Emilio Cobanera

The bond-algebraic approach to dualities

An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the ℤ2 Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan–Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long-standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.

Dualities and the phase diagram of the p-clock model

Abstract A new “bond-algebraic” approach to duality transformations provides a very powerful technique to analyze elementary excitations in the classical two-dimensional XY and p-clock models. By combining duality and Peierls arguments, we establish the existence of non-Abelian symmetries, the phase structure, and transitions of these models, unveil the nature of their topological excitations, and explicitly show that a continuous U(1) symmetry emerges when p ⩾ 5 . This latter symmetry is associated with the appearance of discrete vortices and Berezinskii–Kosterlitz–Thouless-type transitions. We derive a correlation inequality to prove that the intermediate phase, appearing for p ⩾ 5 , is critical (massless) with decaying power-law correlations.

Arbitrary dimensional Majorana dualities and architectures for topological matter

Motivated by the prospect of attaining Majorana modes at the ends of nanowires, we analyze interacting Majorana systems on general networks and lattices in an arbitrary number of dimensions, and derive various universal spin duals. Such general complex Majorana architectures (other than those of simple square or other crystalline arrangements) might be of empirical relevance. As these systems display low-dimensional symmetries, they are candidates for realizing topological quantum order. We prove that (a) these Majorana systems, (b) quantum Ising gauge theories, and (c) transverse-field Ising models with annealed bimodal disorder are all dual to one another on general graphs. As any Dirac fermion (including electronic) operator can be expressed as a linear combination of two Majorana fermion operators, our results further lead to dualities between interacting Dirac fermionic systems. The spin duals allow us to predict the feasibility of various standard transitions as well as spin-glass type behavior in {\it interacting} Majorana fermion or electronic systems. Several new systems that can be simulated by arrays of Majorana wires are further introduced and investigated: (1) the {\it XXZ honeycomb compass} model (intermediate between the classical Ising model on the honeycomb lattice and Kitaevs honeycomb model), (2) a checkerboard lattice realization of the model of Xu and Moore for superconducting

Unified Approach to Quantum and Classical Dualities

(p+ip)

Thermodynamic signatures of edge states in topological insulators

arrays, and a (3) compass type two-flavor Hubbard model with both pairing and hopping terms. By the use of dualities, we show that all of these systems lie in the 3D Ising universality class. We discuss how the existence of topological orders and bounds on autocorrelation times can be inferred by the use of symmetries and also propose to engineer {\it quantum simulators} out of these Majorana networks.

Effective and exact holographies from symmetries and dualities

We show how classical and quantum dualities, as well as duality relations that appear only in a sector of certain theories (emergent dualities), can be unveiled, and systematically established. Our method relies on the use of morphisms of the bond algebra of a quantum Hamiltonian. Dualities are characterized as unitary mappings implementing such morphisms, whose even powers become symmetries of the quantum problem. Dual variables, which have been guessed in the past, can be derived in our formalism. We obtain new self-dualities for four-dimensional Abelian gauge field theories.

Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions.

Topological insulators are states of matter distinguished by the presence of symmetry protected metallic boundary states. These edge modes have been characterised in terms of transport and spectroscopic measurements, but a thermodynamic description has been lacking. The challenge arises because in conventional thermodynamics the potentials are required to scale linearly with extensive variables like volume, which does not allow for a general treatment of boundary effects. In this paper, we overcome this challenge with Hill thermodynamics. In this extension of the thermodynamic formalism, the grand potential is split into an extensive, conventional contribution, and the subdivision potential, which is the central construct of Hills theory. For topologically non-trivial electronic matter, the subdivision potential captures measurable contributions to the density of states and the heat capacity: it is the thermodynamic manifestation of the topological edge structure. Furthermore, the subdivision potential reveals phase transitions of the edge even when they are not manifested in the bulk, thus opening a variety of new possibilities for investigating, manipulating, and characterizing topological quantum matter solely in terms of equilibrium boundary physics.

Holographic symmetries and generalized order parameters for topological matter

Abstract The theoretical basis of the phenomenon of effective and exact dimensional reduction, or holographic correspondence, is investigated in a wide variety of physical systems. We first derive general inequalities linking quantum systems of different spatial (or spatio-temporal) dimensionality, thus establishing bounds on arbitrary correlation functions. These bounds enforce an effective dimensional reduction and become most potent in the presence of certain symmetries. Exact dimensional reduction can stem from a duality that (i) follows from properties of the local density of states, and/or (ii) from properties of Hamiltonian-dependent algebras of interactions. Dualities of the first type (i) are illustrated with large- n vector theories whose local density of states may remain invariant under transformations that change the dimension. We argue that a broad class of examples of dimensional reduction may be understood in terms of the functional dependence of observables on the local density of states. Dualities of the second type (ii) are obtained via bond algebras, a recently developed algebraic tool. We apply this technique to systems displaying topological quantum order, and also discuss the implications of dimensional reduction for the storage of quantum information.

Generalization of Bloch's theorem for arbitrary boundary conditions: Theory

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 fully characterizes the solutions, and may be used to construct an efficiently computable indicator of bulk-boundary correspondence. As an illustration, we show how our approach correctly describes the zero-energy Majorana modes of a time-reversal-invariant s-wave two-band superconductor in a Josephson ring configuration, and predicts that a fractional 4π-periodic Josephson effect can only be observed in phases hosting an odd number of Majorana pairs per boundary.

Quantum Brownian motion in a Landau level

We introduce a universally applicable method, based on the bond-algebraic theory of dualities, to search for generalized order parameters in disparate systems including non-Landau systems with topological order. A key notion that we advance is that of {\em holographic symmetry}. It reflects situations wherein global symmetries become, under a duality mapping, symmetries that act solely on the systems boundary. Holographic symmetries are naturally related to edge modes and localization. The utility of our approach is illustrated by systematically deriving generalized order parameters for pure and matter-coupled Abelian gauge theories, and for some models of topological matter.