Matrix Elements Of Local Operators Research Articles

Statistics of Matrix Elements of Local Operators in Integrable Models

We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements ⟨μ|O|λ⟩ in the same macrostate scale as exp(−cOLln(L)−LMμ,λO), where the probability distribution function for Mμ,λO is well described by Fréchet distributions and cO depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as exp(−dOL2), where dO depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections. Published by the American Physical Society 2024

Eigenstate Correlations, the Eigenstate Thermalization Hypothesis, and Quantum Information Dynamics in Chaotic Many-Body Quantum Systems

We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterize entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalization hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations among sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behavior that are expected to be universal at long distances and low energies. We also propose a maximum-entropy ansatz for the joint distribution of a small number n of eigenstates. In the case n=2, this ansatz reproduces ETH. For n=4 it captures both the growth with time of entanglement between subsystems, as characterized by the purity of the time-evolution operator, and also operator spreading, as characterized by the behavior of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our ansatz with exact diagonalization studies of Floquet quantum circuits. Published by the American Physical Society 2024

On correlation functions for the open XXZ chain with non-longitudinal boundary fields: The case with a constraint

This paper is a continuation of [J. Phys. A: Math. Theor. 55, 405203 (2022)], in which a set of matrix elements of local operators was computed for the XXZ spin-1/2 open chain with a particular case of unparallel boundary fields. Here, we extend these results to the more general case in which both fields are non-longitudinal and related by one constraint, allowing for a partial description of the spectrum by usual Bethe equations. More precisely, the complete spectrum and eigenstates can be characterized within the Separation of Variables (SoV) framework. One uses here the fact that, under the constraint, a part of this SoV spectrum can be described via solutions of a usual, homogeneous, TQ-equation, with corresponding transfer matrix eigenstates coinciding with generalized Bethe states. We explain how to generically compute the action of a basis of local operators on such kind of states, and this under the most general boundary condition on the last site of the chain. As a result, we can compute the matrix elements of some of these basis elements in any eigenstate described by the homogenous TQ-equation. Assuming, following a conjecture of Nepomechie and Ravanini, that the ground state itself can be described in this framework, we obtain multiple integral representations for these matrix elements in the half-infinite chain limit, generalizing those previously obtained in the case of longitudinal boundary fields and in the case of the special boundary conditions considered in [J. Phys. A: Math. Theor. 55, 405203 (2022)].

Form factors of the tricritical three-state Potts model in its scaling limit

We compute the form factors of the order and disorder operators, together with those of the stress–energy tensor, of a two-dimensional three-state Potts model with vacancies along its thermal deformation at the critical point. At criticality, the model is described by the non-diagonal partition function of the unitary minimal model M6,7 of conformal field theories and is accompanied by an internal S 3 symmetry. The off-critical thermal deformation is an integrable massive theory that is still invariant under S 3. The presence of infinitely many conserved quantities, whose spin spectrum is related to the exceptional Lie algebra E 6, allows us to determine the analytic S-matrix, the exact mass spectrum and the matrix elements of local operators of this model in an exact non-perturbative way. We use the spectral representation series of the correlators and the fast convergence of these series to compute several universal ratios of the renormalization group.

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Injecting the UV into the bootstrap: Ising Field Theory

We merge together recent developments in the S-matrix bootstrap program to develop a dual setup in 2 space-time dimensions incorporating scattering amplitudes of massive particles and matrix elements of local operators. In particular, the stress energy tensor allows us to input UV constraints on IR observables in terms of the central charge cUV of the UV Conformal Field Theory. We consider two applications: (1) We establish a rigorous lower bound on cUV of a class of ℤ2 symmetric scalar theories in the IR (including ϕ4); (2) We target Ising Field Theory by, first, minimizing cUV for different values of the magnetic field and, secondly, by determining the allowed range of cubic coupling and one-particle form-factor for fixed cUV = 1/2 and magnetic field.

Exact universal chaos, speed limit, acceleration, Planckian transport coefficient, “collapse” to equilibrium, and other bounds in thermal quantum systems

We introduce “local uncertainty relations” in thermal many body systems. Using these relations, we derive basic bounds. These results include the demonstration of universal non-relativistic speed limits (regardless of interaction range), bounds on acceleration or force/stress, acceleration or material stress rates, transport coefficients (including the diffusion constant and viscosity), electromagnetic or other gauge field strengths, correlation functions of arbitrary spatio-temporal derivatives, Lyapunov exponents, and thermalization times. We further derive analogs of the Ioffe–Regel limit. These bounds are relatively tight when compared to various experimental data. In the ħ→0 limit, all of our bounds either diverge (e.g., the derived speed and acceleration limit) or vanish (as in, e.g., our viscosity and diffusion constant bounds). Our inequalities hold at all temperatures and, as corollaries, imply general power law bounds on response functions at both asymptotically high and low temperatures. Our results shed light on how apparent nearly instantaneous effective “collapse” to energy eigenstates may arise in macroscopic interacting many body quantum systems. We comment on how random off-diagonal matrix elements of local operators (in the eigenbasis of the Hamiltonian) may inhibit their dynamics.

Partial thermalisation of a two-state system coupled to a finite quantum bath

The eigenstate thermalisation hypothesis (ETH) is a statistical characterisation of eigen-energies, eigenstates and matrix elements of local operators in thermalising quantum systems. We develop an ETH-like ansatz of a partially thermalising system composed of a spin-\tfrac{1}{2}12 coupled to a finite quantum bath. The spin-bath coupling is sufficiently weak that ETH does not apply, but sufficiently strong that perturbation theory fails. We calculate (i) the distribution of fidelity susceptibilities, which takes a broadly distributed form, (ii) the distribution of spin eigenstate entropies, which takes a bi-modal form, (iii) infinite time memory of spin observables, (iv) the distribution of matrix elements of local operators on the bath, which is non-Gaussian, and (v) the intermediate entropic enhancement of the bath, which interpolates smoothly between S = 0S=0 and the ETH value of S = \log 2S=log2. The enhancement is a consequence of rare many-body resonances, and is asymptotically larger than the typical eigenstate entanglement entropy. We verify these results numerically and discuss their connections to the many-body localisation transition.

Bootstrapping 2d ϕ4 theory with Hamiltonian truncation data

We combine the methods of Hamiltonian Truncation and the recently proposed generalisation of the S-matrix bootstrap that includes local operators to determine the two-particle scattering amplitude and the two-particle form factor of the stress tensor at s > 0 in the 2d ϕ4 theory. We use the form factor of the stress tensor at s ≤ 0 and its spectral density computed using Lightcone Conformal Truncation (LCT), and inject them into the generalized S-matrix bootstrap set-up. The obtained results for the scattering amplitude and the form factor are fully reliable only in the elastic regime. We independently construct the “pure” S-matrix bootstrap bounds (bootstrap without including matrix elements of local operators), and find that the sinh-Gordon model and its analytic continuation the “staircase model” saturate these bounds. Surprisingly, the ϕ4 two-particle scattering amplitude also very nearly saturates these bounds, and moreover is extremely close to that of the sinh-Gordon/staircase model.

Thouless energy across the many-body localization transition in Floquet systems

The notion of Thouless energy plays a central role in the theory of Anderson localization. We investigate and compare the scaling of Thouless energy across the many-body localization (MBL) transition in a Floquet model. We use a combination of methods that are reliable on the ergodic side of the transition (e.g., spectral form factor) and methods that work on the MBL side (e.g., typical matrix elements of local operators) to obtain a complete picture of the Thouless energy behavior across the transition. On the ergodic side, Thouless energy decreases slowly with the system size, while at the transition it becomes comparable to the level spacing. Different probes yield consistent estimates of Thouless energy in their overlapping regime of applicability, giving the location of the transition point nearly free of finite-size drift. This work establishes a connection between different definitions of Thouless energy in a many-body setting and yields insights into the MBL transition in Floquet systems.

Local Pairing of Feynman Histories in Many-Body Floquet Models

We study many-body quantum dynamics using Floquet quantum circuits in one space dimension as simple examples of systems with local interactions that support ergodic phases. Physical properties can be expressed in terms of multiple sums over Feynman histories, which for these models are paths or many-body orbits in Fock space. A natural simplification of such sums is the diagonal approximation, where the only terms that are retained are ones in which each path is paired with a partner that carries the complex conjugate weight. We identify the regime in which the diagonal approximation holds, and the nature of the leading corrections to it. We focus on the behaviour of the spectral form factor (SFF) and of matrix elements of local operators, averaged over an ensemble of random circuits, making comparisons with the predictions of random matrix theory (RMT) and the eigenstate thermalisation hypothesis (ETH). We show that properties are dominated at long times by contributions to orbit sums in which each orbit is paired locally with a conjugate, as in the diagonal approximation, but that in large systems these contributions consist of many spatial domains, with distinct local pairings in neighbouring domains. The existence of these domains is reflected in deviations of the SFF from RMT predictions, and of matrix element correlations from ETH predictions; deviations of both kinds diverge with system size. We demonstrate that our physical picture of orbit-pairing domains has a precise correspondence in the spectral properties of a transfer matrix that acts in the space direction to generate the ensemble-averaged SFF. In addition, we find that domains of a second type control non-Gaussian fluctuations of the SFF. These domains are separated by walls which are related to the entanglement membrane, known to characterise the scrambling of quantum information.

Low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic XXZ chain

We study the matrix elements of local operators in the eigenstates of the integrable XXZ chain and of the quantum-chaotic model obtained by locally perturbing the XXZ chain with a magnetic impurity. We show that, at frequencies that are polynomially small in the system size, the behavior of the variances of the off-diagonal matrix elements can be starkly different depending on the operator. In the integrable model we find that, as the frequency $\omega\rightarrow0$, the variances are either nonvanishing (generic behavior) or vanishing (for a special class of operators). In the quantum-chaotic model, on the other hand, we find the variances to be nonvanishing as $\omega\rightarrow0$ and to indicate diffusive dynamics. We highlight which properties of the matrix elements of local operators are different between the integrable and quantum-chaotic models independently of the specific operator selected.

Quantum thermalization and Virasoro symmetry

We initiate a systematic study of high energy matrix elements of local operators in 2D CFT. Knowledge of these is required in order to determine whether the generalized eigenstate thermalization hypothesis (ETH) can hold in such theories. Most high energy states are high level Virasoro descendants, and by employing an oscillator representation of the Virasoro algebra we develop an efficient method for computing matrix elements of primary operators in such states. In parameter regimes where we expect (e.g. from AdS/CFT intuition) thermalization to occur, we observe striking patterns in the matrix elements: diagonal matrix elements are smoothly varying and off-diagonal elements, while nonzero, are power-law suppressed compared to the diagonal elements. We discuss the implications of these universal properties of 2D CFTs in regard to their compatibility with generalized ETH.

Entanglement and matrix elements of observables in interacting integrable systems.

We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=E_{α}-E_{β} ({E_{α}} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.

Eigenstate thermalization hypothesis and out of time order correlators.

The eigenstate thermalization hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems. Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out of time order correlators. For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist. The same is true for the peak of the dynamic heterogeneity length χ_{4}, relevant for systems with slow dynamics. These correlations, as well as those between elements of different operators, are encompassed in a generalized form of ETH.

BSM hadronic matrix elements for \u03b5\u2032/\u03b5 and K \u2192 \u03c0\u03c0 decays in the Dual QCD approach

We calculate for the first time all four-quark hadronic matrix elements of local operators possibly contributing to K → ππ decays and in particular to the ratio ε′/ε beyond the Standard Model (BSM). To this end we use the Dual QCD (DQCD) approach. In addition to 7 new mirror operators obtained from the SM ones by flipping the chirality, we count 13 BSM four-quark operators of a given chirality linearly independent of each other and of the aforesaid 14 operators for which hadronic matrix elements are already known. We present results in two bases for all these operators, one termed DQCD basis useful for the calculation of the hadronic matrix elements in the DQCD approach and the other called SD basis suited to the short distance renormalization group evolution above the 1 GeV scale. We demonstrate that the pattern of long distance evolution (meson evolution) matches the one of short distance evolution (quark-gluon evolution), a property which to our knowledge cannot be presently achieved in any other analytical framework. The highlights of our paper are chirally enhanced matrix elements of tensor-tensor and scalar-scalar BSM operators. They could thereby explain the emerging ε′/ε anomaly which is strongly indicated within DQCD with some support from lattice QCD. On the other hand we do not expect the BSM operators to be relevant for the ΔI = 1/2 rule.

Behavior of l-bits near the many-body localization transition

Eigenstates of fully many-body localized (FMBL) systems are described by quasilocal operators $\tau_i^z$ (l-bits), which are conserved exactly under Hamiltonian time evolution. The algebra of the operators $\tau_i^z$ and $\tau_i^x$ associated with l-bits ($\boldsymbol{\tau}_i$) completely defines the eigenstates and the matrix elements of local operators between eigenstates at all energies. We develop a non-perturbative construction of the full set of l-bit algebras in the many-body localized phase for the canonical model of MBL. Our algorithm to construct the Pauli-algebra of l-bits combines exact diagonalization and a tensor network algorithm developed for efficient diagonalization of large FMBL Hamiltonians. The distribution of localization lengths of the l-bits is evaluated in the MBL phase and used to characterize the MBL-to-thermal transition.

An efficient pseudo-spectral method for the description of atomic electronic wave functions – Application to the hydrogen atom in a uniform magnetic field

The mapping of an electronic state on a real-space support lattice may offer advantages over a basis set ansatz in cases where there are linear dependences due to basis set overcompleteness or when strong internal or external fields are present. Such discretization methods are also of interest because they allow for the convenient numerical integration of matrix elements of local operators. We have developed a pseudo-spectral approach to the numerical solution of the time-dependent and time-independent Schrödinger equations describing electronic motion in atoms and atomic ions in terms of a spherical coordinate system. A key feature of this scheme is the construction of a Variational Basis Representation (VBR) for the non-local component and of a Generalized Finite Basis Representation (GFBR) for the local component of the electronic Hamiltonian operator. Radial Hamiltonian eigenfunctions χnl;βr of the H atom-like system and spherical harmonics form the basis set. Two special cases of this approach are explored: In one case, the functions of the field-free H atom are used as the elements of the basis set, and in the second case, each radial basis function has been obtained by variationally optimizing a shielding parameter β to yield a minimum energy for a particular eigenstate of the H atom in a uniform magnetic field.We derive a new quadrature rule of nearly Gaussian accuracy for the computation of matrix elements of local operators between radial basis functions and perform numerical evaluation of local operator matrix elements involving spherical harmonics. With this combination of radial and angular quadrature prescriptions we ensure to a good approximation the discrete orthogonality of Hamiltonian eigenfunctions of H atom-like systems for summation over the grid points. We further show that sets of χnl;βr functions are linearly independent, whereas sets of the polar-angle-dependent components of the spherical harmonics, i.e., the associated Legendre functions, are not and provide a physical interpretation of this mathematical observation.The pseudo-spectral approach presented here is applied to two model systems: the field-free H atom and the H atom in a uniform magnetic field. The results demonstrate the potential of this method for the description of challenging systems such as highly charged atomic ions.

Anomalous thermalization and transport in disordered interacting Floquet systems

Local observables in generic periodically driven closed quantum systems are known to relax to values described by periodic infinite temperature ensembles. At the same time, ergodic static systems exhibit anomalous thermalization of local observables and satisfy a modified version of the eigenstate thermalization hypothesis (ETH), when disorder is present. This raises the question, how does the introduction of disorder affect relaxation in periodically driven systems? In this work, we analyze this problem by numerically studying transport and thermalization in an archetypal example. We find that thermalization is anomalous and is accompanied by subdiffusive transport with a disorder dependent dynamical exponent. Distributions of matrix elements of local operators in the eigenbases of a family of effective time-independent Hamiltonians, which describe the stroboscopic dynamics of such systems, show anomalous departures from predictions of ETH signaling that only a modified version of ETH is satisfied. The dynamical exponent is shown to be related to the scaling of the variance of these distributions with system size.

Many-body-localization: strong disorder perturbative approach for the local integrals of motion

For random quantum spin models, the strong disorder perturbative expansion of the local integrals of motion around the real-spin operators is revisited. The emphasis is on the links with other properties of the many-body-localized phase, in particular the memory in the dynamics of the local magnetizations and the statistics of matrix elements of local operators in the eigenstate basis. Finally, this approach is applied to analyze the many-body-localization transition in a toy model studied previously from the point of view of the entanglement entropy.