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The recent advances in machine learning algorithms have boosted the application of these techniques to the field of condensed matter physics, in order e.g. to classify the phases of matter at equilibrium or to predict the real-time dynamics of a large class of physical models. Typically in these works, a machine learning algorithm is trained and tested on data coming from the same physical model. Here we demonstrate that unsupervised and supervised machine learning techniques are able to predict phases of a non-exactly solvable model when trained on data of a solvable model. In particular, we employ a training set made by single-particle correlation functions of a non-interacting quantum wire and by using principal component analysis, k-means clustering, and convolutional neural networks we reconstruct the phase diagram of an interacting superconductor. We show that both the principal component analysis and the convolutional neural networks trained on the data of the non-interacting model can identify the topological phases of the interacting model with a high degree of accuracy. Our findings indicate that non-trivial phases of matter emerging from the presence of interactions can be identified by means of unsupervised and supervised techniques applied to data of non-interacting systems., Comment: 9 pages, 7 figures. Final version

We investigate the general properties of the dimensional reduction of the Dirac theory, formulated in a Minkowski spacetime with an arbitrary number of spatial dimensions. This is done by applying Hadamard's method of descent, which consists in conceiving low-dimensional theories as a specialization of high-dimensional ones that are uniform along the additional space coordinate. We show that the Dirac equation reduces to either a single Dirac equation or two decoupled Dirac equations, depending on whether the higher-dimensional manifold has even or odd spatial dimensions, respectively. Furthermore, we construct and discuss an explicit hierarchy of representations in which this procedure becomes manifest and can easily be iterated., 22 pages, several figures

We perform a reduction from three to two spatial dimensions of the physics of a spin-1/2 fermion coupled to the electromagnetic field, by applying Hadamard's method of descent. We consider first the free case, in which motion is determined by the Dirac equation, and then the coupling with a dynamical electromagnetic field, governed by the Dirac-Maxwell equations. We find that invariance along one spatial direction splits the free Dirac equation in two decoupled theories. On the other hand, a dimensional reduction in the presence of an electromagnetic field provides a more complicated theory in 2+1 dimensions, in which the method of descent is extended by using the covariant derivative. Equations simplify, but decoupling between different physical sectors occurs only if specific classes of solutions are considered., 21 pages, 1 figure

We study the out-of-equilibrium properties of 1 + 1 dimensional quantum electrodynamics (QED), discretized via the staggered-fermion Schwinger model with an Abelian Zn gauge group. We look at two relevant phenomena: first, we analyze the stability of the Dirac vacuum with respect to particle/antiparticle pair production, both spontaneous and induced by an external electric field; then, we examine the string breaking mechanism. We observe a strong effect of confinement, which acts by suppressing both spontaneous pair production and string breaking into quark/antiquark pairs, indicating that the system dynamics displays a number of out-of-equilibrium features.

These notes collect the lectures given by the author to the "XXIII International Workshop on Geometry and Physics" held in Granada (Spain) in September 2014. The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then to present the geometric structures that underline the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathematics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we concentrate on some quantization procedures, that are founded on the geometric structures of QM — as we have described them in the first part — and represent the ones that are more operatively used in modern theoretical physics. We will discuss first the so-called Coherent State Approach which, mainly complemented by "Feynman Path Integral Technique", is the method which is most widely used in quantum field theory. Finally, we will describe the "Weyl Quantization Approach" which is at the origin of modern tomographic techniques, originally used in optics and now in quantum information theory.

We derive one- and two-dimensional models for classical electromagnetism by making use of Hadamard's method of descent. Low-dimensional electromagnetism is conceived as a specialization of the higher dimensional one, in which the fields are uniform along the additional spatial directions. This strategy yields two independent electromagnetisms in two spatial coordinates, and four independent electromagnetisms in one spatial coordinate., 17 pages, 5 figures

We study the ground-state properties of a class of Zn lattice gauge theories in 1 þ 1 dimensions, in which the gauge fields are coupled to spinless fermionic matter. These models, stemming from discrete representations of the Weyl commutator for the U(1) group, preserve the unitary character of the minimal coupling and have, therefore, the property of formally approximating lattice quantum electrodynamics in one spatial dimension in the large-n limit. The numerical study of such approximated theories is important to determine their effectiveness in reproducing the main features and phenomenology of the target theory, in view of implementations of cold-atom quantum simulators of QED. In this paper, we study the cases n 1⁄4 2 ÷ 8 by means of a DMRG code that exactly implements Gauss’s law. We perform a careful scaling analysis and show that, in absence of a background field, all Zn models exhibit a phase transition which falls in the Ising universality class, with spontaneous symmetry breaking of the CP symmetry. We then perform the large-n limit and find that the asymptotic values of the critical parameters approach the ones obtained for the known phase transition of the zero-charge sector of the massive Schwinger model, which occurs at negative mass.

We present a detailed study of the topological Schwinger model [Phys. Rev. D 99, 014503 (2019)], which describes ($1+1$) quantum electrodynamics of an Abelian $U(1)$ gauge field coupled to a symmetry-protected topological matter sector, by means of a class of ${\mathbb{Z}}_{N}$ lattice gauge theories. Employing density-matrix renormalization group techniques that exactly implement Gauss' law, we show that these models host a correlated topological phase for different values of $N$, where fermion correlations arise through interparticle interactions mediated by the gauge field. Moreover, by a careful finite-size scaling, we show that this phase is stable in the large-$N$ limit and that the phase boundaries are in accordance with bosonization predictions of the $U(1)$ topological Schwinger model. Our results demonstrate that ${\mathbb{Z}}_{N}$ finite-dimensional gauge groups offer a practical route for an efficient classical simulation of equilibrium properties of electromagnetism with topological fermions. Additionally, we describe a scheme for the quantum simulation of a topological Schwinger model exploiting spin-changing collisions in boson-fermion mixtures of ultracold atoms in optical lattices. Although technically challenging, this quantum simulation would provide an alternative to classical density-matrix renormalization group techniques, providing also an efficient route to explore real-time nonequilibrium phenomena.

The interplay of symmetry, topology, and many-body effects in the classification of possible phases of matter poses a formidable challenge that is attracting great attention in condensed-matter physics. Such many-body effects are typically induced by inter-particle interactions involving an action at a distance, such as the Coulomb interaction between electrons in a symmetry-protected topological (SPT) phase. In this work, we show that similar phenomena also appear in high-energy physics, where inter-particle interactions are mediated by gauge bosons, and constrained by a local gauge symmetry. We introduce a variant of the so-called Schwinger model, which describes quantum electrodynamics in (1+1) dimensions (QED$_2$), and show that it can host SPT phases with a topological contribution to the vacuum {\theta} angle, which leads to a new type of topological QED$_2.$ We use bosonization and density-matrix renormalization group techniques to study its rich phase diagram in great detail, and present a scheme for its realization in experiments of ultra-cold atoms in optical lattices., Comment: closer to published version

We study the out-of-equilibrium properties of $1+1$ dimensional quantum electrodynamics (QED), discretized via the staggered-fermion Schwinger model with an Abelian $\mathbb{Z}_{n}$ gauge group. We look at two relevant phenomena: first, we analyze the stability of the Dirac vacuum with respect to particle/antiparticle pair production, both spontaneous and induced by an external electric field; then, we examine the string breaking mechanism. We observe a strong effect of confinement, which acts by suppressing both spontaneous pair production and string breaking into quark/antiquark pairs, indicating that the system dynamics displays a number of out-of-equilibrium features., Comment: 21 pages, 17 figures

We analyze the effects of disorder on the correlation functions of one-dimensional quantum models of fermions and spins with long-range interactions that decay with distance $\ell$ as a power-law $1/\ell^\alpha$. Using a combination of analytical and numerical results, we demonstrate that power-law interactions imply a long-distance algebraic decay of correlations within disordered-localized phases, for all exponents $\alpha$. The exponent of algebraic decay depends only on $\alpha$, and not, e.g., on the strength of disorder. We find a similar algebraic localization for wave-functions. These results are in contrast to expectations from short-range models and are of direct relevance for a variety of quantum mechanical systems in atomic, molecular and solid-state physics.

In the contest of open quantum systems, we study a class of Kraus operators whose definition relies on the defining representation of the symmetric groups. We analyze the induced orbits as well as the limit set and the degenerate cases.

We study the out-of-equilibrium properties of 1+1 dimensional quantum electrodynamics (QED), discretized via the staggered-fermion Schwinger model with an Abelian Zn gauge group. We look at two relevant phenomena: first, we analyze the stability of the Dirac vacuum with respect to particle/antiparticle pair production, both spontaneous and induced by an external electric field; then, we examine the string breaking mechanism. We observe a strong effect of confinement, which acts by suppressing both spontaneous pair production and string breaking into quark/antiquark pairs, indicating that the system dynamics displays a number of out-of-equilibrium features.

We study the ground-state properties of a class of ${\mathbb{Z}}_{n}$ lattice gauge theories in $1+1$ dimensions, in which the gauge fields are coupled to spinless fermionic matter. These models, stemming from discrete representations of the Weyl commutator for the U(1) group, preserve the unitary character of the minimal coupling and have, therefore, the property of formally approximating lattice quantum electrodynamics in one spatial dimension in the large-$n$ limit. The numerical study of such approximated theories is important to determine their effectiveness in reproducing the main features and phenomenology of the target theory, in view of implementations of cold-atom quantum simulators of QED. In this paper, we study the cases $n=2\textdiv{}8$ by means of a DMRG code that exactly implements Gauss's law. We perform a careful scaling analysis and show that, in absence of a background field, all ${\mathbb{Z}}_{n}$ models exhibit a phase transition which falls in the Ising universality class, with spontaneous symmetry breaking of the $CP$ symmetry. We then perform the large-$n$ limit and find that the asymptotic values of the critical parameters approach the ones obtained for the known phase transition of the zero-charge sector of the massive Schwinger model, which occurs at negative mass.

We address some problem related to teaching quantum field theory at the university level. After a discussion of the pros and cons of the canonical quantization approach, we present an alternative teaching proposal. The novelty of this approach rests on the idea of using a multilevel structure, where the levels of phenomenology, formalism and interpretation are related but distinguishable. In this context, the quantization of the electromagnetic field, which is taken as a paradigmatic case in the standard approach, is addressed as a special case and studied only in the last step.

We investigate magnetic properties of strongly interacting bosonic mixtures confined in one dimensional geometries, focusing on recently realized Rb-K gases with tunable interspecies interactions. By combining analytical perturbation theory results with density-matrix-renormalization group calculations, we provide quantitative estimates of the ground state phase diagram as a function of the relevant microscopic quantities, identifying the more favorable experimental regimes in order to access the various magnetic phases. Finally, we qualitatively discuss the observability of such phases in realistic setups when finite temperature effects have to be considered., 9 pages, 7 figures, to be published in EPJ ST special issue on "Novel Quantum Phases and Mesoscopic Physics in Quantum Gases"

We propose and analyze a generalization of the Kitaev chain for fermions with long-range $p$-wave pairing, which decays with distance as a power-law with exponent $\alpha$. Using the integrability of the model, we demonstrate the existence of two types of gapped regimes, where correlation functions decay exponentially at short range and algebraically at long range ($\alpha > 1$) or purely algebraically ($\alpha < 1$). Most interestingly, along the critical lines, long-range pairing is found to break conformal symmetry for sufficiently small $\alpha$. This is accompanied by a violation of the area law for the entanglement entropy in large parts of the phase diagram in the presence of a gap, and can be detected via the dynamics of entanglement following a quench. Some of these features may be relevant for current experiments with cold atomic ions., Comment: 5+3 pages, 4+2 figures

In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant-Kirillov-Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed., Comment: 27 pages; Accepted Manuscript

The expansion dynamics of bosonic gases in optical lattices has recently been the focus of increasing attention, both experimental and theoretical. We consider, by means of numerical Bethe ansatz, the expansion dynamics of initially confined wave packets of two interacting bosons on a lattice. We show that a correspondence between the asymptotic expansion velocities and the projection of the evolved wave function over the bound states of the system exists, clarifying the existing picture for such situations. Moreover, we investigate the role of the lattice in this kind of evolution., 12 pages, 10 figures

We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the 1D Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance $r$ as $1/r^\alpha$, as well as a related class of fermionic Hamiltonians that generalise the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents $\alpha$, based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for $\alpha \lesssim1$, while connected correlation functions can decay with a hybrid exponential and power-law behaviour, with a power that is $\alpha$-dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every $\alpha$. Along the critical lines, for all models breaking of conformal symmetry is argued at low enough $\alpha$. For the fermionic models we show that the edge modes, massless for $\alpha \gtrsim 1$, can acquire a mass for $\alpha < 1$. The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions., Comment: 15 pages, 8 figures

We study the time evolution of entanglement entropy and entanglement spectrum in a finite-size system which crosses a quantum phase transition at different speeds. We focus on the Ising model with a time-dependent magnetic field, which is linearly tuned on a time scale $\tau$ . The time evolution of the entanglement entropy displays different regimes depending on the value of $\tau$, showing also oscillations which depend on the instantaneous energy spectrum. The entanglement spectrum is characterized by a rich dynamics where multiple crossings take place with a gap-dependent frequency. Moreover, we investigate the Kibble-Zurek scaling of entanglement entropy and Schmidt gap., Comment: Accepted for publication in Phys. Rev. B

We study a lattice gauge theory in Wilson's Hamiltonian formalism. In view of the realization of a quantum simulator for QED in one dimension, we introduce an Abelian model with a discrete gauge symmetry $\mathbb{Z}_n$, approximating the $U(1)$ theory for large $n$. We analyze the role of the finiteness of the gauge fields and the properties of physical states, that satisfy a generalized Gauss's law. We finally discuss a possible implementation strategy, that involves an effective dynamics in physical space., 13 pages, 3 figures

Motivated by field-theoretic predictions we investigate the stable excitations that exist in two characteristic gapped phases of a spin-1 model with Ising-like and single-ion anisotropies. The sine-Gordon theory indicates a region close to the phase boundary where a stable breather exists besides the stable particles, that form the Haldane triplet at the Heisenberg isotropic point. The numerical data, obtained by means of the Density Matrix Renormalization Group, confirm this picture in the so-called large-D phase for which we give also a quantitative analysis of the bound states using standard perturbation theory. However, the situation turns out to be considerably more intricate in the Haldane phase where, to the best of our data, we do not observe stable breathers contrarily to what could be expected from the sine-Gordon model, but rather only the three modes predicted by a novel anisotropic extension of the Non-Linear Sigma Model studied here by means of a saddle-point approximation., Comment: 8 pages, 7 eps figures, svjour class

Drawing inspiration from Dirac's work on functions of non-commuting observables, we develop an approach to phase-space descriptions of operators and the Wigner-Weyl correspondence in quantum mechanics, complementary to standard formulations. This involves a two-step process: introducing phase-space descriptions based on placing position dependences to the left of momentum dependences (or the other way around); then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators. The method works uniformly for both continuous Cartesian degrees of freedom and for systems with finite-dimensional state spaces. It is interesting that the kernel encountered is naturally expressible in terms of geometric phases, and its removal involves extracting its square root in a suitable manner.

A new definition and interpretation of geometric phase for mixed state cyclic unitary evolution in quantum mechanics are presented. The pure state case is formulated in a framework involving three selected Principal Fibre Bundles, and the well known Kostant-Kirillov-Souriau symplectic structure on (co) adjoint orbits associated with Lie groups. It is shown that this framework generalises in a natural and simple manner to the mixed state case. For simplicity, only the case of rank two mixed state density matrices is considered in detail. The extensions of the ideas of Null Phase Curves and Pancharatnam lifts from pure to mixed states are also presented., 22 pages, revtex4

Quantum spin-1 chains may develop massless phases in presence of Ising-like and single-ion anisotropies. We have studied c=1 critical phases by means of both analytical techniques, including a mapping of the lattice Hamiltonian onto an O(2) nonlinear sigma model, and a multi-target DMRG algorithm which allows for accurate calculation of excited states. We find excellent quantitative agreement with the theoretical predictions and conclude that a pure Gaussian model, without any orbifold construction, describes correctly the low-energy physics of these critical phases. This combined analysis indicates that the multicritical point at large single-ion anisotropy does not belong to the same universality class as the Takhtajan-Babujian Hamiltonian as claimed in the past. A link between string-order correlation functions and twisting vertex operators, along the c=1 line that ends at this point, is also suggested.

Recent neutron scattering experiments on CsNiCl_3 reveal some features which are not well described by the nonlinear sigma model nor by numerical simulations on isolated S=1 spin chains. In particular, in real systems the intensity of the continuum of multiparticle excitations, at T=6K, is about 5 times greater than predicted. Also the gap is slightly higher and the correlation length is smaller. We propose a theoretical scenario where the interchain interaction is approximated by a staggered magnetic field, yielding to a correct prediction of the observed quantities.

We investigate both numerically and analytically the behavior of a spin-1 antiferromagnetic isotropic Heisenberg chain in an external uniform magnetic field. Extensive DMRG studies of chains up to N = 80 sites extend previous analyses and obtain the zero-temperature correlation functions. We argue that, despite the presence of a magnetic field, the model is still well described by an O(3) nonlinear σ-model. A saddle-point analysis, that also includes Gaussian fluctuations, is developed for the latter both at zero and finite temperatures. Its predictions are compared with those of the numerical analysis and with the existing experimental literature.

We present here a nonlinear sigma-model (NL$\sigma$M) study of a spin-1 antiferromagnetic Heisenberg chain in an external commensurate staggered magnetic field. We find, already at the mean-field level, excellent agreement with recent and very accurate Density Matrix Renormalization Group (DMRG) studies, and that up to the highest values of the field for which a comparison is possible, for the staggered magnetization and the transverse spin gap. Qualitative but not quantitative agreement is found between the NL$\sigma$M predictions for the longitudinal spin gap and the DMRG results. The origin of the discrepancies is traced and discussed. Our results allow for extensions to higher-spin chains that have not yet been studied numerically, and the predictions for a spin-2 chain are presented and discussed. Comparison is also made with previous theoretical approaches that led instead to predictions in disagreement with the DMRG results.

Noncommutative lattices have been recently used as finite topological approximations in quantum physical models. As a first step in the construction of bundles and characteristic classes over such noncommutative spaces, we shall study their K-theory. We shall do it algebraically, by studying the algebraic K-theory of the associated algebras of ‘continuous functions’ which turn out to be noncommutative approximately finite dimensional (AF) C ∗ -algebra . We also work out several examples.

Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.

We study the t-J model with the inclusion of the so called three-site term which comes out from the t/U --> 0 expansion of the Hubbard model. We find that this singlet pair hopping term has no qualitative effect on the structure of the pure mean field phase diagram for nonmagnetic states. In accordance with experimental data on high-T_c materials and some numerical studies, we also find wide regions of phase coexistence whenever the coupling J is greater than a critical value J_c. We show that J_c varies linearly with the temperature T, going to zero at T=0.

We study the so-called nonmagnetic phases (dimer and flux states) in the t-J model below half filling. We present a new phase diagram, at zero and finite temperature, that includes broad areas of phase coexistence (dimer-flux or flux-uniform), in accordance with experimental and numerical data on the possibility of separation into hole-rich and hole-poor regions. We also briefly comment on some techniques used in the literature to discuss phase separation in the t-J model., Comment: 17 pages, Latex, 6 figures

We show that the elliptic parametrization of the coupling constants of the quantum XYZ spin chain can be analytically extended outside of their natural domain, to cover the whole phase diagram of the model, which is composed of 12 adjacent regions, related to one another by a spin rotation. This extension is based on the modular properties of the elliptic functions and we show how rotations in parameter space correspond to the double covering PGL(2,Z)of the modular group, implying that the partition function of the XYZ chain is invariant under this group in parameter space, in the same way as a Conformal Field Theory partition function is invariant under the modular group acting in real space. The encoding of the symmetries of the model into the modular properties of the partition function could shed light on the general structure of integrable models.