For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of A. In particular, this gives rise to a gauge symmetry described by the action of Z. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries that do not commute with the topological invariants represented by elements of Z are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem.
For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of A. In particular, this gives rise to a gauge symmetry described by the action of Z. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries that do not commute with the topological invariants represented by elements of Z are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem.
The relation between the manifold topology, observables, and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated with positions and displacements on the manifold. The guiding, physically motivated, principles are (i) locality, i.e., the generating role of the algebras localized in small, topologically trivial, regions, (ii) diffeomorphism covariance, which guarantees the intrinsic character of the analysis, and (iii) the exclusion of additional local degrees of freedom with respect to the Schrödinger representation. The locally normal representations of the resulting observable algebra are classified by unitary representations of the fundamental group of the manifold, which actually generate an observable, “topological,” subalgebra. The result is confronted with the standard approach based on the introduction of the universal covering M̃ of M and on the decomposition of L2(M̃) according to the spectrum of the fundamental group, which plays the role of a gauge group. It is shown that in this way one obtains all the representations of the observables iff the fundamental group is amenable. The implications on the observability of the permutation group in particle statistics are discussed.
Gauge theories and topological phases play a fundamental role in different areas of physics. The first ones are at the basis of the Standard Model in the field of particle physics, describing the electroweak and strong interactions through a non-Abelian gauge theory. In condensed matter and statistical physics, gauge theories arise as low-energy effective descriptions of strongly correlated phenomena, such as quantum spin liquids and quantum Hall effect. In this realm there is a strong connection with topological phases and order, as emergent gauge fermions and bosons often describe collective excitations of new exotic states of matter of spin models. The discretization on a lattice is a possible way of dealing with the strongly coupled nature of these theories, due to the fact that this formulation at finite volume provides natural regularizing cut-offs, i.e. the lattice size and spacing. This allows for the investigation of different non-perturbative properties both numerically and analytically. Despite the success of these methods, there are various aspects which remain intractable due to the sign or complex action problems, like the out of equilibrium real time evolution or the analysis of quantum chromodynamics with finite chemical potential. In this respect, quantum simulators of many-body systems to simulate high energy physics arise as promising alternatives to face these problems in the near future. They are quantum systems that can be controlled and used to simulate more complicated systems, whose properties could not be analysed with classical computational, experimental or theoretical tools. In the last decades, there has been a huge development in the fields of quantum optics and atomic physics, allowing for the realization highly precise and controllable platforms by means of trapped ions, superconducting circuits, Rydberg atoms and ultracold atoms in optical lattices. This work of Thesis is part of these fields and has different purposes, all of them connected with the study of gauge theories and topological phases. Firstly, we want to develop a reformulation of lattice gauge theories in terms of gauge invariant fields, in a way to deal solely with physical variables directly in the action. Among several possible advantages, a crucial point is that this can be particularly helpful for the construction of consistent approximation schemes, such as mean-field theories, in order to understand and capture some of the physical features of the theory and make the mean-field approximation consistent with the Elitzur theorem, stating that a local gauge symmetry can not be spontaneously broken. As a second point, we face the problem of simulating higher dimensional gauge theories with ultracold atoms. One of the challenges in more than one dimension is indeed the realization of plaquette interaction terms in the Hamiltonian of lattice gauge theories: these can be engineered through four correlated hoppings in perturbation theory, or by means of constrained hoppings in the dual formulation. In this respect, our main target is to set up an ultracold atomic platform generating the plaquette term in two dimensions using only two correlated hoppings, and protecting gauge invariance through angular momentum conservation. The last point we address in the Thesis is related to the analysis of particles moving in static background gauge potentials, i.e. when the considered gauge field has no dynamical term in the action of the system. Indeed, the dynamics of quantum particles in the presence of static gauge fields gives rise to intriguing physical phenomena. In particular, using ultracold atom setups, the realization of artificial gauge potentials can be used to investigate the physics of topological semimetals, such as Weyl or Dirac type. In the last ten years, a lot of attention has been paid to their characterization, due to the appearence of clear theoretical predictions and very well-controlled experimental techniques. We investigate here the relation between topological phase transitions and van Hove singularities, i.e. the discontinuities in the energy derivative of the density of states, in three-dimensional gapless systems. In such materials, topological phase transitions can be defined by changes of the topological invariants of the Fermi sheets, happening at specific singular points. Moreover, Fermi surface singularities result in the presence of the so-called van Hove points. We then present a general argument to relate topological phase transitions and van Hove singularities, and show observable consequences that are related to the transport properties of the system. We exemplify our argument in Weyl systems by analyzing the three-dimensional Hofstadter model for various commensurate fluxes, which offers the opportunity to consider different kinds of Weyl metals and to understand the features of their density of states.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.
Abstract In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by aG-crossed braided extension C⊆CG× C⊆CG× , we show that physical considerations require that a connected étale algebraA∈ C C admit aG-equivariant algebra structure for symmetry to be preserved under condensation ofA. Given any categorical actionG→EqBr( C C ) such thatg(A) ≅Afor allg∈G, we show there is a short exact sequence whose splittings correspond toG-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation ofA, while inequivalent splittings of the sequence correspond to different SETOs resulting from the anyon-condensation transition. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of CAloc CAloc , and gauging this symmetry commutes with anyon condensation.
Group theory of the Morse potential.- Four Euclidean conformal group approach to the multiphoton processes in the H-atom.- Landau theory, symmetry breaking and the chain criterion.- The configuration dN in cubic symmetry: A symmetry adapted weak field approach.- Group theory of the effective potential.- Bloch theorem for crystals with structural distortions.- Global symmetries of spin systems defined on abelian manifolds.- Phases and conjugacy classes.- The probability density in non relativistic quantum mechanics and the (extended) Galilei group.- Quaternionic quantum mechanics and Adler's chromostatics.- Coherent angular momentum states for the isotropic oscillator.- How to measure the canonical momentum p and operators p + f(q).- A uniqueness result for the segal quantization of a classical system with symmetries.- On the singular points of the matrix elements of the resolvent operator.- Limitations on quantum measuring processes.- Classical and quantum physics on phase space: The state spaces.- Quantum dynamical semigroups and complete positivity. An application to isotropic spin relaxation.- On geometric quantization of the rigid body.- A unified theory of simple dynamical systems.- Geometry of the time-dependent variational principle in quantum mechanics.- Variational characterization of Hamiltonian vectorfields as lie algebra of canonical diffeomorphisms.- The lie theory of extended groups in classical mechanics - Is it of relevance to quantum mechanics?.- On the structure of phase space.- Quantization as mapping and as deformation.- Metric and group structures from quantum physics.- Spontaneous symmetry breaking and restricted superposition principle.- Many-body quantum mechanics as symplectic dynamics.- A nonlinear superposition principle for Riccati equations of the conformal type.- Backlund transformations, conservation laws and lie symmetries for the generalized nonlinear Schrodinger equation.- Killing tensors and variable separation for Hamilton-Jacobi equations.- Lie pseudogroups and gauge theory.- Conformal invariance and symmetry breaking for the one-dimensional wave equation.- The flavour sequence and superselection rules.- The Zitterbewegung of the electron and its compact phase space.- New experimental evidence for a new interpretation of flavor SU(3).- Baryon magnetic moments in broken SU(6).- The Racah-Wigner algebra and the masses of exotic hadrons.- Simple construction of a relativistic wave equation with arbitrary mass and spin and belinfante's 1/S-conjecture.- Poincare group, symplectic group, and relativistic extended hadrons consisting of spin 1/2 quarks.- An operator S matrix theory of the strongly interacting particles.- Generalized Galilei invariant partial wave expansions of the scattering amplitude for collisions between two particles with arbitrary spin.- Ternary algebras as the basis of a dynamical theory of subconstituents.- Solitons as sections in non trivial bundles.- Effects of heavy colored higgs scalars in grand unified theories.- Categories of nonlinear group realizations: A possible explanation for the multiple states of charge.- Classical solutions of non-linear ?-models and their quantum fluctuations.- Higgs representations for SO(n) gauges theories.- Multivortex configurations in the Abelian Higgs model.- Quark confinement in field theories with discrete gauge symmetry Z(3).- Proliferation of effective fields by spontaneous symmetry breakdown.- Explicit solutions for the scalar coupled Yang-Mills system in Minkowski space.- Dynamical supersymmetry SU(6/4) in nuclei.- Group theory of the IBA and its applications.- Bases for irreps of U(6) and three body problems.- Intrinsic sphape of nuclei.- Group theory and nuclear high spin phenomena.- SU3 symmetry in resonating group calculations. Toward a microscopic theory of 12C + 12C molecular resonances.- Collective motion, composite particle structure, and symplectic groups in nuclei.- Group theory and confrontation of nuclear collective models.- Matrix elements of generators of U(6) in a U(6)?U(3)?U(2) basis.- Hartree-Fock one-body dynamics and U(n) co-adjoint orbits.- Group-theoretical aspects of the many-body problem in nuclear theory.- The continuing quest for exact solutions of Einstein's field eouations - - new goals for the eighties.- All non-twisting N's with cosmological constant.- Can quantum effects prevent spacetime collapse?.- Complex plane representation of the Geroch group and a proof of a geroch conjecture.- The Geroch group and soliton solutions of the stationary axially symmetric Einstein's equations.- Geometry of projective asymptotic twistor space.- A truly relativistic approach of the concept of time.- Bases in Sp(2n) representation space.- Semiunitary projective representations of non-connected lie groups: A criterion for the choice of the unitary subgroup.- SU(mn) ? SU(m) x SU(n) isoscalar factors and S(f1+f2) ? S(f1) x S(f2) isoscalar factors.- Structure of enveloping algebras of low-rank groups.- Computer programs for the reduction of symmetrised nth powers of space group irreducible representations using gard's subgroup method.- Some remarks on the uniqueness and reality of Clebsch-Gordan coefficients for corepresentations.- On some special relations involving 3-jm symbols.- Two problems related to the Stone-von Neumann-Mackey uniqueness theorem.- Non-complex representations and their relation to antiunitary symmetry.- Representation theory of compact groups.- Representation theory of compact groups / Shift operators for discrete unitary representations of O(p,q).- Representation theory of compact groups / Maximal Abelian subalgebras of sp(2n,?) and their applications in physics.- Representation theory of compact groups / Polynomial space group tensors.- A permutation for SUn consistent with duality.- Integral transform representation of SL (2,R).- The theory of G? supermanifolds.- Casimir operations and the relations of the simplest supersymmetric superalgebra.- On realizations of graded lie algebras and on properties of d-polynomials.- On gauge formulations of gravitation theories.- Lagrangian formalism for superfields with internal symmetry.- Causal propagation in second order Lorentz-invariant wave equations.- The exceptional group E8 for grand unification.- Completely integrable N-body problems in three-dimension and their relativistic generalization.- Ward-Takashi identities as a necessary and sufficient condition for gauce invariance.- Supersymmetric generalization of Riemannian symmetric pairs.- Composite gauge fields and Riemannian geometry.- Gauge-theory ghosts and unitarity - Progress in the geometrical treatment.- Color in internal supersymmetry unification.- Canonical (possibly Lagrangian) realizations of the Poincare' group with increasing mass-spin trajectories.- Similarity analysis of wave propagation in axially nonlinear nonhomogeneous systems.