Non-integrable Phases Research Articles

Entanglement and thermodynamic entropy in a clean many-body-localized system

Whether or not the thermodynamic entropy is equal to the entanglement entropy of an eigenstate is of fundamental interest, and is closely related to the eigenstate thermalization hypothesis (ETH). However, this has never been exploited as a diagnostic tool in many-body- localized (MBL) systems. In this work, we perform this diagnostic test on a clean interacting system (subjected to a static electric field) that exhibits three distinct phases: integrable, non-integrable ergodic and non-integrable MBL. We find that in the non-integrable phase, the equivalence between the thermodynamic entropy and the entanglement entropy of individual eigenstates holds. In sharp contrast, in the integrable and non-integrable MBL phases, the entanglement entropy shows large eigenstate-to-eigenstate fluctuations, and differs from the thermodynamic entropy. Thus the non-integrable MBL phase violates ETH similar to an integrable system; however, a key difference is that the magnitude of the entanglement entropy in the MBL phase is significantly smaller than in the integrable phase, where the entanglement entropy is of the same order of magnitude as in the non-integrable phase, but with a lot of eigenstate-to-eigenstate fluctuations. Quench dynamics from an initial CDW state independently supports the validity of the ETH in the ergodic phase and its violation in the MBL phase.

Evolutions of Yang Phase Under Cyclic Condition and Adiabatic Condition

There are three non-integrable phases in literatures: Berry phase,Aharonov–Anandan phase, and Yang phase. This article discussesthe evolutions of Yang phase under the cyclic condition and theadiabatic condition for the general time-dependent harmonicoscillator, thus reveals the intimate relations betweenthese three non-integrable phases.

Second-quantized formulation of geometric phases

The level crossing problem and associated geometric terms are neatly formulated by the second quantized formulation. This formulation exhibits a hidden local gauge symmetry related to the arbitrariness of the phase choice of the complete orthonormal basis set. By using this second quantized formulation, which does not assume adiabatic approximation, a convenient exact formula for the geometric terms including off-diagonal geometric terms is derived. The analysis of geometric phases is then reduced to a simple diagonalization of the Hamiltonian, and it is analyzed both in the operator and path integral formulations. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval $T$. The integrability of Schr\"{o}dinger equation and the appearance of the seemingly non-integrable phases are thus consistent. The topological proof of the Longuet-Higgins' phase-change rule, for example, fails in the practical Born-Oppenheimer approximation where a large but finite ratio of two time scales is involved and $T$ is identified with the period of the slower system. The difference and similarity between the geometric phases associated with level crossing and the exact topological object such as the Aharonov-Bohm phase become clear in the present formulation. A crucial difference between the quantum anomaly and the geometric phases is also noted.

Remark on Relations Between Different Non-integrable Phases

There are three non-integrable phases in literatures: Berry phase,Aharonov–Anandan phase, and Yang phase. This article discusses thedefinitions and relations between these three non-integrable phases.

Effects of fermion masses and twisting on non-integrable phases on compact extra dimensions

The effective potential for the Wilson loop in the SU(2) gauge theory with N f massive fundamental and N a massive adjoint fermions on S 1 × M 4 is computed at the one-loop level, assuming periodic boundary conditions for the gauge field and general boundary conditions for fermions. It is shown that there are critical values for the bare mass, and the boundary condition parameter for the adjoint fermions, beyond which the symmetry pattern changes. However, neither bare mass, nor the boundary condition parameter for the fundamental fermion play any role on the vacuum structure, thus the symmetry breaking pattern. When the two different types of fermions with equal masses exist together the pattern of the fundamental fermion dominate, and SU(2) gauge symmetry remains intact independent of the fermion masses.

Triality, biquaternion and vector representation of the Dirac equation

The triality properties of Dirac spinors are studied, including a construction of the algebra of (complexified) biquaternion. A bilinear law of composition for biquaternion is defined by means of Levi-civita symbol and Lorentzian metric only. It is proved that there exists a vector-representation of Dirac spinors. The massive Dirac equation in the vector-representation is actually self-dual. A chiral transformations for spinors is equivalent to U(1) transformations for corresponding complex vectors. The Dirac’s idea of non-integrable phases is used to study the behavior of massive term.

Gauge symmetry breaking through the Hosotani mechanism in softly broken supersymmetric QCD

Gauge symmetry breaking through the Hosotani mechanism (the dynamics of nonintegrable phases) in softly broken supersymmetric QCD with ${N}_{F}^{\mathrm{fd}}$ flavors is studied. For $N=\mathrm{even},$ there is a single $\mathrm{SU}(N)$ symmetric vacuum state, while for $N=\mathrm{odd},$ there is a doubly degenerate $\mathrm{SU}(N)$ symmetric vacuum state in the model. We also study generalized supersymmetric QCD by adding ${N}_{F}^{\mathrm{adj}}$ numbers of massless adjoint matter. The gauge symmetry breaking pattern such as $\mathrm{SU}(3)\ensuremath{\rightarrow}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}U(1)$ is possible for appropriate choices of the matter content and values of the supersymmetry breaking parameter. The massless state of the adjoint Higgs scalar is also discussed in the models.

Dynamics of nonintegrable phases in softly broken supersymmetric gauge theory with massless adjoint matter

We study $\mathrm{SU}(N)$ supersymmetric Yang-Mills theory with massless adjoint matter defined on ${M}^{3}\ensuremath{\bigotimes}{S}^{1}.$ The $\mathrm{SU}(N)$ gauge symmetry is broken maximally to ${U(1)}^{N\ensuremath{-}1},$ independent of the number of flavor and the boundary conditions of the fields associated with the Scherk-Schwarz mechanism of supersymmetry breaking. The mass of the Higgs scalar is generated through quantum corrections in the extra dimension. The quantum correction can become manifest by a finite Higgs boson mass at low energies even in the limit of small extra dimensions thanks to the supersymmetry breaking parameter of the Scherk-Schwarz mechanism.

Recurrent dynamical symmetry breaking and restoration by Wilson lines at finite densities on a torus

In this paper we derive the general expression of a one-loop effective potential of the nonintegrable phases of Wilson lines for an SU(N) gauge theory with a massless adjoint fermion defined on the spactime manifold ${R}^{1,d\ensuremath{-}3}\ifmmode\times\else\texttimes\fi{}{T}^{2}$ at finite temperature and fermion density. The phase structure of the vacuum is presented for the case with $d=4$ and $N=2$ at zero temperature. It is found that gauge symmetry is broken and restored alternately as the fermion density increases, a feature not found in the Higgs mechanism. It is the manifestation of the quantum effects of the nonintegrable phases.

Erratum: Softly broken supersymmetric gauge theories through compactifications [Phys. Rev. D58, 026004 (1998)

Effects of boundary conditions of fields for compactified space directions on the supersymmetric gauge theories are discussed. For general and possible boundary conditions the supersymmetry is explicitly broken to yield universal soft supersymmetry breaking terms, and the gauge symmetry of the theory can also be broken through the dynamics of non-integrable phases, depending on number and the representation under the gauge group of matters. The 4-dimensional supersymmetric QCD is studied as a toy model when one of the space coordinates is compactified on $S^1$.

Softly broken supersymmetric gauge theories through compactifications

The effects of boundary conditions of fields for compactified space directions on supersymmetric gauge theories are discussed. For general and possible boundary conditions supersymmetry is explicitly broken to yield universal soft supersymmetry breaking terms, and the gauge symmetry of the theory can also be broken through the dynamics of nonintegrable phases, depending on the number and the representation under the gauge group of matter. Four-dimensional supersymmetric QCD is studied as a toy model when one of the space coordinates is compactified on ${S}^{1}.$

Quantum group symmetry in multiple Chern - Simons theory on a torus

We discuss $w_\infty$ and $sl_q(2)$ symmetries in multiple Chern-Simons theory on a torus. It is shown that these algebraic structures arise from the dynamics of the non-integrable phases of the Chern-Simons fields. The generators of these algebras are constructed from the Wilson line operators corresponding to these phases. The vacuum states form the basis of cyclic representation of $sl_q (2)$.

Comment on "Operator algebra in Chern-Simons theory on a torus"

It is shown that "nonintegrable phases of Wilson line integrals" are not true dynamical variables in Chern-Simons field theory.

NONINTEGRABLE PHASE DYNAMICS IN TWO-DIMENSIONAL GAUGE MODELS

In two-dimensional gauge models the nonintegrable phases of the line Wilson integrals are the only physical gauge field degrees of freedom, providing the gauge field diminishes rather rapidly at spatial infinities. We construct a “physical” quantum Hamiltonian in terms of the physical degrees of freedom for the Schwinger model and the chiral Schwinger model. We show that in the latter case the physical quantum picture can be formulated in two equivalent ways: either in terms of the fermionic matter degrees of freedom moving in a linearly rising background electric field, or in terms of matter fields with exotic statistics.

DYNAMICS OF MAGNETIC FIELDS IN MAXWELL, YANG–MILLS AND CHERN–SIMONS THEORIES ON THE TORUS

We reconsider the problem of uniform magnetic fields passing perpendicularly through a two-torus. We focus on dynamical effects from nonintegrable phases on the torus at nonzero B and from magnetic fields themselves in the vacuum. The spectrum is computed and is shown to be always independent of the nonintegrable phases on the torus. It is concluded that a Chern–Simons term will always be induced by radiative corrections to fermions on the torus when B ≠ 0. The special case of an electromagnetically uncharged anyon gas is noted and shown to be a system whose spectrum can depend on the nonintegrable phases in the two torus directions, subject to a consistency requirement. In three and four dimensions dynamical symmetry breaking of non-Abelian fields and associated condensate formation is possible by radiative corrections. The classification of non-Abelian magnetic fields in terms of "flux integers" is discussed, and a method for obtaining such integers for an arbitrary gauge algebra is presented which provides a rigorous generalization of 't Hooft's su(2) classification. Finally, we remark on the off-shell effective action in Chern–Simons theory.

ANYON EQUATION ON A TORUS

Starting from the quantum field theory of nonrelativistic matter on a torus interacting with Chern-Simons gauge fields, we derive the Schrödinger equation for an anyon system. The nonintegrable phases of the Wilson line integrals on a torus play an essential role. In addition to generating degenerate vacua, they enter in the definition of a many-body Schrödinger wave function in quantum mechanics, which can be defined as a regular function of the coordinates of anyons. It obeys a non-Abelian representation of the braid group algebra, being related to Einarsson’s wave function by a singular gauge transformation.

Radiatively induced Chern-Simons terms on the torus.

A Chern-Simons term is induced by integrating over fermions in the effective action of a (2+1)-dimensional field theory on a torus. It is shown that the Chern-Simons coefficient is functionally dependent on the nonintegrable phases of the torus as well as on the finite temperature and density. In the non-Abelian case such nonintegrable phases inhibit the generation of a Chern-Simons term in the most general case.

Symmetry breaking by Wilson lines and finite-temperature effects

In this paper we study gauge symmetry breaking and restoration in an SU( N) gauge theory defined on space-time manifold R 1, d−2 × S 1 when fermions obey non-trivial boundary conditions. We derive the one-loop effective potentials for the non-integrable phases of Wilson lines at both zero and finite temperature, using the zeta-function technique. It is shown that gauge symmetry can be broken with fermions in the adjoint representation, but is restored at high temperature. Detailed analyses are given for the groups SU(2), SU(3) and SU(5) on R 1,2 × S 1.

A note on non-integrable phases and coherent states

The relationship between non-integrable phases and generalised coherent states is discussed.

Yang-Mills theory on a circle

In SU(N) gauge theory on the manifold S1×R1 (circle×time) the non-integrable phases of the Wilson line integrals are the only physical gauge field degrees of freedom. The dynamics of such phases are completely solved in pure gauge field theory in the Coulomb gauge. It is also shown that the presence of fermions drastically alters the structure of the vacuum.