Ramis Movassagh

Criticality without frustration for quantum spin-1 chains.

Frustration-free (FF) spin chains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled the ground state of a FF quantum spin-s chain with nearest-neighbor interactions can be for small values of s. While FF spin-1/2 chains are known to have unentangled ground states, the case s=1 remains less explored. We propose the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior. The ground state can be viewed as the uniform superposition of balanced strings of left and right brackets separated by empty spaces. Entanglement entropy of one half of the chain scales as 1/2 log n+O(1), where n is the number of spins. We prove that the energy gap above the ground state is polynomial in 1/n. The proof relies on a new result concerning statistics of Dyck paths which might be of independent interest.

Supercritical entanglement in local systems: Counterexample to the area law for quantum matter

Entanglement between two quantum systems is a non-classical correlation between them. Entanglement is a feature of quantum mechanics which does not appear classically, and it serves as a resource for quantum technologies. In condensed matter theory, the area law says that the amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary of the subsystem [1]. A system that obeys an area law can be simulated more efficiently than an arbitrary quantum system, and an area law provides useful information about the low-energy physics of the system [1–3]. The area law has only been proved for one-dimensional systems with a constant-energy spectral gap [4]. However, it is widely believed that for physically reasonable quantum systems, the area law could not be violated by more than a logarithmic factor in the system’s size [5]. Here, we introduce a class of exactly solvable one dimensional models that have exponentially more entanglement than previously expected, and violate the area law by a square root factor. In addition to using recent advances in quantum information theory, we have drawn upon various branches of mathematics in our work. We hope that the tools we have developed may be useful for other problems in physics as well.Significance We introduce a class of exactly solvable models with surprising properties. We show that even simple quantum matter is much more entangled than previously believed possible. One then expects more complex systems to be substantially more entangled. For over two decades it was believed that the area law is violated by at most a logarithm in the system’s size for quantum matter (i.e., interactions satisfying physical reasonability criteria clearly stated in the article). In this work we introduce a class of physically reasonable models that we can prove violate the area law by a square root, i.e., exponentially more than the logarithm. Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an “area law”: The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system’s size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system’s size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially more entanglement than suggested by the area law, and violate the area law by a square-root factor. This work suggests that simple quantum matter is richer and can provide much more quantum resources (i.e., entanglement) than expected. In addition to using recent advances in quantum information and condensed matter theory, we have drawn upon various branches of mathematics such as combinatorics of random walks, Brownian excursions, and fractional matching theory. We hope that the techniques developed herein may be useful for other problems in physics as well.

Quantum interference in polyenes.

The explicit form of the zeroth Greens function in the Hückel model, approximated by the negative of the inverse of the Hückel matrix, has direct quantum interference consequences for molecular conductance. We derive a set of rules for transmission between two electrodes attached to a polyene, when the molecule is extended by an even number of carbons at either end (transmission unchanged) or by an odd number of carbons at both ends (transmission turned on or annihilated). These prescriptions for the occurrence of quantum interference lead to an unexpected consequence for switches which realize such extension through electrocyclic reactions: for some specific attachment modes the chemically closed ring will be the ON position of the switch. Normally the signs of the entries of the Greens function matrix are assumed to have no physical significance; however, we show that the signs may have observable consequences. In particular, in the case of multiple probe attachments - if coherence in probe connections can be arranged - in some cases new destructive interference results, while in others one may have constructive interference. One such case may already exist in the literature.

Error Analysis of Free Probability Approximations to the Density of States of Disordered Systems

Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solving the electronic structure of disordered systems. We use free probability to approximate the ensemble-averaged density of states without exact diagonalization. We present an error analysis that quantifies the accuracy using a generalized moment expansion, allowing us to distinguish between different approximations. We identify an approximation that is accurate to the eighth moment across all noise strengths, and contrast this with perturbation theory and isotropic entanglement theory.

The Green’s function for the Hückel (tight binding) model

Applications of the Huckel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green’s function, G, of the N×N Huckel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)). We then extend the results to d− dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green’s function and discuss applications related to transpo...

Density of states of quantum spin systems from isotropic entanglement.

We propose a method that we call isotropic entanglement (IE), which predicts the eigenvalue distribution of quantum many body (spin) systems with generic interactions. We interpolate between two known approximations by matching fourth moments. Though such problems can be QMA-complete, our examples show that isotropic entanglement provides an accurate picture of the spectra well beyond what one expects from the first four moments alone. We further show that the interpolation is universal, i.e., independent of the choice of local terms.

Generic Local Hamiltonians are Gapless

We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. Examples include many-body localization and random spin models. We calculate the scaling of the gap with the systems size when the local terms are distributed according to a Gaussian β orthogonal random matrix ensemble. As a corollary, there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. When the local eigenvalue distribution is discrete, in addition to the lack of an energy gap in the limit, we prove that the ground state has finite size degeneracies. The proofs are simple and constructive. This work excludes the important class of truly translationally invariant Hamiltonians where the local terms are all equal.

Sample covariance based estimation of Capon algorithm error probabilities

The method of interval estimation (MIE) provides a strategy for mean squared error (MSE) prediction of algorithm performance at low signal-to-noise ratios (SNR) below estimation threshold where asymptotic predictions fail. MIE interval error probabilities for the Capon algorithm are known and depend on the true data covariance and assumed signal array response. Herein estimation of these error probabilities is considered to improve representative measurement errors for parameter estimates obtained in low SNR scenarios, as this may improve overall target tracking performance. A statistical analysis of Capon error probability estimation based on the data sample covariance matrix is explored herein.

Quantum Interference, Graphs, Walks, and Polynomials

In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Greens function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Hückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Greens function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for nonalternants from cancellation of odd- and even-length walk terms. We report some progress, but not a complete resolution, of the problem of understanding the coefficients in the expansion of the Greens function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. Furthermore, we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Greens function.

Eigenpairs of Toeplitz and Disordered Toeplitz Matrices with a Fisher–Hartwig Symbol

Toeplitz matrices have entries that are constant along diagonals. They model directed transport, are at the heart of correlation function calculations of the two-dimensional Ising model, and have applications in quantum information science. We derive their eigenvalues and eigenvectors when the symbol is singular Fisher–Hartwig. We then add diagonal disorder and study the resulting eigenpairs. We find that there is a “bulk” behavior that is well captured by second order perturbation theory of non-Hermitian matrices. The non-perturbative behavior is classified into two classes: Runaways type I leave the complex-valued spectrum and become completely real because of eigenvalue attraction. Runaways type II leave the bulk and move very rapidly in response to perturbations. These have high condition numbers and can be predicted. Localization of the eigenvectors are then quantified using entropies and inverse participation ratios. Eigenvectors corresponding to Runaways type II are most localized (i.e., super-exponential), whereas Runaways type I are less localized than the unperturbed counterparts and have most of their probability mass in the interior with algebraic decays. The results are corroborated by applying free probability theory and various other supporting numerical studies.