesis, but there are exceptions, such as many-body localized (MBL) systems and quantum many-body scars. Here, we present the study of a weak violation of MBL due to a special state embedded in a spectrum of MBL states. The special state is not MBL since it displays logarithmic scaling of the entanglement entropy and of the bipartite fluctuations of particle number with subsystem size. In contrast, the bulk of the spectrum becomes MBL as disorder is introduced. We establish this by studying the entropy as a function of disorder strength and by observing that the level spacing statistics undergoes a transition from Wigner-Dyson to Poisson statistics as the disorder strength is increased.

Minimal Active Space: NOSCF and NOSI in Multistate Density Functional Theory.

In this Perspective, we introduce a minimal active space (MAS) for the lowest N eigenstates of a molecular system in the framework of multistate density functional theory (MSDFT), consisting of no more than N2 nonorthgonal Slater determinants. In comparison with some methods in wave function theory in which one seeks to expand the ever increasing size of an active space to approximate the wave functions, it is possible to have an upper bound in MSDFT because the auxiliary states in a MAS are used to represent the exact N-dimensional matrix density function D(r). Here, we partition the total Hamiltonian matrix functional into an orbital-dependent part, including multistate kinetic energy Tms and Coulomb-exchange energy EHx plus an external potential energy ∫dr v(r)D(r), and a correlation matrix density functional . The latter accounts for the part of correlation energy not explicitly included in the minimal active space. A major difference from Kohn-Sham DFT is that state interactions are necessary to represent the N-matrix density D(r) in MSDFT, rather than a noninteracting reference state for the scalar ground-state density ρo(r). Two computational approaches are highlighted. We first derive a set of nonorthogonal multistate self-consistent-field (NOSCF) equations for the variational optimization of . We introduce the multistate correlation potential, as the functional derivative of , which includes both correlation effects within the MAS and that from the correlation matrix functional. Alternatively, we describe a nonorthogonal state interaction (NOSI) procedure, in which the determinant functions are optimized separately. Both computational methods are useful for determining the exact eigenstate energies and for constructing variational diabatic states, provided that the universal correlation matrix functional is known. It is hoped that this discussion would stimulate developments of approximate multistate density functionals both for the ground and excited states.

Diffraction-free beam propagation at the exceptional point of non-Hermitian Glauber Fock lattices

We construct localized beams in a non-Hermitian Glauber Fock (NGF) lattice of coupled waveguides and show that they can propagate over a long distance withalmost no diffraction. We specifically obtain the diffraction-free beams in a finite NGF lattice at the exceptional point (EP) by using the exact eigenstates of the semi-infinite unidirectional NGF lattice. We provide a numerical approach to finding other lattices that are capable of supporting non-diffracting beams at EPs.