Optical Abelian lattice gauge theories

We discuss a general framework for the realization of a family of Abelian lattice gauge theories, i.e., link models or gauge magnets, in optical lattices. We analyze the properties of these models that make them suitable for quantum simulations. Within this class, we study in detail the phases of a U(1)$U\left(1\right)$-invariant lattice gauge theory in 2+1$2+1$ dimensions, originally proposed by P. Orland. By using exact diagonalization, we extract the low-energy states for small lattices, up to 4×4$4\times 4$. We confirm that the model has two phases, with the confined entangled one characterized by strings wrapping around the whole lattice. We explain how to study larger lattices by using either tensor network techniques or digital quantum simulations with Rydberg atoms loaded in optical lattices, where we discuss in detail a protocol for the preparation of the ground-state. We propose two key experimental tests that can be used as smoking gun of the proper implementation of a gauge theory in optical lattices. These tests consist in verifying the absence of spontaneous (gauge) symmetry breaking of the ground-state and the presence of charge confinement. We also comment on the relation between standard compact U(1)$U\left(1\right)$ lattice gauge theory and the model considered in this paper.     

Deformed quantum double realization of the toric code and beyond

Quantum double models, such as the toric code, can be constructed from transfer matrices of lattice gauge theories with discrete gauge groups and parametrized by the center of the gauge group algebra and its dual. For general choices of these parameters the transfer matrix contains operators acting on links which can also be thought of as perturbations to the quantum double model driving it out of its topological phase and destroying the exact solvability of the quantum double model. We modify these transfer matrices with perturbations and extract exactly solvable models which remain in a quantum phase, thus nullifying the effect of the perturbation. The algebra of the modified vertex and plaquette operators now obey a deformed version of the quantum double algebra. The Abelian cases are shown to be in the quantum double phase whereas the non-Abelian phases are shown to be in a modified phase of the corresponding quantum double phase. These are illustrated with the groups Z_n${\mathbb{Z}}_n$ and S₃$S_3$. The quantum phases are determined by studying the excitations of these systems namely their fusion rules and the statistics. We then go further to construct a transfer matrix which contains the other Z₂${\mathbb{Z}}_2$ phase namely the double semion phase. More generally for other discrete groups these transfer matrices contain the twisted quantum double models. These transfer matrices can be thought of as being obtained by introducing extra parameters into the transfer matrix of lattice gauge theories. These parameters are central elements belonging to the tensor products of the algebra and its dual and are associated to vertices and volumes of the three dimensional lattice. As in the case of the lattice gauge theories we construct the operators creating the excitations in this case and study their braiding and fusion properties.     

Canonical quantization and the spectral action; a nice example

We study the canonical quantization of the theory given by Chamseddine–Connes spectral action on a particular finite spectral triple with algebra _M2(C)⊕C$M_2\left(\mathbb{C}\right)\oplus \mathbb{C}$. We define a quantization of the natural distance associated with this noncommutative space and show that the quantum distance operator has a discrete spectrum. We also show that it would be the same for any other geometric quantity. Finally we propose a physical Hilbert space for the quantum theory. This spectral triple had been previously considered by Rovelli as a toy model, but with a different action which was not gauge invariant. The results are similar in the two cases, but the gauge invariance of the spectral action manifests itself by the presence of a non-trivial degeneracy structure for our distance operator.     

The propagation kernel for strong coupling gravity

The strong coupling limit of Einstein gravity in d+1$d+1$ dimensions gives rise to a quantum theory where after factorization of the conformal factor mode SL(d,R)/SO(d)$\mathrm{SL}(d,\mathbb{R})/\mathrm{SO}(d)$ nonlinear sigma-models are spatially coupled by the diffeomorphism constraint. A functional integral representation for the theory?s propagation kernel is derived in completions of the proper time gauge which manifestly invokes only physical gauge invariant degrees of freedom. In the weak field limit it reduces to the propagation kernel of massless and transversal-traceless free fields. For strong fields a covariant normal coordinate expansion is developed which covers the configuration manifold globally. Its leading order approximant resembles a semiclassical propagation kernel but without the need to solve the classical constraints. The results have implications for the ground state structure of quantum gravity.     

Quantum critical properties in the topological Ginzburg–Landau theory of self-dual Josephson junction arrays

This paper examines the multicritical behavior of a generalized U(_N1)×U(_N2)$U\left(N_1\right)\times U\left(N_2\right)$ Ginzburg–Landau theory containing two multicomponent complex fields which couple differently to two gauge fields described by two Maxwell terms and one mixed-Chern–Simons term. This model is relevant to the dynamics of Cooper pairs and vortices in a self-dual Josephson junction array system near its superconductor–insulator transition. We develop a renormalization group flow at fixed dimension and obtain the beta functions at one loop when both disorder fields are critical. Two sets of infrared-stable charged fixed points solutions are found for N>_Nc$N>N_c$: partially charged solutions with respect to the gauge fields exist with _Nc=35.6$N_c=35.6$, and fully charged solutions exist with _Nc=12.16$N_c=12.16$. We show that fine tuning the ratio of the two energy scales in the model has the effect of reducing the critical number _Nc$N_c$ and thus enlarges the region where the quantum phase transition is continuous. It is also found that the decoupled fixed point which is stable in the neutral case is no longer attainable in the presence of fluctuating gauge fields. We probe the conductivity at the critical point and show that it has a universal character determined by the renormalization group infrared-stable fixed-point values of the gauge couplings.     

Vortex properties of mesoscopic superconducting samples

In this work we investigated theoretically the vortex properties of mesoscopic samples of different geometries, submitted to an external magnetic field. We use both London and Ginzburg–Landau theories and also solve the non-linear Time Dependent Ginzburg–Landau equations to obtain vortex configurations, equilibrium states and the spatial distribution of the superconducting electron density in a mesoscopic superconducting triangle and long prisms with square cross-section. For a mesoscopic triangle with the magnetic field applied perpendicularly to sample plane the vortex configurations were obtained by using Langevin dynamics simulations. In most of the configurations the vortices sit close to the corners, presenting twofold or three-fold symmetry. A study of different meta-stable configurations with same number of vortices is also presented. Next, by taking into account de Gennes boundary conditions via the extrapolation length, b, we study the properties of a mesoscopic superconducting square surrounded by different metallic materials and in the presence of an external magnetic field applied perpendicularly to the square surface. It is determined the b  -limit for the occurrence of a single vortex in a mesoscopic square of area d²$d^2$, for 4ξ(0)?d?10ξ(0)$4\xi (0)?d?10\xi (0)$.     

Emergence of helicity ±2 modes (gravitons) from qubit models

We construct two quantum qubit models (or quantum spin models) on three-dimensional lattice in space, L-type model and N-type model. We show that, under a controlled approximation, all   the low energy excitations of the L-type model are described by one set of helicity ±2 modes with ω∝k³$\omega \propto k^3$ dispersion. We also argue that all   the low energy excitations of the N-type model are described by one set of helicity ±2 modes with ω∝k$\omega \propto k$ dispersion. In both model, the low energy helicity ±2 modes can be described by a symmetric tensor field _hμν$h_{\mu \nu }$ in continuum limit, and the gaplessness of the helicity ±2 modes is protected by an emergent linearized diffeomorphism gauge symmetry _hμν→_hμν+_∂μfν+_∂νfμ$h_{\mu \nu }\to h_{\mu \nu }+{\partial }_{\mu }{f}_{\nu }+{\partial }_{\nu }{f}_{\mu }$ at low energies. Thus the linearized quantum gravity emerge from our lattice models  . It turns out that the low energy effective Lagrangian density of the L-type model is invariant under the linearized diffeomorphism gauge transformation. Such a property protects the gapless ω∝k³$\omega \propto k^3$ helicity ±2 modes. In contrast, the low energy effective Lagrangian of the N-type model changes by a boundary term under the linearized diffeomorphism gauge transformation. Such a property protects the gapless ω∝k$\omega \propto k$ helicity ±2 modes. From many-body physics point of view, the ground states of the our two qubit model represent new states of quantum matter, whose low energy excitations are all described by one set of gapless helicity ±2 modes.     

Quantum criticality and Yang–Mills gauge theory

We present a family of nonrelativistic Yang–Mills gauge theories in D+1$D+1$ dimensions whose free-field limit exhibits quantum critical behavior with gapless excitations and dynamical critical exponent z=2$z=2$. The ground state wavefunction is intimately related to the partition function of relativistic Yang–Mills in D   dimensions. The gauge couplings exhibit logarithmic scaling and asymptotic freedom in the upper critical spacetime dimension, equal to 4+1$4+1$. The theories can be deformed in the infrared by a relevant operator that restores Poincaré invariance as an accidental symmetry. In the large-N limit, our nonrelativistic gauge theories can be expected to have weakly curved gravity duals.     

Metastable supersymmetry breaking without scales

We construct new examples of models of metastable D=4$D=4$N=1$N=1$ supersymmetry breaking in which all scales are generated dynamically. Our models rely on Seiberg duality and on the ISS mechanism of supersymmetry breaking in massive SQCD. Some of the electric quark superfields arise as composites of a strongly coupled gauge sector. This allows us to start with a simple cubic superpotential and an asymptotically free gauge group in the ultraviolet, and end up with an infrared effective theory which breaks supersymmetry dynamically in a metastable state.     

A compact 341 model at TeV scale

We build a gauge model based on the SU(3)_c⊗SU(4)_L⊗U(1)_X$SU{\left(3\right)}_c\otimes SU{\left(4\right)}_L\otimes U{\left(1\right)}_X$ symmetry where the scalar spectrum needed to generate gauge boson and fermion masses has a smaller scalar content than usually assumed in literature. We compute the running of its abelian gauge coupling and show that a Landau pole shows up at the TeV scale, a fact that we use to consistently implement those fermion masses that are not generated by Yukawa interactions, including neutrino masses. This is appropriately achieved by non renormalizable effective operators, suppressed by the Landau pole scale. Also, SU(3)_c⊗SU(3)_L⊗U(1)_N$SU{\left(3\right)}_c\otimes SU{\left(3\right)}_L\otimes U{\left(1\right)}_N$ models embedded in this gauge structure are bound to be strongly coupled at this same energy scale, contrary to what is generally believed, and neutrino mass generation is rather explained through the same effective operators used in the larger gauge group. Besides, their nice features, as the existence of cold dark matter candidates and the ability to reproduce the observed standard model Higgs-like phenomenology, are automatically inherited by our model. Finally, our results imply that this model is constrained to be observed or discarded soon, since it must be realized at the currently probed energy scale in LHC.     

Non-Abelian fusion rules from an Abelian system

We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n$n$ dimensional vector space which we call H_n$H_n$. The Z_p${\mathbb{Z}}_p$ gauge particles act on the vertex particles and thus H_n$H_n$ can be thought of as a C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n$n$ and p$p$, though we believe this feature holds for all n>p$n>p$. We will see that non-Abelian anyons of the quantum double of C(S₃)$\mathbb{C}\left(S_3\right)$ are obtained as part of the vertex excitations of the model with n=6$n=6$ and p=3$p=3$. Ising anyons are obtained in the model with n=4$n=4$ and p=2$p=2$. The n=3$n=3$ and p=2$p=2$ case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Z_p${\mathbb{Z}}_p$. This makes them possible candidates for realizing quantum computation.     

Tractors,mass, and Weyl invariance

Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describe massive and gauge invariant theories using Weyl invariance. The key idea is to introduce a new scalar field which is constant when evaluated at the scale corresponding to the metric of physical interest. This technique relies on being able to efficiently construct Weyl invariant theories. This is achieved using tractor calculus—a mathematical machinery designed for the study of conformal geometry. From a physics standpoint, this amounts to arranging fields in multiplets with respect to the conformal group but with novel Weyl transformation laws. Our approach gives a mechanism for generating masses from Weyl weights. Breitenlohner–Freedman stability bounds for Anti-de Sitter theories arise naturally as do direct derivations of the novel Weyl invariant theories given by Deser and Nepomechie. In constant curvature spaces, partially massless theories—which rely on the interplay between mass and gauge invariance—are also generated by our method. Another simple consequence is conformal invariance of the maximal depth partially massless theories. Detailed examples for spins s?2$s?2$ are given including tractor and component actions, on-shell and off-shell approaches and gauge invariances. For all spins s?2$s?2$ we give tractor equations of motion unifying massive, massless, and partially massless theories.