QUASI-TOPOLOGICAL QUANTUM FIELD THEORIES AND Z(2) LATTICE GAUGE THEORIES

We consider a two-parameter family of Z(2) gauge theories on a lattice discretization T(M) of a three-manifold M and its relation to topological field theories. Familiar models such as the spin-gauge model are curves on a parameter space Gamma. We show that there is a region Gamma(0) subset of Gamma where the partition function and the expectation value h < W-R(gamma)> i of the Wilson loop can be exactly computed. Depending on the point of Gamma(0), the model behaves as topological or quasi-topological. The partition function is, up to a scaling factor, a topological number of M. The Wilson loop on the other hand, does not depend on the topology of gamma. However, for a subset of Gamma(0), < W-R(gamma)> depends on the size of gamma and follows a discrete version of an area law. At the zero temperature limit, the spin-gauge model approaches the topological and the quasi-topological regions depending on the sign of the coupling constant.

Su(3) lattice gauge theories and spin chains

I modelli su reticolo con simmetrie SU(n) sono attualmente oggetto di studio sia dal punto di vista sperimentale, sia dal punto di vista teorico; particolare impulso alla ricerca in questo campo è stato dato dai recenti sviluppi in campo sperimentale per quanto riguarda la tecnica dell’intrappolamento di atomi ultrafreddi in un reticolo ottico. In questa tesi viene studiata, sia con tecniche analitiche sia con simulazioni numeriche, la generalizzazione del modello di Heisenberg su reticolo monodimensionale a simmetria SU(3). In particolare, viene proposto un mapping tra il modello di Heisenberg SU(3) e l’Hamiltoniana con simmetria SU(2) bilineare-biquadratica con spin 1. Vengono inoltre presentati nuovi risultati numerici ottenuti con l’algoritmo DMRG che confermano le previsioni teoriche in letteratura sul modello in esame. Infine è proposto un approccio per la formulazione della funzione di partizione dell’Hamiltoniana bilineare-biquadratica a spin-1 servendosi degli stati coerenti per SU(3).

Atomic Quantum Simulation of U(N) and SU(N) Abelian Lattice Gauge Theories

Using ultracold alkaline-earth atoms in optical lattices, we construct a quantum simulator for U(N) and SU(N) lattice gauge theories with fermionic matter based on quantum link models. These systems share qualitative features with QCD, including chiral symmetry breaking and restoration at nonzero temperature or baryon density. Unlike classical simulations, a quantum simulator does not suffer from sign problems and can address the corresponding chiral dynamics in real time.

Ultracold Quantum Gases and Lattice Systems: Quantum Simulation of Lattice Gauge Theories

Abelian and non-Abelian gauge theories are of central importance in many areas of physics. In condensed matter physics, AbelianU(1) lattice gauge theories arise in the description of certain quantum spin liquids. In quantum information theory, Kitaev’s toric code is a Z(2) lattice gauge theory. In particle physics, Quantum Chromodynamics (QCD), the non-Abelian SU(3) gauge theory of the strong interactions between quarks and gluons, is nonperturbatively regularized on a lattice. Quantum link models extend the concept of lattice gauge theories beyond the Wilson formulation, and are well suited for both digital and analog quantum simulation using ultracold atomic gases in optical lattices. Since quantum simulators do not suffer from the notorious sign problem, they open the door to studies of the real-time evolution of strongly coupled quantum systems, which are impossible with classical simulation methods. A plethora of interesting lattice gauge theories suggests itself for quantum simulation, which should allow us to address very challenging problems, ranging from confinement and deconfinement, or chiral symmetry breaking and its restoration at finite baryon density, to color superconductivity and the real-time evolution of heavy-ion collisions, first in simpler model gauge theories and ultimately in QCD.

Two-dimensional Lattice Gauge Theories with Superconducting Quantum Circuits

A quantum simulator of U(1) lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

Quantum Simulation of Non-Abelian Lattice Gauge Theories

We use quantum link models to construct a quantum simulator for U(N) and SU(N) lattice gauge theories. These models replace Wilson’s classical link variables by quantum link operators, reducing the link Hilbert space to a finite number of dimensions. We show how to embody these quantum link models with fermionic matter with ultracold alkaline-earth atoms using optical lattices. Unlike classical simulations, a quantum simulator does not suffer from sign problems and can thus address the corresponding dynamics in real time. Using exact diagonalization results we show that these systems share qualitative features with QCD, including chiral symmetry breaking and we study the expansion of a chirally restored region in space in real time.

Chiral symmetry breaking in QCD-like gauge theories with a confining propagator and dynamical gauge boson mass generation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

We regularize compact and non-compact Abelian Chern–Simons–Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group R, each local Hilbert space is analogous to the one of a charged “particle” moving in the link-pair group space R2 in a constant “magnetic” background field. In the compact case, the link-pair group space is a torus U(1)2 threaded by k units of quantized “magnetic” flux, with k being the level of the Chern–Simons theory. The holonomies of the torus U(1)2 give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from U(1) to Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern–Simons limit of a large “photon” mass, this results in a Z(k)-symmetric variant of Kitaev’s toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle . Non-Abelian U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.

Renormalisation of the Fayet-Illiopoulos term in supersymmetric spontaneously broken U( 1) gauge theories

It is shown that the Fayet-Illiopoulos D term in N= 1 supersymmetric spontaneously broken U( 1) gauge theories may get one-loop corrections, even when trace U( 1) charges are zero. However, these corrections are only logarithmically divergent and hence do not affect the naturalness of the theory.

Renormalisation of the Fayet-Illiopoulos term in supersymmetric spontaneously broken U( 1) gauge theories

It is shown that the Fayet-Illiopoulos D term in N= 1 supersymmetric spontaneously broken U( 1) gauge theories may get one-loop corrections, even when trace U( 1) charges are zero. However, these corrections are only logarithmically divergent and hence do not affect the naturalness of the theory.

Cancellation of quadratically divergent mass corrections in globally supersymmetric spontaneously broken gauge theories

The one-loop quadratically divergent mass corrections in globally supersymmetric gauge theories with spontaneously broken abelian and non-abelian gauge symmetry are studied. Quadratically divergent mass corrections are found to persist in an abelian model with an ABJ anomaly. However, additional supermultiplets necessary to cancel the ABJ anomaly, turn out to be sufficient to eliminate the quadratic divergences as well, rendering the theory natural. Quadratic divergences are shown to vanish also in the case of an anomaly free model with spontaneously broken non-abelian gauge symmetry.