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We point out that in superconductors there may exist localized quantum excitations of the electric and condensate fields surrounding a static charge, which cannot be interpreted as simply the ground state of the screened charge plus some number of massive photons. This is illustrated via a lattice Monte Carlo calculation of the energy spectrum of a pair of separated static charges in an effective Ginzburg-Landau model of superconductivity., 8 pages, 8 figures

We show that the Higgs and confinement phases of a gauge Higgs theory, with the Higgs field in the fundamental representation of the gauge group, are distinguished both by a broken or unbroken realization of the global center subgroup of the gauge group, and by the type of confinement in each phase. This is color confinement in the Higgs phase, and a stronger property, which we call "separation-of-charge" confinement, in the confining phase., Comment: 9 pages, 5 figures. Review article for the journal Symmetry. v2: Reference added

We explore certain properties of the Hartree-Fock approximation to the ground state of the two-dimensional Hubbard model, emphasizing the fact that in the Hartree approach there is an enormous multiplicity of self-consistent solutions which are nearly degenerate in energy, reminiscent of a spin glass, but which may differ substantially in other bulk properties. It is argued that this multiplicity is physically relevant at low temperatures. We study the localization properties of the one-particle wavefunctions comprising the Hartree-Fock states, and find that these are unlocalized at small and moderate values of U/t, in particular in the stripe region, but become highly localized at values corresponding to strong repulsion. We also find rectangular domains as well as stripes in the stripe region of the phase diagram, and study pair correlations in the neighborhood of half-filling., Comment: 29 pages, 16 figures. v2: to be published in Annals of Physics. Title changed to Multiplicity, localization, and domains in the Hartree-Fock ground state of the two-dimensional Hubbard model. Substantial revision to the introduction

We present evidence that seemingly elementary and non-composite objects, namely isolated static fermions in certain gauge Higgs theories, have a mass spectrum corresponding to localized excitations of the surrounding gauge and Higgs fields.

Static quark-antiquark states in QCD, at finite quark separation, have a spectrum of metastable states corresponding to string-like excitations of the gauge field. In this article I suggest that there may also exist an excitation spectrum of heavy fermions in some gauge Higgs theories deep in the Higgs phase. In this situation there are no color electric flux tubes connecting quarks with antiquarks. There may, nonetheless, exist stable excitations of the bosonic fields surrounding an isolated fermion, below the particle production threshold. I present numerical evidence indicating the existence of such excitations in an SU(3) gauge Higgs theory, with the scalar field in the fundamental representation of the gauge group., Comment: 10 pages, 8 figures

The static quark potential arises from vacuum fluctuations of the gauge fields; this is clear from the fact that the potential is extracted from the vacuum expectation value of a Wilson loop. In a Euclidean functional integral, the orientation of a rectangular R × T loop is obviously irrelevant to the expectation value, and in particular it can be oriented in a plane at a fixed time.

We propose that the Higgs phase of a gauge Higgs theory is the phase of spontaneously broken custodial symmetry, and present a new gauge invariant order parameter for custodial symmetry breaking which is very closely analogous to the Edwards-Anderson order parameter for spin glasses. Custodial symmetry is a global symmetry acting on the Higgs field alone, and we show here that the spin glass transition in gauge Higgs theories, from a QCD-like phase to a Higgs phase of broken custodial symmetry, coincides with the transition between two distinct types of confinement. These are color confinement in the Higgs phase, and a stronger version of confinement, which we have termed "separation-of-charge" confinement, in the QCD-like phase., Comment: 17 pages, 1 figure

We show that the gauge and scalar fields which surround static ions in the Ginzburg-Landau model of superconductivity have a spectrum of excitations which are potentially observable. This ties in with earlier results along these lines found in a variety of gauge Higgs theories. Excitations of this type would appear as a mass spectrum of the “elementary” particles in the theory.

We compute the Coulomb interaction energy of dense sets of static quarks in a compact volume (much smaller than the lattice volume) containing one quark per lattice site. The quark color charges are combined into either a set of three-quark nucleon states, or into a non-factorizable "one big hadron" state. In both cases we find that the energy per quark is roughly constant as the volume of quarks increases. A surprise is that if we construct the nucleon states from sets of three quarks chosen at random in the volume, then the energy per quark remains roughly constant, even as the average distance between quarks in a nucleon state grows as the volume increases. This energy dependence of a nucleon in a dense medium is at odds with the behavior of an isolated nucleon as quark separation increases, and for static quarks it is not easily explicable in terms of some version of Debye screening., Comment: 8 pages, 6 figures

It is suggested that the center vortex confinement mechanism, familiar in hadronic physics, may have some relevance to high-T c phenomena. We focus specifically on the transition from the superconducting phase to the pseudogap phase. There is evidence of a vortex liquid in the latter phase, in which the pairing responsible for superconductivity still exists, but superconductivity itself does not. An analogy, drawn from particle physics, may be the Higgs to confinement phase transition in an SU(N) gauge theory, where the confined phase is a vortex liquid, and the Higgs phase is a phase of a broken global Z N symmetry. We illustrate this idea with numerical simulations of a spatially asymmetric U(1) gauge-Higgs model, with lattice artifact monopoles suppressed. We show the existence of a Higgs (superconductor) to confinement (vortex liquid) phase, explicitly identifying vortices in lattice configurations generated in the confined phase, and showing that they produce an area-law falloff in planar Wilson loops, which may be measurable experimentally. The superconducting phase is a phase of broken global Z 2 symmetry.

The property of color confinement ("C confinement"), meaning that all asymptotic particle states are color neutral, holds not only in QCD, but also in gauge-Higgs theories deep in the Higgs regime. In this talk we describe a new and stronger confinement criterion, separation-of-charge confinement or "S$_c$ confinement," which is an extension of the Wilson area-law criterion to gauge + matter theories. We will show that there is a transition between S$_c$ and C confinement in the phase plane of gauge-Higgs theories, and we will also explain what symmetry is actually broken in the Higgs phase of a gauge-Higgs theory., 12 pages, 7 figures. Talk presented at the Quark Confinement and the Hadron Spectrum XIII conference (Confinement 2018), Maynooth, Ireland, August 2018

In SU($N$) gauge-Higgs theories, with a single Higgs field in the fundamental representation, there exists in addition to the local gauge symmetry a global SU(2) symmetry, at $N=2$, and a global U(1) symmetry, for $N \ne 2$. We construct a gauge-invariant order parameter for the breaking of these global symmetries in the Higgs sector, and calculate numerically the transition lines, in coupling-constant space, for SU(2) and SU(3) gauge theories with unimodular Higgs fields. The order parameter is non-local, and therefore its non-analyticity does not violate the theorem proved by Osterwalder and Seiler. We then show that there exists a transition, in gauge-Higgs theories, between two types of confinement: ordinary color neutrality in the Higgs region, and a stronger condition, which we have called "separation-of-charge confinement," in the confinement region. We conjecture that the symmetry-breaking transition coincides with the transition between these two physically different types of confinement., 20 pages, 14 figures. v2: new section VII, other minor revisions. Version to appear in PRD

We consider the four point connected correlator representing a static quark-antiquark pair separated by a spatial distance R, propagating for a Euclidean time T. This function is computed by lattice Monte Carlo in SU(2) pure gauge theory at lattice couplings $\beta=2.2$ and $\beta=2.5$ in both Coulomb and Landau gauges. The Coulomb gauge correlator is well behaved, and is dominated at large T by a state whose energy grows linearly as $\sigma R$, with $\sigma$ the known asymptotic string tension. The connected correlator in Landau gauge behaves differently. At intermediate R there is clear evidence of a linear potential, but the corresponding string tension extrapolates to zero at large T. At large R the connected correlator becomes negative; moreover there are strong finite size effects. These numerical results suggest that unphysical states dominate the large Euclidean time behavior of this Landau gauge correlator., Comment: 6 pages, 10 figures

We present a phase diagram for SU(3) lattice gauge theory with 695 MeV dynamical staggered fermions, in the plane of temperature and chemical potential, obtained from effective Polyakov line actions. The derivation is via the method of relative weights, and the effective theories are solved at finite chemical potential by mean field theory., 8 pages, 4 figures, 1 table; presented at the 34th International Symposium on Lattice Field Theory (Lattice 2017), June 18 - 24, 2017, Granada, Spain; to appear in AIP Conference Proceedings

We apply the relative weights method to SU(3) gauge theory with staggered fermions of mass 695 MeV at a set of temperatures in the range $151 \le T \le 267$ MeV, to obtain an effective Polyakov line action at each temperature. We then apply a mean-field method to search for phase transitions in the effective theory at finite densities. The result is a transition line in the plane of temperature and chemical potential, with an endpoint at high temperature, as expected, but also a second endpoint at a lower temperature. We cannot rule out the possibilities that a transition line reappears at temperatures lower than the range investigated, or that the second endpoint is absent for light quarks., 11 pages, 9 figures. v2: references added, minor cosmetic changes, and a small numerical correction in the appendix. v3: significant revisions in procedure, figures added, some comparison with other results in the literature. v4: corrected one citation and minor misprint

In Coulomb gauge a longitudinal electric field is generated instantaneously with the creation of a static quark-antiquark pair. The field due to the quarks is a sum of two contributions, one from the quark and one from the antiquark, and there is no obvious reason that this sum should fall off exponentially with distance from the sources. We show here, however, from numerical simulations in pure SU(2) lattice gauge theory, that the color Coulomb electric field does in fact fall off exponentially with transverse distance away from a line joining static quark-antiquark sources, indicating the existence of a color Coulomb flux tube, and the absence of long-range Coulomb dipole fields.

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Based on the author's long-standing course, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing a detailed introduction to quantum mechanics.

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information

Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information