Holographic Symmetries and Generalized Order Parameters for Topological Matter

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 system's 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.Comment: v2, 10 pages, 3 figures. Accepted for publication in Physical Review B Rapid Communication

Unified approach to Quantum and Classical 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 were guessed in the past- can be derived in our formalism. We obtain new self-dualities for four-dimensional Abelian gauge field theories.Comment: 4+3 pages, 3 figure

The non-Abelian Duality Problem

We exploit a new theory of duality transformations to construct dual representations of models incompatible with traditional duality transformations. Hence we obtain a solution to the long-standing problem of non-Abelian dualities that hinges on two key observations: (i) from the point of view of dualities, whether the group of symmetries of a model is or is not Abelian is unimportant, and (ii) the new theory of dualities that we exploit includes traditional duality transformations, but also introduces in a natural way more general transformations.Comment: 5 pages (2 figures) plus Supplemental materia

Thermal conductance as a probe of the non-local order parameter for a topological superconductor with gauge fluctuations

We investigate the effect of quantum phase slips on a helical quantum wire coupled to a superconductor by proximity. The effective low-energy description of the wire is that of a Majorana chain minimally coupled to a dynamical $\mathbb{Z}_2$ gauge field. Hence the wire emulates a matter-coupled gauge theory, with fermion parity playing the role of the gauged global symmetry. Quantum phase slips lift the ground state degeneracy associated with unpaired Majorana edge modes at the ends of the chain, a change that can be understood as a transition between the confined and the Higgs-mechanism regimes of the gauge theory. We identify the quantization of thermal conductance at the transition as a robust experimental feature separating the two regimes. We explain this result by establishing a relation between thermal conductance and the Fredenhagen-Marcu string order-parameter for confinement in gauge theories. Our work indicates that thermal transport could serve as a measure of non-local order parameters for emergent or simulated topological quantum order.Comment: 5 pages, 2 figures; v2: different introduction, added references, updated figure 2; published version to appear in PR

Statistical translation invariance protects a topological insulator from interactions

We investigate the effect of interactions on the stability of a disordered, two-dimensional topological insulator realized as an array of nanowires or chains of magnetic atoms on a superconducting substrate. The Majorana zero-energy modes present at the ends of the wires overlap, forming a dispersive edge mode with thermal conductance determined by the central charge $c$ of the low-energy effective field theory of the edge. We show numerically that, in the presence of disorder, the $c=1/2$ Majorana edge mode remains delocalized up to extremely strong attractive interactions, while repulsive interactions drive a transition to a $c=3/2$ edge phase localized by disorder. The absence of localization for strong attractive interactions is explained by a self-duality symmetry of the statistical ensemble of disorder configurations and of the edge interactions, originating from translation invariance on the length scale of the underlying mesoscopic array.Comment: 5+2 pages, 8 figure

Phase transitions in the Zp\mathbb{Z}_p and U(1) clock models

Quantum phase transitions are studied in the non-chiral $p$-clock chain, and a new explicitly U(1)-symmetric clock model, by monitoring the ground-state fidelity susceptibility. For $p\ge 5$, the self-dual $\mathbb{Z}_p$-symmetric chain displays a double-hump structure in the fidelity susceptibility with both peak positions and heights scaling logarithmically to their corresponding thermodynamic values. This scaling is precisely as expected for two Beresinskii-Kosterlitz-Thouless (BKT) transitions located symmetrically about the self-dual point, and so confirms numerically the theoretical scenario that sets $p=5$ as the lowest $p$ supporting BKT transitions in $\mathbb{Z}_p$-symmetric clock models. For our U(1)-symmetric, non-self-dual minimal modification of the $p$-clock model we find that the phase diagram depends strongly on the parity of $p$ and only one BKT transition survives for $p\geq 5$. Using asymptotic calculus we map the self-dual clock model exactly, in the large $p$ limit, to the quantum $O(2)$ rotor chain. Finally, using bond-algebraic dualities we estimate the critical BKT transition temperatures of the classical planar $p$-clock models defined on square lattices, in the limit of extreme spatial anisotropy. Our values agree remarkably well with those determined via classical Monte Carlo for isotropic lattices. This work highlights the power of the fidelity susceptibility as a tool for diagnosing the BKT transitions even when only discrete symmetries are present.Comment: 17 pages, 14 figure

Topological phases in two-dimensional arrays of parafermionic zero modes.

Theoretical Physic

Dualities and the phase diagram of the pp-clock model

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 $p \geq 5$. 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 $p\geq 5$, is critical (massless) with decaying power-law correlations.Comment: 48 pages, 5 figures. Submitted to Nuclear Physics