A new "bond-algebraic" approach to duality transformations provides a very powerful technique to analyze elementary excitations in the classical twodimensional 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.

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 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 like 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 2 Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations like dual variables and Jordan-Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits 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.

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.

What distinguishes trivial superfluids from topological superfluids 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 fermion parity switches that distinctively characterize topological superconductivity (fermion superfluidity) in generic 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. DOI: 10.1103/PhysRevLett.113.267002 PACS numbers: 74.20.-z, 03.65.Vf, 74.45.+c, 74.90.+n In recent years, the physics of Majorana zero-energy modes has become a key subfield of condensed matter physics [1][2][3][4]. On the theory side, a central result is the bulkboundary correspondence [5] that associates Majorana zero modes to the boundary of (or defects in) a topologically nontrivial superconductor, with the Kitaev chain as a prototypical example [6]. The mathematical formalism underlying this correspondence relies on the symmetries and topological invariants of the Bogoliubov-de Gennes equation [7], a mean-field description of the superconducting state in which the conservation of the number of fermions (a continuous symmetry) is broken down to a discrete symmetry, the conservation of fermion-number parity. Majorana zero modes are directly linked to the spontaneous breaking of this residual discrete symmetry [8].As the experimental side of Majorana physics continues to develop [9][10][11][12][13][14], it becomes crucial to unveil how much of the mean-field picture survives beyond its natural limits. This has motivated recent studies [15][16][17][18][19][20][21], with the focus on the anomalous 2Φ 0 ¼ h=e flux periodicity of the Josephson effect-the hallmark of a topological superconductor [6].A main thrust of this Letter is the characterization of interacting many-body, number-conserving, topological superconductors, or superfluids, leading to a subsequent analysis on the meaning of Majorana zero modes beyond mean field. The theoretical study of any interacting quantum system is hampered by the exponential growth of the Hilbert space with the number of particles. An additional complication of superconducting systems is the lack of simple principles to guide the design of particlenumber conserving models, in which the phase of the order parameter is not a good quantum number. To overcome both obstacles, we have constructed an exactly solvable, number-conserving variation of the Kitaev chain. Because our model belongs to a class of integrable pairing models [22-24] based on the s-wave reduced BCS Hamiltonian firs...

The LaAlO 3 =SrTiO 3 interface hosts a two-dimensional electron system that is unusually sensitive to the application of an in-plane magnetic field. Low-temperature experiments have revealed a giant negative magnetoresistance (dropping by 70%), attributed to a magnetic-field induced transition between interacting phases of conduction electrons with Kondo-screened magnetic impurities. Here we report on experiments over a broad temperature range, showing the persistence of the magnetoresistance up to the 20 K rangeindicative of a single-particle mechanism. Motivated by a striking correspondence between the temperature and carrier density dependence of our magnetoresistance measurements we propose an alternative explanation. Working in the framework of semiclassical Boltzmann transport theory we demonstrate that the combination of spin-orbit coupling and scattering from finite-range impurities can explain the observed magnitude of the negative magnetoresistance, as well as the temperature and electron density dependence. The mobile electrons at the LaAlO 3 =SrTiO 3 (LAO=STO) interface [1] display an exotic combination of superconductivity [2,3] and magnetic order [4][5][6][7]. The onset of superconductivity at sub-Kelvin temperatures appears in an interval of electron densities where the effect of Rashba spin-orbit coupling on the band structure at the Fermi level is strongest [8,9], but whether this correlation implies causation remains unclear.Transport experiments above the superconducting transition temperature have revealed a very large ("giant") drop in the sheet resistance of the LAO=STO interface upon application of a parallel magnetic field [10][11][12][13]. An explanation has been proposed [13,14] in terms of the Kondo effect: Variation of the electron density or magnetic field drives a quantum phase transition between a highresistance correlated electronic phase with screened magnetic impurities and a low-resistance phase of polarized impurity moments. The relevance of spin-orbit coupling for magnetotransport is widely appreciated [10,[14][15][16][17][18][19], but it was generally believed to be too weak an effect to provide a single-particle explanation of the giant magnetoresistance.In this work we provide experimental data (combining magnetic field, gate voltage, and temperature profiles for the resistance of the LAO=STO interface) and theoretical calculations that support an explanation fully within the single-particle context of Boltzmann transport. The key ingredients are the combination of spin-orbit coupling, band anisotropy, and finite-range electrostatic impurity scattering.The thermal insensitivity of the giant magnetoresistance [10,11], in combination with a striking correspondence that we have observed between the gate voltage and temperature dependence of the effect, are features that are difficult to reconcile with the thermally fragile Kondo interpretationbut fit naturally in the semiclassical Boltzmann description.We first present the experimental data and then turn to the theoretical de...