arXiv subjects

Gerardo Ortiz

Publications and source records attributed to Gerardo Ortiz.

At least 55 records · Page 3Linked to original sources

arXiv2017

Non-Abelian fermion parity interferometry of Majorana bound states in a Fermi sea

We study the quantum dynamics of Majorana and regular fermion bound states coupled to a quasi-one-dimensional metallic lead. The dynamics following the quench in the coupling to the lead exhibits a series of dynamical revivals as the bound state propagates in the lead and reflects from the boundaries. We show that the nature of revivals for a single Majorana bound state depends uniquely on the presence of a resonant level in the lead. When two spatially separated Majorana modes are coupled to the lead, the revivals depend only on the phase difference between their host superconductors. Remarkably, the quench in this case effectively performs a fermion-parity interferometry between Majorana bound states, revealing their unique non-Abelian braiding. Using both analytical and numerical techniques, we find the pattern of fermion parity transfers following the quench, study its evolution in the presence of disorder and interactions, and thus, ascertain the fate of Majorana bound states in a Fermi sea.

cond-mat.str-el↗

arXiv2017

A generalization of Bloch's theorem for arbitrary boundary conditions: Theory

We present a generalization of Bloch's theorem to finite-range lattice systems of independent fermions, in which translation symmetry is broken only by arbitrary boundary conditions, by providing exact, analytic expressions for all energy eigenvalues and eigenstates. By transforming the single-particle Hamiltonian into a corner-modified banded block-Toeplitz matrix, a key step is a bipartition of the lattice, which splits the eigenvalue problem into a system of bulk and boundary equations. The eigensystem inherits most of its solutions from an auxiliary, infinite translation-invariant Hamiltonian that allows for non-unitary representations of translation symmetry. A reformulation of the boundary equation in terms of a boundary matrix ensures compatibility with the boundary conditions, and determines the allowed energy eigenstates. We show how the boundary matrix captures the interplay between bulk and boundary properties, leading to efficient indicators of bulk-boundary correspondence. Remarkable consequences of our generalized Bloch theorem are the engineering of Hamiltonians that host perfectly localized, robust zero-energy edge modes, and the predicted emergence, e.g. in Kitaev's chain, of localized excitations whose amplitudes decay exponentially with a power-law prefactor. We further show how the theorem yields diagonalization algorithms for the class of Hamiltonians under consideration, and use the proposed bulk-boundary indicator to characterize the topological response of a multi-band time-reversal invariant s-wave superconductor under twisted boundary conditions, showing how a fractional Josephson effect can occur without a fermionic parity switch. Finally, we establish connections to the transfer matrix method and demonstrate, using the paradigmatic Kitaev's chain example, that a non-diagonalizable transfer matrix signals the presence of solutions with a power-law prefactor.

cond-mat.stat-mech↗

arXiv2017

Exact solution of corner-modified banded block-Toeplitz eigensystems

Motivated by the challenge of seeking a rigorous foundation for the bulk-boundary correspondence for free fermions, we introduce an algorithm for determining exactly the spectrum and a generalized-eigenvector basis of a class of banded block quasi-Toeplitz matrices that we call corner-modified. Corner modifications of otherwise arbitrary banded block-Toeplitz matrices capture the effect of boundary conditions and the associated breakdown of translational invariance. Our algorithm leverages the interplay between a non-standard, projector-based method of kernel determination (physically, a bulk-boundary separation) and families of linear representations of the algebra of matrix Laurent polynomials. Thanks to the fact that these representations act on infinite-dimensional carrier spaces in which translation symmetry is restored, it becomes possible to determine the eigensystem of an auxiliary projected block-Laurent matrix. This results in an analytic eigenvector Ansatz, independent of the system size, which we prove is guaranteed to contain the full solution of the original finite-dimensional problem. The actual solution is then obtained by imposing compatibility with a boundary matrix, also independent of system size. As an application, we show analytically that eigenvectors of short-ranged fermionic tight-binding models may display power-law corrections to exponential decay, and demonstrate the phenomenon for the paradigmatic Majorana chain of Kitaev.

quant-ph↗

arXiv2016

On the Role of Self-Adjointness in the Continuum Formulation of Topological Quantum Phases

Topological quantum phases of matter are characterized by an intimate relationship between the Hamiltonian dynamics away from the edges and the appearance of bound states localized at the edges of the system. Elucidating this correspondence in the continuum formulation of topological phases, even in the simplest case of a one-dimensional system, touches upon fundamental concepts and methods in quantum mechanics that are not commonly discussed in textbooks, in particular the self-adjoint extensions of a Hermitian operator. We show how such topological bound states can be derived in a prototypical one-dimensional system. Along the way, we provide a pedagogical exposition of the self-adjoint extension method as well as the role of symmetries in correctly formulating the continuum, field-theory description of topological matter with boundaries. Moreover, we show that self-adjoint extensions can be characterized generally in terms of a conserved local current associated with the self-adjoint operator.

cond-mat.mes-hall↗

arXiv2016

Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions

We present a procedure for exactly diagonalizing finite-range quadratic fermionic Hamiltonians with arbitrary boundary conditions in one of D dimensions, and periodic in the remaining D-1. The key is a Hamiltonian-dependent separation of the bulk from the boundary. By combining information from the two, we identify a matrix function that completely characterizes the solutions, and allow to identify an indicator of bulk-boundary correspondence. As an illustration, we show how our method correctly describes the zero-energy Majorana modes of a time-reversal-invariant s-wave superconductor in a Josephson ring configuration, and predicts that a fractional 4 pi Josephson effect can only be observed in phases hosting an odd number of Majorana pairs.

cond-mat.supr-con↗

arXiv2016

Staircase of crystal phases of hard-core bosons on the Kagome lattice

We study the quantum phase diagram of a system of hard-core bosons on the Kagome lattice with nearest-neighbor repulsive interactions, for arbitrary densities, by means of the hierarchical mean field theory and exact diagonalization techniques. This system is isomorphic to the spin S=1/2 XXZ model in presence of an external magnetic field, a paradigmatic example of frustrated quantum magnetism. In the non-frustrated regime, we find two crystal phases at densities 1/3 and 2/3 that melt into a superfluid phase when increasing the hopping amplitude, in semi-quantitative agreement with quantum Monte Carlo computations. In the frustrated regime and away from half-filling, we find a series of plateaux with densities commensurate with powers of 1/3. The broader density plateaux (at densities 1/3 and 2/3) are remnants of the classical degeneracy in the Ising limit. For densities near half-filling, this staircase of crystal phases melts into a superfluid, which displays finite chiral currents when computed with clusters having an odd number of sites. Both the staircase of crystal phases and the superfluid phase prevail in the non-interacting limit, suggesting that the lowest dispersionless single-particle band may be at the root of this phenomenon.

cond-mat.str-el↗

arXiv2016

Robust Topological Degeneracy of Classical Theories

We challenge the hypothesis that the ground states of a physical system whose degeneracy depends on topology must necessarily realize topological quantum order and display non-local entanglement. To this end, we introduce and study a classical rendition of the Toric Code model embedded on Riemann surfaces of different genus numbers. We find that the minimal ground state degeneracy (and those of all levels) depends on the topology of the embedding surface alone. As the ground states of this classical system may be distinguished by local measurements, a characteristic of Landau orders, this example illustrates that topological degeneracy is not a sufficient condition for topological quantum order. This conclusion is generic and, as shown, it applies to many other models. We also demonstrate that certain lattice realizations of these models, and other theories, display a ground state entropy (and those of all levels) that is "holographic", i.e., extensive in the system boundary. We find that clock and $U(1)$ gauge theories display topological (in addition to gauge) degeneracies.

cond-mat.stat-mech↗

arXiv2016

What is a particle-conserving Topological Superfluid?

We establish a criterion for characterizing superfluidity in interacting, particle-number conserving systems of fermions as topologically trivial or non-trivial. Because our criterion is based on the concept of many-body fermionic parity switches, it is directly associated to the observation of the fractional Josephson effect and indicates the emergence of zero-energy modes that anticommute with fermionic parity. We tested these ideas on the Richardson-Gaudin-Kitaev chain, a particle-number conserving system that is solvable by way of the algebraic Bethe ansatz, and reduces to a long-range Kitaev chain in the mean-field approximation. Guided by its closed-form solution, we introduce a procedure for constructing many-body Majorana zero-energy modes of gapped topological superfluids in terms of coherent superpositions of states with different number of fermions. We discuss their significance and the physical conditions required to enable quantum control in the light of superselection rules.

cond-mat.supr-con↗

arXiv2015

Zero modes, Bosonization and Topological Quantum Order: The Laughlin State in Second Quantization

We introduce a "second-quantized" representation of the ring of symmetric functions to further develop a purely second-quantized -- or "lattice" -- approach to the study of zero modes of frustration free Haldane-pseudo-potential-type Hamiltonians, which in particular stabilize Laughlin ground states. We present three applications of this formalism. We start demonstrating how to systematically construct all zero-modes of Laughlin-type parent Hamiltonians in a framework that is free of first-quantized polynomial wave functions, and show that they are in one-to-one correspondence with dominance patterns. The starting point here is the pseudo-potential Hamiltonian in "lattice form", stripped of all information about the analytic structure of Landau levels (dynamical momenta). Secondly, as a by-product, we make contact with the bosonization method, and obtain an alternative proof for the equivalence between bosonic and fermionic Fock spaces. Finally, we explicitly derive the second-quantized version of Read's non-local (string) order parameter for the Laughlin state, extending an earlier description by Stone. Commutation relations between the local quasi-hole operator and the local electron operator are generalized to various geometries.

cond-mat.str-el↗

arXiv2015

Why are all dualities conformal? Theory and practical consequences

We relate duality mappings to the "Babbage equation" F(F(z)) = z, with F a map linking weak- to strong-coupling theories. Under fairly general conditions F may only be a specific conformal transformation of the fractional linear type. This deep general result has enormous practical consequences. For example, one can establish that weak- and strong- coupling series expansions of arbitrarily large finite size systems are trivially related, i.e., after generating one of those series the other is automatically determined through a set of linear constraints between the series coefficients. This latter relation partially solve or, equivalently, localize the computational complexity of evaluating the series expansion to a simple fraction of those coefficients. As a bonus, those relations also encode non-trivial equalities between different geometric constructions in general dimensions, and connect derived coefficients to polytope volumes. We illustrate our findings by examining various models including, but not limited to, ferromagnetic and spin-glass Ising, and Ising gauge type theories on hypercubic lattices in 1< D <9 dimensions.

cond-mat.stat-mech↗

arXiv2014

Dynamical generation of Floquet Majorana flat bands in s-wave superconductors

We present quantum control techniques to engineer flat bands of symmetry-protected Majorana edge modes in s-wave superconductors. Specifically, we show how periodic control may be employed for designing time-independent effective Hamiltonians, which support $Floquet$ $Majorana$ $flat$ $bands$, starting from equilibrium conditions that are either topologically trivial or only support individual Majorana pairs. In the first approach, a suitable modulation of the chemical potential simultaneously induces Majorana flat bands and dynamically $activates$ a pre-existing chiral symmetry which is responsible for their protection. In the second approach, the application of effective parity kicks dynamically $generates$ a desired chiral symmetry by suppressing chirality-breaking terms in the static Hamiltonian. Our results demonstrate how the use of time-dependent control enlarges the range of possibilities for realizing gapless topological superconductivity, potentially enabling access to topological states of matter that have no known equilibrium counterpart.

cond-mat.supr-con↗

arXiv2014

Quasi-Critical Brain Dynamics on a Non-Equilibrium Widom Line

Is the brain really operating at a critical point? We study the non-equilibrium properties of a neural network which models the dynamics of the neocortex and argue for optimal quasi-critical dynamics on the Widom line where the correlation length is maximal. We simulate the network and introduce an analytical mean-field approximation, characterize the non-equilibrium phase transition, and present a non-equilibrium phase diagram, which shows that in addition to an ordered and disordered phase, the system exhibits a quasiperiodic phase corresponding to synchronous activity in simulations which may be related to the pathological synchronization associated with epilepsy.

q-bio.NC↗

arXiv2014

Symmetry and Topological Order

We prove sufficient conditions for Topological Quantum Order at both zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries (that notably extend and differ from standard local gauge symmetries) and their associated defects, thus providing a unifying framework based on a symmetry principle. These symmetries may be actual invariances of the system, or may emerge in the low-energy sector. Prominent examples of Topological Quantum Order display Gauge-Like Symmetries. New systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges), discuss associated braiding, and show the insufficiency of the energy spectrum, topological entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. General symmetry considerations illustrate that not withstanding spectral gaps, thermal fluctuations may impose restrictions on certain suggested quantum computing schemes and lead to "thermal fragility". Our results allow us to go beyond standard topological field theories and engineer systems with Topological Quantum Order.

cond-mat.str-el↗

arXiv2014

Fast Quantum Methods for Optimization

Discrete combinatorial optimization consists in finding the optimal configuration that minimizes a given discrete objective function. An interpretation of such a function as the energy of a classical system allows us to reduce the optimization problem into the preparation of a low-temperature thermal state of the system. Motivated by the quantum annealing method, we present three strategies to prepare the low-temperature state that exploit quantum mechanics in remarkable ways. We focus on implementations without uncontrolled errors induced by the environment. This allows us to rigorously prove a quantum advantage. The first strategy uses a classical-to-quantum mapping, where the equilibrium properties of a classical system in $d$ spatial dimensions can be determined from the ground state properties of a quantum system also in $d$ spatial dimensions. We show how such a ground state can be prepared by means of quantum annealing, including quantum adiabatic evolutions. This mapping also allows us to unveil some fundamental relations between simulated and quantum annealing. The second strategy builds upon the first one and introduces a technique called spectral gap amplification to reduce the time required to prepare the same quantum state adiabatically. If implemented on a quantum device that exploits quantum coherence, this strategy leads to a quadratic improvement in complexity over the well-known bound of the classical simulated annealing method. The third strategy is not purely adiabatic; instead, it exploits diabatic processes between the low-energy states of the corresponding quantum system. For some problems it results in an exponential speedup (in the oracle model) over the best classical algorithms.

quant-ph↗

arXiv2014

Composite fermion-boson mapping for fermionic lattice models

We present a mapping of elementary fermion operators onto a quadratic form of composite fermionic and bosonic operators. The mapping is an exact isomorphism as long as the physical constraint of one composite particle per cluster is satisfied. This condition is treated on average in a composite particle mean-field approach, which consists of an ansatz that decouples the composite fermionic and bosonic sectors. The theory is tested on the one- and two-dimensional Hubbard models. Using a Bogoliubov determinant for the composite fermions and either a coherent or Bogoliubov state for the bosons, we obtain a simple and accurate procedure for treating the Mott insulating phase of the Hubbard model with mean-field computational cost.

cond-mat.str-el↗

arXiv2014

Degenerate Adiabatic Perturbation Theory: Foundations and Applications

We present details and expand on the framework leading to the recently introduced degenerate adiabatic perturbation theory [Phys. Rev. Lett. 104, 170406 (2010)], and on the formulation of the degenerate adiabatic theorem, along with its necessary and sufficient conditions given in [Phys. Rev. A 85, 062111 (2012)]. We start with the adiabatic approximation for degenerate Hamiltonians that paves the way to a clear and rigorous statement of the associated degenerate adiabatic theorem, where the non-abelian geometric phase (Wilczek-Zee phase) plays a central role to its quantitative formulation. We then describe the degenerate adiabatic perturbation theory, whose zeroth order term is the degenerate adiabatic approximation, in its full generality. The parameter in the perturbative power series expansion of the time-dependent wave function is directly associated to the inverse of the time it takes to drive the system from its initial to its final state. With the aid of the degenerate adiabatic perturbation theory we obtain rigorous necessary and sufficient conditions for the validity of the adiabatic theorem of quantum mechanics. Finally, to illustrate the power and wide scope of the methodology, we apply the framework to a degenerate Hamiltonian, whose closed form time-dependent wave function is derived exactly, and also to other non-exactly-solvable Hamiltonians whose solutions are numerically computed.

quant-ph↗

arXiv2014

Many-body characterization of topological superconductivity: The Richardson-Gaudin-Kitaev chain

What distinguishes trivial from topological superluids in interacting many-body systems where the number of particles is conserved? Building on a class of integrable pairing Hamiltonians, we present a number-conserving, interacting variation of the Kitaev model, the Richardson-Gaudin-Kitaev chain, that remains exactly solvable for periodic and antiperiodic boundary conditions. Our model allows us to identify fermionic parity switches that distinctively characterize topological superconductivity in interacting many-body systems. Although the Majorana zero-modes in this model have only a power-law confinement, we may still define many-body Majorana operators by tuning the flux to a fermion parity switch. We derive a closed-form expression for an interacting topological invariant and show that the transition away from the topological phase is of third order.

cond-mat.mes-hall↗

arXiv2014

Commensurate and Incommensurate States of Topological Quantum Matter

We prove numerically and by dualities the existence of modulated, commensurate and incommensurate states of topological quantum matter in simple systems of parafermions, motivated by recent proposals for the realization of such systems in mesoscopic arrays. In two space dimensions, we obtain the simplest representative of a topological universality class that we call Lifshitz. It is characterized by a topological tricritical point where a non-locally ordered homogeneous phase meets a disordered phase and a third phase that displays modulations of a non-local order parameter.

cond-mat.str-el↗