Integrable Hamiltonian Research Articles

Path integral for chord diagrams and chaotic-integrable transitions in double scaled SYK

We study transitions from chaotic to integrable Hamiltonians in the double scaled Sachdev-Ye-Kitaev (SYK) and p-spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bilocal (GΣ) Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition. We also analyze the phase diagram for generic deformations, which in some cases includes a zero-temperature phase transition. Published by the American Physical Society 2024

Extending GPU-accelerated Gaussian integrals in the TeraChem software package to f type orbitals: Implementation and applications.

The increasing availability of graphics processing units (GPUs) for scientific computing has prompted interest in accelerating quantum chemical calculations through their use. However, the complexity of integral kernels for high angular momentum basis functions often limits the utility of GPU implementations with large basis sets or for metal containing systems. In this work, we report the implementation of f function support in the GPU-accelerated TeraChem software package through the development of efficient kernels for the evaluation of Hamiltonian integrals. The high efficiency of the resulting code is demonstrated through density functional theory (DFT) calculations on increasingly large organic molecules and transition metal complexes, as well as coupled cluster singles and doubles calculations on water clusters. Preliminary investigations into Ni(I) catalysis with DFT and the photochemistry of MnH(CH3) with complete active space self-consistent field are also carried out. Overall, our GPU-accelerated software appears to be well-suited for fast simulation of large transition metal containing systems, as well as organic molecules.

Supersymmetric Integrable Hamiltonian Systems, Conformal Lie Superalgebras K(1, N = 1, 2, 3), and Their Factorized Semi-Supersymmetric Generalizations

We successively reanalyzed modern Lie-algebraic approaches lying in the background of effective constructions of integrable super-Hamiltonian systems on functional N=1,2,3- supermanifolds, possessing rich supersymmetries and endowed with suitably related compatible Poisson structures. As an application, we describe countable hierarchies of new nonlinear Lax-type integrable N=2,3-semi-supersymmetric dynamical systems and constructed their central extended superconformal Lie superalgebra K(1|3) and its finite-dimensional coadjoint orbits, generated by the related Casimir functionals. Moreover, we generalized these results subject to the suitably factorized super-pseudo-differential Lax-type representations and present the related super-Poisson brackets and compatible suitably factorized Hamiltonian superflows. As an interesting point, we succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of the centrally extended super-pseudo-differential operator Lie superalgebras on the N=1,2,3-supercircle.

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.

Nonexistence of integrable nonlinear magnetic fields with invariants quadratic in momenta

Nonlinear, completely integrable Hamiltonian systems that serve as blueprints for novel particle accelerators at the intensity frontier are promising avenues for research, as Fermilab’s Integrable Optics Test Accelerator (IOTA) example clearly illustrates. Here, we show that only very limited generalizations are possible when no approximations in the underlying Hamiltonian or Maxwell equations are allowed, as was the case for IOTA. Specifically, no such systems exist with invariants quadratic in the momenta, precluding straightforward generalization of the Courant–Snyder theory of linear integrable systems in beam physics. We also conjecture that no such systems exist with invariants of higher degree in the momenta. This leaves solenoidal magnetic fields, including their nonlinear fringe fields, as the only completely integrable static magnetic fields, albeit with invariants that are linear in the momenta. The difficulties come from enforcing Maxwell equations; without constraints, we show that there are many solutions. In particular, we discover a previously unknown large family of integrable Hamiltonians.

The R-mAtrIx Net

We provide a novel neural network architecture that can: i) output R-matrix for a given quantum integrable spin chain, ii) search for an integrable Hamiltonian and the corresponding R-matrix under assumptions of certain symmetries or other restrictions, iii) explore the space of Hamiltonians around already learned models and reconstruct the family of integrable spin chains which they belong to. The neural network training is done by minimizing loss functions encoding Yang–Baxter equation, regularity and other model-specific restrictions such as hermiticity. Holomorphy is implemented via the choice of activation functions. We demonstrate the work of our neural network on the spin chains of difference form with two-dimensional local space. In particular, we reconstruct the R-matrices for all 14 classes. We also demonstrate its utility as an Explorer, scanning a certain subspace of Hamiltonians and identifying integrable classes after clusterisation. The last strategy can be used in future to carve out the map of integrable spin chains with higher dimensional local space and in more general settings where no analytical methods are available.

Efficient screening and catalytic mechanism of TM@β-Te for nitrogen reduction reaction

At present, nitrogen reduction reaction (NRR) has become one critical method for NH3 production through the single-atom catalyst (SAC). A variety of TM@β-Te SACs, constructed by 28 transition metal (TM) atoms anchoring on monovacant β-Te substrate, are systematically investigated by using DFT calculations. An efficient “six-step” screening scheme is proposed to screen out potential NRR catalysts, during which the implicit and explicit solvent models are involved to evaluate competitive reactions of H2O and H+. The final selected (Ta, W)@β-Te catalysts have limiting potentials (UL) of −0.32 V and −0.37 V, respectively. Three intrinsic descriptors of φ (the ratio of number of valence electrons and electronegativity of TM atom), ΔG∗N(adsorption energy of a single N atom on the substrate) and ICOHP (Crystal orbital Hamiltonian integrals for TM-N bonds) are proposed and their correlations with UL are investigated to expound the catalytic activity and mechanism. The relationships of dN≡N ∼ φ, ΔG∗N2∼ φ, ΔG∗N∼ φ and ICOHP ∼ φ resemble “volcanoes”, where (Ta, W)@β-Te are located near the vertex as potential catalysts, which can be verified by scaling relationship of ICOHP and ΔG∗N. Moreover, the UL ∼ φ, UL ∼ ΔG∗Nand UL ∼ ICOHP can be used to determine the potential catalysts from rough catalyst screening to accurate determination of potential determining step (PDS) in catalytic reaction. The constant potential model is adopted to investigate the influences of electrode potentials on adsorption strength of NRR intermediates. This work is informative for the research of NRR process and catalyst design of monoalkenes, which offers an efficient and dependable approach for screening excellent NRR catalysts and revealing the origin of catalytic activity.

First-order quantum phase transition in the two-qubit squeezed Rabi model

We study the ground state of the two-qubit squeezed Rabi model. Two special transformations are found to diagonalize the system Hamiltonian when each qubit’s frequency is close to the field frequency, where both the squeezing and counterrotating-wave interactions are removed, leading to an effective integrable Hamiltonian. The analytical ground state is determined and matches with numerical solutions well for a range of squeezing strengths and qubit-field detunings in the ultrastrong-coupling regime. We demonstrate that the ground state exhibits a first-order quantum phase transition at a phase boundary linearly induced by the squeezed light. We characterize the two-qubit negativity analytically and find that its two-qubit entanglement increases with the increasing squeezing strength nonlinearly. The average photon numbers of the field mode and variances of position and momentum quadratures are also analyzed and discovered to have a nonlinear relation with the squeezing strength. Finally, we discuss the experimental scheme and realization possibility of the predicted results.

Global properties of generic real–analytic nearly–integrable Hamiltonian systems

We introduce a new class Gsn of generic real analytic potentials on Tn and study global analytic properties of natural nearly–integrable Hamiltonians 12|y|2+εf(x), with potential f∈Gsn, on the phase space M=B×Tn with B a given ball in Rn. The phase space M can be covered by three sets: a ‘non–resonant’ set, which is filled up to an exponentially small set of measure e−cK (where K is the maximal size of resonances considered) by primary maximal KAM tori; a ‘simply resonant set’ of measure εKa and a third set of measure εKb which is ‘non perturbative’, in the sense that the H–dynamics on it can be described by a natural system which is not nearly–integrable. We then focus on the simply resonant set – the dynamics of which is particularly interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori) – and show that on such a set the secular (averaged) 1 degree–of–freedom Hamiltonians (labeled by the resonance index k∈Zn) can be put into a universal form (which we call ‘Generic Standard Form’), whose main analytic properties are controlled by only one parameter, which is uniform in the resonance label k.

Nested gausslet basis sets

We introduce nested gausslet bases, an improvement on previous gausslet bases that can treat systems containing atoms with much larger atomic numbers. We also introduce pure Gaussian distorted gausslet bases, which allow the Hamiltonian integrals to be performed analytically, as well as hybrid bases in which the gausslets are combined with standard Gaussian-type bases. All these bases feature the diagonal approximation for the electron–electron interactions so that the Hamiltonian is completely defined by two Nb × Nb matrices, where Nb ≈ 104 is small enough to permit fast calculations at the Hartree–Fock level. In constructing these bases, we have gained new mathematical insight into the construction of one-dimensional diagonal bases. In particular, we have proved an important theorem relating four key basis set properties: completeness, orthogonality, zero-moment conditions, and diagonalization of the coordinate operator matrix. We test our basis sets on small systems with a focus on high accuracy, obtaining, for example, an accuracy of 2 × 10−5 Ha for the total Hartree–Fock energy of the neon atom in the complete basis set limit.

Guiding centre motion for particles in a ponderomotive magnetostatic end plug

The Hamiltonian dynamics of a single particle in a rotating plasma column, interacting with an magnetic multipole is perturbatively solved for up to second order, using the method of Lie transformations. First, the exact Hamiltonian is expressed in terms of canonical action-angle variables, and then an approximate integrable Hamiltonian is introduced, using another set of actions and angles, which describe the centre of oscillation for the particle. The perturbation introduces an effective ponderomotive potential, which to leading order is positive. At the second order, the pseudopotential consists of a sum of terms of the Miller form, and can have either sign. Additionally, at second order, the ponderomotive interaction introduces a modification to the particle effective mass, when considering the motion along the column axis. It is found that particles can be axially confined by the ponderomotive potentials, but acquire radial excursions which scale as the confining potential. The radial excursions of the particle along its trajectory are investigated, and a condition for the minimal rotation frequency for which the particle remains radially confined is derived. Last, we comment on the changes to the aforementioned solution to the pseudopotentials and particle trajectory in the case of resonant motion, that is, a motion which has the same periodicity as the perturbation.

Ground state wavefunctions of elliptic relativistic integrable Hamiltonians

We derive ground state eigenfunctions and eigenvalues of various relativistic elliptic integrable models. The models we discuss appear in computations of superconformal indices of four-dimensional theories obtained by compactifying six-dimensional models on Riemann surfaces. These include, among others, the Ruijsenaars-Schneider model and the van Diejen model. The derivation of the eigenfunctions builds on physical inputs, such as conjectured Lagrangian across dimensions IR dualities and assumptions about the behavior of the indices in the limit of compactifications on surfaces with large genus/number of punctures/flux.

Finite temperature properties of an integrable zigzag ladder chain

We consider the interaction-round-a-face version of the six-vertex model for arbitrary anisotropy parameter, which allow us to derive an integrable one-dimensional quantum Hamiltonian with three-spin interactions. We apply the quantum transfer matrix approach for the face model version of the six-vertex model. The integrable quantum Hamiltonian shares some thermodynamical properties with the Heisenberg XXZ chain, but has different ordering and critical exponents. Two gapped phases are the dimerized antiferromagnetic order and the usual antiferromagnetic (Néel) order for positive nearest neighbor Ising coupling. In between these, there is an extended critical region, which is a quantum spin-liquid with broken parity symmetry inducing an oscillatory behavior at the long distance <σ1zσℓ+1z> correlation. At finite temperatures, the numerical solution of the non-linear integral equations allows for the determination of the correlation length as well as for the momentum of the oscillation.

On the Role of the Integrable Toda Model in One-Dimensional Molecular Dynamics

We prove that the common Mie–Lennard-Jones (MLJ) molecular potentials, appropriately normalized via an affine transformation, converge, in the limit of hard-core repulsion, to the Toda exponential potential. Correspondingly, any Fermi–Pasta–Ulam (FPU)-like Hamiltonian, with MLJ-type interparticle potential, turns out to be 1/n-close to the Toda integrable Hamiltonian, n being the exponent ruling repulsion in the MLJ potential. This means that the dynamics of chains of particles interacting through typical molecular potentials, is close to integrable in an unexpected sense. Theoretical results are accompanied by a numerical illustration; numerics shows, in particular, that even the very standard 12–6 MLJ potential is closer to integrability than the FPU potentials which are more commonly used in the literature.

Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum

We study dynamical phase transitions occurring in the stationary state of the dynamics of integrable many-body non-hermitian Hamiltonians, which can be either realized as a no-click limit of a stochastic Schrödinger equation or using spacetime duality of quantum circuits. In two specific models, the Transverse Field Ising Chain and the Long Range Kitaev Chain, we observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian: bounded entanglement entropy when the imaginary part of the quasi-particle spectrum is gapped and a logarithmic growth for gapless imaginary spectrum. This observation suggests the possibility to generalize the area-law theorem to non-Hermitian Hamiltonians.

Solubility of NaCl under anisotropic stress state.

Salt solubility is generally determined under isotropic stress conditions. Yet, in the context of salt weathering of porous media, mechanical constraints on the in-pore growth of salt crystals are likely to be orientation-dependent, resulting in an anisotropic stress state on the crystal. In this paper, we determine by molecular simulation the solubility of NaCl in water when the crystal is subjected to anisotropic stress. Such anisotropy causes the chemical potential of the crystal to be orientation-dependent, and proper thermodynamic formulation requires describing the chemical potential as a tensor. The solute and crystal chemical potentials are computed from free energy calculations using Hamiltonian thermodynamic integration, and the usual condition of solubility is reformulated to account for the tensorial nature of the crystal chemical potential. We investigate in detail how the uniaxial compression of the crystal affects its solubility. The molecular simulation results led to revisiting the Correns law under anisotropic stress. Regarding the solute, the non-ideal behavior of the liquid phase is captured using Pitzer's ion interaction approach up to high concentrations of interest for in-pore crystallization and beyond the concentrations addressed in the existing literature. Regarding NaCl crystals, the validity of the generalized Gibbs-Duhem equation for a tensorial chemical potential is carefully verified, and it is found that crystallization progresses almost orthogonally to the crystal surface even under high shear stresses. Comparing uniaxial and isotropic compression highlights the major differences in solubility caused by stress anisotropy, and the revisited Correns law offers an appropriate framework to capture this phenomenon.

Bethe/Gauge correspondence for linear quiver theories with ABCD gauge symmetry and spin chains

This note is an extension of [8] there the supersymmetric vacua of three-dimensional N=2 gauge theories with matter are shown to be in one-to-one correspondence with the eigenstate of XXZ integrable spin chain Hamiltonians with open boundary conditions. We consider the A2 quiver gauge theory, which is the simplest non-trivial quiver gauge theory, and sl3 open XXZ spin chain with diagonal boundary condition. We demonstrate the correspondence between the vacuum equations of different gauge groups and Bethe Ansatz equations with different boundary parameters. Not only that, but we furthermore push forward the program to the general Ar quiver gauge theory.

Periodic orbit theory of Bethe-integrable quantum systems: an N-particle Berry–Tabor trace formula

One of the fundamental results of semiclassical theory is the existence of trace formulae showing how spectra of quantum mechanical systems emerge from massive interference among amplitudes related with time-periodic structures of the corresponding classical limit. If it displays the properties of Hamiltonian integrability, this connection is given by the celebrated Berry–Tabor trace formula, and the periodic structures it is built on are tori supporting closed trajectories in phase space. Here we show how to extend this connection into the domain of quantum many-body systems displaying integrability in the sense of the Bethe ansatz, where a classical limit cannot be rigorously defined due to the presence of singular potentials. Formally following the original derivation of Berry and Tabor (1976 Proc. R. Soc. A 349 101), but applied to the Bethe equations without underlying classical structure, we obtain a many-particle trace formula for the density of states of N interacting bosons on a ring, the Lieb–Liniger model. Our semiclassical expressions are in excellent agreement with quantum mechanical results for and 4 particles. For N = 2 we relate our results to the quantization of billiards with mixed boundary conditions. Our work paves the way towards the treatment of the important class of integrable many-body systems by means of semiclassical trace formulae pioneered by Michael Berry in the single-particle context.

Generic KAM Hamiltonians are not quantum ergodic

We show that under generic conditions, the quantisation of a $1$-parameter family of KAM perturbations $P(x,\xi;t)$ of a completely integrable and Kolmogorov non-degenerate Gevrey smooth Hamiltonian is not quantum ergodic, at least for a full measure subset of the parameter $t\in (0,\delta)$.

An embedded Hamiltonian dynamic evolutionary neural network model for high-dimensional data recognition

Aiming at the limitation of Hamiltonian neural network in high-dimensional data processing with non-observable physical quantity system, an embedded Hamiltonian dynamic evolutionary neural network model (EHDN) is proposed in this paper. First, the dynamic physical quantities of high dimensional nonlinear systems are represented by constructing Hamiltonian functions. Then a symplectic integrator is designed to realize the time-varying evolution of network features based on Hamiltonian regular equation and variational numerical integration. Finally, the Hamiltonian dynamic evolutionary neural network is embedded in the convolutional neural network (CNN) to realize the Hamiltonian simulation evolution of convolutional features. Experimental results on two different datasets, CIFAR and Fashion-Mnist, show that compared with existing state-of-the-art (SOTA) methods, EHDN model can combine the advantages of Hamiltonian dynamics and convolutional network to improve the recognition accuracy and training stability in high-dimensional image classification tasks.