Ground State Wave Function Research Articles

Natural orbitals renormalization group approach to a spin- 12 impurity interacting with two helical liquids

Using the natural orbitals renormalization group, we studied the problem of a localized spin- 1/2 impurity coupled to two helical liquids via the Kondo interaction in a quantum spin Hall insulator, based on the Kane-Mele model defined in a finite zigzag graphene nanoribbon. We investigated the influence of the Kondo couplings with the helical liquids on both the static and dynamic properties of the ground state. The number and distinct spatial structures of the active natural orbitals (ANOs), which play essential roles in constructing the ground-state wave function, were first analyzed. Our numerical results indicate that two ANOs emerge, equal to the number of helical liquids. Specifically, at the coupling symmetry point, both ANOs are fully active with their spatial structures being respectively constituted by the different helical liquids. In comparison, when deviating from the symmetry point, only one ANO remains fully active, which is dominantly constructed by the helical liquid with the larger Kondo coupling. Local screening of the impurity, described by the impurity spin polarization and susceptibility, was further studied. It shows that at the coupling symmetry point, the impurity is maximally polarized and the spin susceptibility reaches the maximum. On the contrary, the impurity tends to be screened without polarization when the Kondo couplings deviate well from the symmetry point. The Kondo screening cloud, manifested by the spin correlation between the impurity and the conduction electrons, was finally explored. It is demonstrated that the Kondo cloud is mainly formed by the helical liquid with the larger Kondo coupling to the impurity. On the other hand, the spin-orbital coupling breaks the symmetry in spatial distribution of the spin correlation, leading to anisotropy in the Kondo cloud.

Quantum entanglement and the thermal hadron

This paper tests how effectively the bound states of strongly-interacting gauge theories are amenable to an emergent description as a thermal ensemble. This description can be derived from a conjectured minimum free-energy principle, with the entanglement entropy of two-parton subsystems playing the role of thermodynamic entropy. This allows us to calculate the ground-state hadron spectrum and wave function over a wide range of parton masses without solving the Schr\"odinger equation. We carry out this analysis for certain illustrative models in ($1+1$) dimensions and discuss prospects for higher dimensions.

Solving Schrodinger equations using a physically constrained neural network* *Supported by the National Natural Science Foundation of China (12035006, 12075098), the Natural Science Foundation of Hubei Province (2019CFB563), the Hubei Province Department of Education (D20201108), Hubei Province Department of Science and Technology (2021BLB171)

Deep neural networks (DNNs) and auto differentiation have been widely used in computational physics to solve variational problems. When a DNN is used to represent the wave function and solve quantum many-body problems using variational optimization, various physical constraints have to be injected into the neural network by construction to increase the data and learning efficiency. We build the unitary constraint to the variational wave function using a monotonic neural network to represent the cumulative distribution function (CDF) . Using this constrained neural network to represent the variational wave function, we solve Schrodinger equations using auto-differentiation and stochastic gradient descent (SGD) by minimizing the violation of the trial wave function to the Schrodinger equation. For several classical problems in quantum mechanics, we obtain their ground state wave function and energy with very low errors. The method developed in the present paper may pave a new way for solving nuclear many-body problems in the future.

Strongly Bound Dibaryon with Maximal Beauty Flavor from Lattice QCD.

We report the first lattice QCD study of the heavy dibaryons in which all six quarks have the bottom (beauty) flavor. Performing a state-of-the-art lattice QCD calculation we find clear evidence for a deeply bound Ω_{bbb}-Ω_{bbb} dibaryon in the ^{1}S_{0} channel, as a pole singularity in the S-wave Ω_{bbb}-Ω_{bbb} scattering amplitude with a binding energy -81( _{-16}^{+14}) MeV. With such a deep binding, Coulomb repulsion serves only as a perturbation on the ground state wave function of the parametrized strong potential and may shift the strong binding only by a few percent. Considering the scalar channel to be the most bound for single flavored dibaryons, we conclude this state is the heaviest possible most deeply bound dibaryon in the visible universe.

Controlling Anderson localization of a Bose-Einstein condensate via spin-orbit coupling and Rabi fields in bichromatic lattices

We perform theoretical studies of the interplay between disorder, spin-orbit coupling (SOC), and Rabi fields, and show that both SOC and Rabi fields can be used to dramatically control the degree of Anderson localization of a Bose-Einstein condensate in bichromatic lattices. We obtain ground-state phase diagrams in the SOC and Rabi field plane for different values of disorder strength and use realistic experimental parameters compatible with $^{39}\mathrm{K}$. We find cases of fixed disorder and SOC (Rabi field), where the Rabi field (SOC) reduces the threshold for localization and controls the localization length. We also show regimes of fixed disorder and Rabi field, where the extent of the ground-state wave function is periodic in the SOC, leading to alternating regions of stronger and weaker localization as SOC changes. Lastly, we describe examples of fixed disorder and SOC, where tuning the Rabi field leads to a strong localization peak.

Theory and phenomenology of stressed wave-dark-matter soliton

Soliton in the hostile turbulent wave dark matter ($\mathrm{\ensuremath{\Psi}}\mathrm{DM}$) halo of a galaxy agitates with various kinds of excitation, and the soliton even breathes heavily under great stress. A theory of collective excitation for a $\mathrm{\ensuremath{\Psi}}\mathrm{DM}$ soliton is presented. The collective excitation has different degrees of coupling to negative energy modes, where lower-order excitation generally necessitates more negative energy coupling. A constrained variational principle is developed to assess the frequencies and mode structures of small-amplitude perturbations. The predicted frequencies are in good agreement with those found in simulations. Soliton breathing at amplitudes on the verge of breakup is also a highlight of this work. Even in this extreme nonlinear regime, the wave function perturbation amplitudes are moderate. The simulation data show a stable oscillation with frequency weakly dependent on the oscillation amplitude and hint at a self-consistent quasilinear model for the wave function that accounts for modifications in the ground-state wave function and the equilibrium density. The mock solution, constructed from the simulation data, can shed light on the dynamics of the large-amplitude breathing soliton and support the quasilinear model, as evidenced by its ability to predict well the nonlinear eigenfrequency shifts and large-amplitude breathing frequency observed in simulations.

Fourier-flow model generating Feynman paths

As an alternative but unified and more fundamental description for quantum physics, Feynman path integrals generalize the classical action principle to a probabilistic perspective, under which the physical observables' estimation translates into a weighted sum over all possible paths. The underlying difficulty is to tackle the whole path manifold from finite samples that can effectively represent the Feynman propagator dictated probability distribution. Modern generative models in machine learning can handle learning and representing probability distribution with high computational efficiency. In this study, we propose a Fourier-flow generative model to simulate the Feynman propagator and generate paths for quantum systems. As demonstration, we validate the path generator on the harmonic and anharmonic oscillators. The latter is a double-well system without analytic solutions. To preserve the periodic condition for the system, the Fourier transformation is introduced into the flow model to approach a Matsubara representation. With this novel development, the ground-state wave function and low-lying energy levels are estimated accurately. Our method offers a new avenue to investigate quantum systems with machine learning assisted Feynman Path integral solving.

The Dunkl oscillator in the momentum representation and coherent states

We discuss quantum mechanical systems with Dunkl derivatives by constructing the Dunkl-Heisenberg relation in the momentum representation by means of the reflection operator for momentum and we obtain the corresponding position quantum eigenfunction. We examine the one-dimensional Dunkl oscillator in the momentum space in terms of ν-deformed Hermite polynomials. We obtain the energy levels as well as the ground-state and excited wave functions in terms of the ν-deformed Hermite polynomials. We also describe some properties of the ν-deformed Hermite polynomials. We apply the method to the construction of coherent states.

2-D Quantum Ising Model Fidelity and Order Parameter

This paper explores an interesting relationship between the ground-state fidelity quantum phase transitions and bifurcations per lattice site, and proposes a universal order parameter for the quantum Ising model for a square lattice of infinite-size in two spatial dimensions as a prototype model with symmetry breaking order. The approach is based on computing ground-state wave functions using a tensor network algorithm utilizing a representation with infinite projected entangled-pair states. The results can be applied to any systems with symmetry breaking order, because in the conventional Landau-Ginzburg-Wilson paradigm a quantum system subjected to a phase transition exhibits spontaneous symmetry breaking quantified by a local order parameter. A reduced fidelity bifurcation between two different reduced density matrices is also explored.

Solving the nuclear pairing model with neural network quantum states.

We present a variational Monte Carlo method that solves the nuclear many-body problem in the occupation number formalism exploiting an artificial neural network representation of the ground-state wave function. A memory-efficient version of the stochastic reconfiguration algorithm is developed to train the network by minimizing the expectation value of the Hamiltonian. We benchmark this approach against widely used nuclear many-body methods by solving a model used to describe pairing in nuclei for different types of interaction and different values of the interaction strength. Despite its polynomial computational cost, our method outperforms coupled-cluster and provides energies that are in excellent agreement with the numerically exact full configuration-interaction values.

The reflected entanglement spectrum for free fermions

We consider the reflected entropy and the associated entanglement spectrum for free fermions reduced to two intervals in 1 + 1 dimensions. Working directly in the continuum theory the reflected entropy can be extracted from the spectrum of a singular integral equation whose kernel is determined by the known free fermion modular evolved correlation function. We find the spectrum numerically and analytically in certain limits. For intervals that almost touch the reflected entanglement spectrum approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the ground state wave function.

The hierarchy of Davydov's Ansätze: From guesswork to numerically "exact" many-body wave functions.

This Perspective presents an overview of the development of the hierarchy of Davydov's Ansätze and a few of their applications in many-body problems in computational chemical physics. Davydov's solitons originated in the investigation of vibrational energy transport in proteins in the 1970s. Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lending unambiguous evidence to the absence of formal quantum phase transitions in Holstein systems. The multiple Davydov Ansätze have been proposed, with increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian, the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic properties using Davydov's Ansätze as its trial states. A quantity proven to disappear for large multiplicities, the Ansatz relative deviation is introduced to quantify how closely the Schrödinger equation is obeyed. Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology, this Perspective provides applications of Davydov's Ansätze to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, where accurate dynamic and spectroscopic properties of the many-body systems are given by the Davydov trial states.

Neural network enhanced measurement efficiency for molecular groundstates

It is believed that one of the first useful applications for a quantum computer will be the preparation of groundstates of molecular Hamiltonians. A crucial task involving state preparation and readout is obtaining physical observables of such states, which are typically estimated using projective measurements on the qubits. At present, measurement data is costly and time-consuming to obtain on any quantum computing architecture, which has significant consequences for the statistical errors of estimators. In this paper, we adapt common neural network models (restricted Boltzmann machines and recurrent neural networks) to learn complex groundstate wavefunctions for several prototypical molecular qubit Hamiltonians from typical measurement data. By relating the accuracy ɛ of the reconstructed groundstate energy to the number of measurements, we find that using a neural network model provides a robust improvement over using single-copy measurement outcomes alone to reconstruct observables. This enhancement yields an asymptotic scaling near ɛ −1 for the model-based approaches, as opposed to ɛ −2 in the case of classical shadow tomography.

Topological superconducting vortex from trivial electronic bands

Superconducting vortices are promising traps to confine non-Abelian Majorana quasi-particles. It has been widely believed that bulk-state topology, of either normal-state or superconducting ground-state wavefunctions, is crucial for enabling Majorana zero modes in solid-state systems. This common belief has shaped two major search directions for Majorana modes, in either intrinsic topological superconductors or trivially superconducting topological materials. Here we show that Majorana-carrying superconducting vortex is not exclusive to bulk-state topology, but can arise from topologically trivial quantum materials as well. We predict that the trivial bands in superconducting HgTe-class materials are responsible for inducing anomalous vortex topological physics that goes beyond any existing theoretical paradigms. A feasible scheme of strain-controlled Majorana engineering and experimental signatures for vortex Majorana modes are also discussed. Our work provides new guidelines for vortex-based Majorana search in general superconductors.

Correlation in momentum space of Tonks–Girardeau gas

In momentum space, we investigate the correlation properties of the ground state of Tonks–Girardeau gases. With Bose–Fermi mapping method, the exact ground state wavefunction in coordinate space can be obtained based on the wavefunction of spin-polarized Fermions. By Fourier transformation we obtain the ground state wavefunction in momentum space, and therefore the pair correlation and the reduced one-body density matrix (ROBDM) in momentum space, whose diagonal part is the momentum distribution. The ROBDM in momentum space is the Fourier transformation of the ROBDM in coordinate space and the pair correlation in momentum space is the Fourier transformation of the reduced two-body density matrix in coordinate space. The correlations in momentum space display larger values only in small momentum region and vanish in most other regions. The lowest natural orbital and occupation distribution in momentum space are also obtained.

Bosonic topological insulators at the superconductor-to-superinsulator transition

We review the topological gauge theory of the superconductor-to-superinsulator transition. The possible intermediate Bose metal phase intervening between these two states is a bosonic topological insulator. We point out that the correct treatment of a bosonic topological insulator requires a normally neglected, additional dimensionless parameter, which arises because of the non-commutativity between the infinite gap limit and phase space reduction. We show that the bosonic topological insulator is a functional first Landau level. The additional parameter drives two Berezinskii–Kosterlitz–Thouless (BKT) quantum transitions to superconducting and superinsulating phases, respectively. The two BKT correlation scales account for the emergent granularity observed around the transition. Finally, we derive the ground state wave function for a system of charges and vortices in the Bose metal phase.

Entanglement bipartitioning and tree tensor networks

AbstractWe propose the entanglement bipartitioning approach to design an optimal network structure of the tree tensor network (TTN) for quantum many-body systems. Given an exact ground-state wavefunction, we perform sequential bipartitioning of spin-cluster nodes so as to minimize the mutual information or the maximum loss of the entanglement entropy associated with the branch to be bipartitioned. We demonstrate that entanglement bipartitioning of up to 16 sites gives rise to nontrivial tree network structures for S = 1/2 Heisenberg models in one and two dimensions. The resulting TTNs enable us to obtain better variational energies, compared with standard TTNs such as the uniform matrix product state and perfect binary tree tensor network.

Isotope Effects on Ground and Excited States of Ethyl Cation, H+(C2H4)

The structure and spectra of ethyl cation, H+(C2H4), and its deuterated analogues are investigated using diffusion Monte Carlo (DMC). These calculations all show that the ground state wave function for H+(C2H4) is localized near the minimum energy configuration in which the excess proton is in a bridging configuration, although the amplitude of the vibrational motions of the bridging proton is large. Deuteration of the bridging proton reduces the amplitude of this motion, while deuteration of only the ethylenic hydrogen atoms in H+(C2D4) has little effect on the amplitude of the motion of the bridging proton. Excited states that are accessed by spectroscopically observed transitions in H+(C2H4) are calculated using fixed-node DMC. The calculated and measured frequencies for the states with one quantum of excitation in the ethylenic CH stretching vibrations show good agreement. We also explore the excited state with one quantum of excitation in the proton transfer vibration of the bridging proton and obtain a frequency of 616 cm-1 for H+(C2H4). This frequency increases to 629 cm-1 in H+(C2D4). Deuteration decreases this frequency to 491 and 495 cm-1 in D+(C2H4) and D+(C2D4), respectively. The effects of partial deuteration on the frequencies of the CH stretching vibrations, and the corresponding probability amplitudes are also explored. Finally, we report the vibrationally averaged rotational constants for the four isotopologues of ethyl cation considered in this study.

Projected Hybrid Density Functionals: Method and Application to Core Electron Ionization.

This work introduces a new class of hybrid density functional theory (DFT) approximations, which incorporate different fractions of nonlocal exact exchange in predefined states such as core atomic orbitals (AOs). These projected hybrid density functionals are related to range-separated hybrid functionals, which incorporate different fractions of nonlocal exchange at different electron-electron separations. This work derives projected hybrids using the Adiabatic Projection formalism. One projects the electron-electron interaction operator onto the chosen predefined states, introduces the projected operator into the noninteracting Kohn-Sham reference system, and employs a formally exact density functional to model the remaining electron-electron interactions. Projected hybrids, like range-separated hybrids, approximate the partially interacting reference system's ground-state wave function as a single Slater determinant. Projected hybrids are readily implemented into existing density functional codes, requiring only a projection of the one-particle density matrices and exchange operators entering existing routines. This work also presents an application to core electron ionization. Projecting onto core atomic orbitals allows us to introduce additional nonlocal exchange into atomic core regions. This reduces the impact of self-interaction error on computed core electron properties. Benchmark studies are reported for PBE0c70, a core-projected variant of the Perdew-Burke-Ernzerhof global hybrid PBE0, in which the fraction of nonlocal exchange is increased from 25% to 70% in atomic core regions. PBE0c70-predicted core orbital energies accurately recover nonrelativistic core-electron binding energies of second-period elements Li-Ne and third-period elements Na-Ar, without degrading the good performance of PBE0 for atomization energies and valence ionization potentials.

Is there a better way of representing stationary wave functions than basis expansion?

We extend our previously developed approach for finding the exact analytical form of ground state wave function of single-electron Schrödinger equations to target arbitrary excited states of 1D model chemical systems of hydrogen atom and H 2 + $$ {\mathrm{H}}_2^{+} $$ with soft Coulomb potential. Our representation of the wave functions is formally unified by a factorized formula involving a polynomial prefactor encoding the nodal information, a non-integer power prefactor, and an exponentially decaying term modulated by a modulator function. We show that our method leads to faster energy convergence than the basis expansion method, and hence is a better way of representing the wave function. This conclusion is also validated in the real-world H 2 + $$ {\mathrm{H}}_2^{+} $$ .