Quantum Link Models Research Articles

Order-by-disorder and quantum Coulomb phase in quantum square ice.

We show that quantum square ice-namely, the two-dimensional version of proton or spin ice with tunable quantum tunneling of the electric or magnetic dipole moment-exhibits a quantum spin-liquid phase supporting fractionalized spinons. This phase corresponds to a thermally induced, deconfined quantum Coulomb phase of a two-dimensional lattice gauge theory. It emerges at finite, yet exceedingly low temperatures from the melting of two distinct order-by-disorder phases appearing in the ground state: a plaquette valence-bond solid for low tunneling; and a canted Néel state for stronger tunneling. The latter phases appear via the highly nonlinear effect of quantum fluctuations within the degenerate manifold of ice-rule states, and they can be identified as the two competing ground states of a discrete lattice gauge theory (quantum link model) emerging as the effective Hamiltonian of the system within degenerate perturbation theory.

Tensor Networks for Lattice Gauge Theories and Atomic Quantum Simulation

We show that gauge invariant quantum link models, Abelian and non-Abelian, can be exactly described in terms of tensor networks states. Quantum link models represent an ideal bridge between high-energy to cold atom physics, as they can be used in cold-atoms in optical lattices to study lattice gauge theories. In this framework, we characterize the phase diagram of a (1+1)-d quantum link version of the Schwinger model in an external classical background electric field: the quantum phase transition from a charge and parity ordered phase with non-zero electric flux to a disordered one with a net zero electric flux configuration is described by the Ising universality class.

The (2 + 1)-d U(1) quantum link model masquerading as deconfined criticality**Dedicated to the memory of Bernard B Beard (1957–2012).

The (2 + 1)-d U(1) quantum link model is a gauge theory, amenable to quantum simulation, with a spontaneously broken SO(2) symmetry emerging at a quantum phase transition. Its low-energy physics is described by a (2 + 1)-d RP(1) effective field theory, perturbed by an SO(2) breaking operator, which prevents the interpretation of the emergent pseudo-Goldstone boson as a dual photon. At the quantum phase transition, the model mimics some features of deconfined quantum criticality, but remains linearly confining. Deconfinement only sets in at high temperature.

Superconducting Circuits for Quantum Simulation of Dynamical Gauge Fields

We describe a superconducting-circuit lattice design for the implementation and simulation of dynamical lattice gauge theories. We illustrate our proposal by analyzing a one-dimensional U(1) quantum-link model, where superconducting qubits play the role of matter fields on the lattice sites and the gauge fields are represented by two coupled microwave resonators on each link between neighboring sites. A detailed analysis of a minimal experimental protocol for probing the physics related to string breaking effects shows that, despite the presence of decoherence in these systems, distinctive phenomena from condensed-matter and high-energy physics can be visualized with state-of-the-art technology in small superconducting-circuit arrays.

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, Abelian U(1) lattice gauge theories arise in the description of certain quantum spin liquids. In quantum information theory, Kitaev's toric code is a lattice gauge theory. In particle physics, Quantum Chromodynamics (QCD), the non‐Abelian gauge theory of the strong interactions between quarks and gluons, is non‐perturbatively 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.

Atomic quantum simulation of U(N) and SU(N) non-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.

Atomic Quantum Simulation of Dynamical Gauge Fields Coupled to Fermionic Matter: From String Breaking to Evolution after a Quench

Using a Fermi-Bose mixture of ultracold atoms in an optical lattice, we construct a quantum simulator for a U(1) gauge theory coupled to fermionic matter. The construction is based on quantum links which realize continuous gauge symmetry with discrete quantum variables. At low energies, quantum link models with staggered fermions emerge from a Hubbard-type model which can be quantum simulated. This allows us to investigate string breaking as well as the real-time evolution after a quench in gauge theories, which are inaccessible to classical simulation methods.

Algebraic approach to interacting quantum systems

We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different (spin-particle-gauge) {\it languages} by means of exact mappings (isomorphisms) that we name {\it dictionaries} and prove a fundamental theorem establishing when two arbitrary languages can be connected. These mappings serve to unravel symmetries which are hidden in one representation but become manifest in another. In addition, we establish a formal link between seemingly unrelated physical phenomena by changing the language of our model description. This link leads to the idea of {\it universality} or equivalence. Moreover, we introduce the novel concept of {\it emergent symmetry} as another symmetry guiding principle. By introducing the notion of {\it hierarchical languages}, we determine the quantum phase diagram of lattice models (previously unsolved) and unveil hidden order parameters to explore new states of matter. Hierarchical languages also constitute an essential tool to provide a unified description of phases which compete and coexist. Overall, our framework provides a simple and systematic methodology to predict and discover new kinds of orders. Another aspect exploited by the present formalism is the relation between condensed matter and lattice gauge theories through quantum link models. We conclude discussing applications of these dictionaries to the area of quantum information and computation with emphasis in building new models of computation and quantum programming languages.

Quantum link models with many rishon flavors and with many colors

Quantum link models are a novel formulation of gauge theories in terms of discrete degrees of freedom. These degrees of freedom are described by quantum operators acting in a finite-dimensional Hilbert space. We show that for certain representations of the operator algebra, the usual Yang-Mills action is recovered in the continuum limit. The quantum operators can be expressed as bilinears of fermionic creation and annihilation operators called rishons. Using the rishon representation the quantum link Hamiltonian can be expressed entirely in terms of color-neutral operators. This allows us to study the large N tc limit of this model. In the 't Hooft limit we find an area law for the Wilson loop and a mass gap. Furthermore, the strong coupling expansion is a topological expansion in which graphs with handles and boundaries are suppressed.

Low-energy effective theories of quantum spin and quantum link models

Quantum spin and quantum link models provide an unconventional regularization of field theory in which classical fields arise via dimensional reduction of discrete variables. This D-theory regularization leads to the same continuum theories as the conventional approach. We show this by deriving the low-energy effective Lagrangians of D-theory models using coherent state path integral techniques. We illustrate our method for the $(2+1)$-d Heisenberg quantum spin model which is the D-theory regularization of the 2-d O(3) model. Similarly, we prove that in the continuum limit a $(2+1)$-d quantum spin model with $SU(N)_L\times SU(N)_R\times U(1)_{L=R}$ symmetry is equivalent to the 2-d principal chiral model. Finally, we show that $(4+1)$-d SU(N) quantum link models reduce to ordinary 4-d Yang-Mills theory.

Higher-spin cluster algorithms: the Heisenberg spin and U(1) quantum link models

I discuss here how the highly-efficient spin- 1 2 cluster algorithm for the Heisenberg antiferromagnet may be extended to higher-dimensional representations; some numerical results are provided. The same extensions can be used for the U (1) flux cluster algorithm, but have not yielded signals of the desired Coulomb phase of the system.

QCD as a quantum link model

QCD is constructed as a lattice gauge theory in which the elements of the link matrices are represented by non-commuting operators acting in a Hilbert space. The resulting quantum link model for QCD is formulated with a fifth Euclidean dimension, whose extent resembles the inverse gauge coupling of the resulting four-dimensional theory after dimensional reduction. The inclusion of quarks is natural in Shamir's variant of Kaplan's fermion method, which does not require fine-tuning to approach the chiral limit. A rishon representation in terms of fermionic constituents of the gluons is derived and the quantum link Hamiltonian for QCD with a U(N) gauge symmetry is expressed in terms of glueball, meson and constituent quark operators. The new formulation of QCD is promising both from an analytic and from a computational point of view.

A surface cluster algorithm for the U(1) quantum link model

We discuss the implementation in continuous time of a cluster algorithm for the quantum link model for U(1) gauge theory (a d + 1 dimensional D-theory). Clusters in this quantum link model are 2 d surfaces, and the algorithm is seen to be a generalization of the 1 d loop cluster algorithms for spin models. Problems sampling different winding number sectors are discussed.

Confinement, chiral symmetry breaking and continuum limits in quantum link models

Using the example of compact U(1) lattice gauge theory we argue that quantum link models can be used to reproduce the physics of conventional Hamiltonian lattice gauge theories. In addition to the usual gauge coupling g, these models have a new parameter j which naturally cuts-off large electric flux quanta on each link while preserving exact U(1) gauge invariance. The j → ∞ limit recovers the conventional Hamiltonian. At strong couplings, the theory shows confinement and chiral symmetry breaking for all non-trivial values of j. The phase diagram of the 3+1 dimensional theory suggests that a coulomb phase is present at large but finite j. Setting g = 0, a new approach to the physics of compact U(1) gauge theory on the lattice emerges. In this case the parameter j takes over the role of the gauge coupling, and j → ∞ describes free photons.

Correlation functions from quantum cluster algorithms: application to U(1) quantum spin and quantum link models

We demonstrate how correlation functions for non-diagonal operators can be measured with the loop-cluster algorithm for quantum spin models. We introduce the U(1) quantum link model and present the construction of a cluster algorithm for the model. We further show how an improved estimator for Wilson loops can be realised with the flux-cluster algorithm.

Quantum spins and quantum links: The D-theory approach to field theory

A new non-perturbative approach to quantum field theory is proposed. Instead of performing a path integral over configurations of classical fields, D-theory works with discrete quantized variables. Classical spin fields are replaced by quantum spins, and classical gauge fields are replaced by quantum links. The classical fields of a d-dimensional quantum field theory reappear as low-energy effective degrees of freedom of the discrete variables, provided the ( d+1)-dimensional D-theory is massless. When the extent of the extra Euclidean dimension becomes small in units of the correlation length, an ordinary d-dimensional quantum field theory emerges by dimensional reduction. The D-theory formulation of some spin models and gauge theories is constructed explicitly. In particular, QCD emerges as a quantum link model.

Green’s functions from quantum cluster algorithms

We show that cluster algorithms for quantum models have a meaning independent of the basis chosen to construct them. Using this idea, we propose a new method for measuring with little effort a whole class of Green’s functions, once a cluster algorithm for the partition function has been constructed. To explain the idea, we consider the quantum XY model and compute its two point Green’s function in various ways, showing that all of them are equivalent. We also provide numerical evidence confirming the analytic arguments. Similar techniques are applicable to other models. In particular, in the recently constructed quantum link models, the new technique allows us to construct improved estimators for Wilson loops and may lead to a very precise determination of the glueball spectrum.

D-theory: field theory via dimensional reduction of discrete variables

A new non-perturbative approach to quantum field theory — D-theory — is proposed, in which continuous classical fields are replaced by discrete quantized variables which undergo dimensional reduction. The 2-d classical O((3) model emerges from the (2 + 1)-d quantum Heisenberg model formulated in terms of quantum spins. Dimensional reduction is demonstrated explicity by simulating correlation lengths up to 350,000 lattice spacings using a loop cluster algorithm. In the framework of D-theory, gauge theories are formulated in terms of quantum links — the gauge analogs of quantum spins. Quantum links are parallel transporter matrices whose elements are non-commuting operators. They can be expressed as bilinears of anticommuting fermion constituents. In quantum link models dimensional reduction to four dimensions occurs, due to the presence of a 5-d Coulomb phase, whose existence is confirmed by detailed simulations using standard lattice gauge theory. Using Shamir's variant of Kaplan's fermion proposal, in quantum link QCD quarks appear as edge states of a 5-d slab. This naturally protects their chiral symmetries without fine-tuning. The first efficient cluster algorithm for a gauge theory with a continuous gauge group is formulated for the U(1) quantum link model. Improved estimators for Wilson loops are constructed, and dimensional reduction to ordinary lattice QED is verified numerically.

Quantum Spins and Quantum Links: The D-Theory Approach to Field Theory

Abstract A new non-perturbative approach to quantum field theory is proposed. Instead of performing a path integral over configurations of classical fields, D-theory works with discrete quantized variables. Classical spin fields are replaced by quantum spins, and classical gauge fields are replaced by quantum links. The classical fields of a d -dimensional quantum field theory reappear as low-energy effective degrees of freedom of the discrete variables, provided the ( d +1)-dimensional D-theory is massless. When the extent of the extra Euclidean dimension becomes small in units of the correlation length, an ordinary d -dimensional quantum field theory emerges by dimensional reduction. The D-theory formulation of some spin models and gauge theories is constructed explicitly. In particular, QCD emerges as a quantum link model.

Quantum link models: A discrete approach to gauge theories

Quantum link models: A discrete approach to gauge theories