Invariants of ground state phases in one dimension. Afternoon session: see Bruno Nachtergaele's paragraph
A ground state phase of a quantum spin chain can be defined as the equivalence class of Hamiltonians that can be smoothly deformed into each other without closing the spectral gap above the ground state energy. In this talk, I will present a number of invariants characterizing such phases in the framework of finitely correlated states in one dimension, with or without symmetry requirements.
Non-relativistic interacting bosons at zero temperature exhibit two interesting critical phases: the celebrated condensate phase and the critical theory at zero density, known as quantum critical point. From a theoretical point of view these theories are particularly challenging in dimension two, which is in both cases critical in the sense of Renormalization Group (RG). This is the method for understanding from first principles the emergence of scaling laws in interacting many body systems at low or zero temperatures, as well as for computing thermodynamic and correlation functions.
In collaboration with A. Giuliani we proved renormalizability of the quantum critical point and of the condensed phase in two dimensions, both in the ultraviolet and in the infrared, and developed a theory valid at all orders in renormalized perturbation theory, with explicit bounds on the generic order. In this talk I will present these results and compare them with the existing literature. While the results we obtained for the quantum critical case match with previous ones and extend them to all orders, we think that our findings call into question the stability of the two dimensional condensate at zero temperature.
We consider a two-parameter family of relative R´enyi entropies that uniﬁes the study of all known relative entropies (or divergences). These include the quantum relative entropy, the recently deﬁned quantum Rényi divergences, as well as the original quantum relative Rényi entropies. Consequently, the data-processing inequality (i.e. monotonicity under quantum operations) for all these quantities follows directly from the data-processing inequality for these new relative entropies. These new relative R´enyi entropies stem from the quantum entropic functionals deﬁned by Jakšić et. al. in the context of non-equilibrium statistical mechanics, and they satisfy the quantum generalizations of Rényi’s axioms for a divergence. This work has been done jointly with Koenraad Audenaert.
In the study of the intricate dynamics of many-body systems it is often convenient, or actually unavoidable, to resort to simpler approximate descriptions. For quantum-mechanical many-body systems of bosons it is possible to use effective one-particle equations to track the microscopic evolution of many-particle states in appropriate regimes. Many sophisticated results are available for the case of fixed volume, however, regimes in the direction of the thermodynamic limit seem fairly unexplored. I will report on a recent joint work with J. Fröhlich, P. Pickl, and A. Pizzo in which we consider an interacting Bose-gas at zero temperature in the regime of large volume V, large gas density rho, and small ratio V/rho; the coupling constant of the pair-potential being comparable to 1/rho. We derive an effective non-linear equation for the time evolution of coherent order one excitations above the ground state of the condensate and provide an explicit error bound in terms of density and volume. The effective equation allows an immediate discussion of the dispersion of small excitations. For repulsive potentials we recover Bogolyubov's well-known formula for the speed of sound in the gas, and for attractive potentials we show a dynamical instability.
While physical states of one infraparticle have been constructed in the setting of non-relativistic QED by Chen, Froehlich and Pizzo, the case of two or more such excitations is not under control in any rigorous framework of QFT. Since all the electrically charged particles are infraparticles, this is a rather compelling open problem of mathematical scattering theory. In this talk I will discuss the status of this problem in the setting of algebraic QFT and non-relativistic QED. In particular, I will cover my recent joint work with A. Pizzo on scattering of two electrons in the massless infrared-regular Nelson model.
We consider the quantum Heisenberg ferromagnet in three dimensions. We present the first rigorous proof of validity of the spin-wave approximation at low temperatures, at the level of the system's free energy. The proof combines a bosonic formulation of the model induced by the Holstein-Primakoff representation with probabilistic estimates, localization bounds and functional inequalities. Joint work with Michele Correggi and Robert Seiringer.
Gian Michele Graf
Afternoon session: Bulk-edge duality for topological insulators
Topological insulators are materials, which are conducting at their edges, though not in the bulk. Their Hamiltonians can not be deformed into those of ordinary insulators while retaining (fermionic) time-reversal invariance and the insulating property. After reviewing some of the reasons for the interest they recently raised, indices will be defined for insulators, telling apart the two types. We will do so first for two-dimensional insulators: Once based on the electronic bulk states, and once on the edge states. We explain why the indices are equivalent (duality). The result requires only translation invariance of the material in direction of the edge; but in case of full translation
invariance the bulk index can be expressed in terms of Bloch states and the duality understood by means of Levinson's theorem. Finally we will discuss the duality in the three-dimensional case.
(Joint work with M. Porta)
On the BCS gap equation for superfluid fermionic gases
Afternoon session: Non-equilibrium steady states and entropy production
The jellium is a model, introduced by Wigner (1934), for a gas of electrons moving in a uniform neutralizing background of positive charge. Wigner suggested that the repulsion between electrons might lead to a broken translational symmetry. For classical one-dimensional systems this fact was proven by Kunz (1974), while in the quantum setting, Brascamp and Lieb (1975) proved translation symmetry breaking at low densities. Here, we prove translation symmetry breaking for the quantum one-dimensional jellium at all densities.
We consider families of random non-unitary contraction operators defined as deformations of CMV matrices which appear naturally in the study of random quantum walks on trees or lattices. We establish several results about the location and nature of the spectrum of such non-normal operators as a function of their parameters. We relate these results to the analysis of certain random quantum walks, the dynamics of which can be studied by means of iterates of such random non-unitary contraction operators. This is joint work with Eman Hamza.
We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the variance and the two-point correlation function are governed by a universal power law behaviour that differs from the Wigner-Dyson-Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii, and describes the eigenvalue density correlations in general metallic samples with weak disorder. Our result rigorously establishes the Altshuler--Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an arithmetic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner.
Photonic crystals are to light what crystalline solids are to electrons: here, electromagnetic waves propagate in a material whose electric permittivity and magnetic permeability are periodic matrix-valued functions. In two recent publications, we have rigorously established effective light dynamics for adiabatically perturbed non-gyrotropic photonic crystals. Our results also give a rigorous dynamical justification for the absence of topological effects in non-gyrotropic photonic crystals. The central technical tool is space-adiabatic perturbation theory.
Compared to quantum problems, initial states need to satisfy two additional physical constraints: electromagnetic waves are real and source free (in the absence of charges and currents). Hence, we have had to show that the effective dynamics preserve these constraints. The crucial insight is that complex conjugation induces a particle-hole-type symmetry which exchanges incoming and outgoing Bloch waves; this symmetry implies that frequency bands come in pairs whose Bloch waves are complex conjugates of one another. Since one needs symmetric pairs of (complex) Bloch functions to express (real) electromagnetic waves, this is a bona fide multiband problem.
We conclude the talk by discussing the problem of establishing semiclassical ray optics equations for the two-band case. (Joint work with Giuseppe De Nittis)
Afternoon session: Effective behavior of mean-field Bose gases, with Jan Philip Solovej
One of the important aspects of many-body quantum mechanics of electrons is the analysis of two-body density matrices. While the characterization of one-body density matrices is well known and simple to state, that of two-body matrices is far from simple -- indeed, it is not fully known. In this talk I will present joint work with Eric Carlen in which we study the possible entropy of such matrices. We find, inter
alia, that minimum entropy is achieved for Slater determinant N-body parent functions. Thus, from the entropic point of view, Slater determinants play the same role as condensates play for bosons.
Afternoon session: Gapped ground state phases of quantum lattice systems, with Sven Bachmann
Seven Seeds for Discussion:
1. If the spectral gap above the ground state(s) of a family of quantum lattice models with short-range interactions depending on a continuous parameter lambda, has a positive lower bound independent of lambda and the size of the system, the ground states are said to belong to the same gapped quantum phase. We will discuss the construction of quasi-local automorphisms (as thermodynamic limits of unitary dynamics) that related the ground states within a gapped quantum phase.
2. Following 1, it is natural to ask when there exists a smooth interpolation between two models within a gapped ground state. We discuss results that have recently been proved that answer this question for some special cases (PVBS, AKLT, FCS) and argue that the edge ground state dimension is characteristic of the equivalence class in one dimension. Higher dimensional classes are richer, some of them exhibiting so-called topological order.
3. More generally, the relevance of the definition of a gapped ground state phase depends on the invariants that it entails. Apart from the number of edge states discussed in 2, automorphic equivalence is conjectured to preserve the long-range entanglement structure that can be characterized by the entanglement entropy. Furthermore, if, in addition, the interactions along the path are all invariant under the action of a symmetry group, further invariants arise which refine the classification of phases.
4. Understanding phases is only one aspect of the issue. A striking consequence of the classical theory of phases at finite temperature is the universality at the phase transition. We will discuss the type of universality conjectured to arise in the transitions between ground state phases.
5. Quantum spin systems with random interactions are often gapless but the if there is localization in an interval of energies above the ground state the system is said to have a mobility gap. We will discuss how such a system displays some of the same properties as systems with a spectral gap.
6. Many-body localization in random quantum spin systems (mostly conjectures!).
7. The particle nature of elementary excitations above a gapped ground state.
Giuseppe de Nittis
In this work we provide a classification of type AI topological insulators in dimension d=1,2,3,4 which is based on the equivariant homotopy property of "Real" vector bundles. This allows us to produce a finest classification that covers also the unstable regime which is usually not accessible via K-theoretical techniques. We prove the absence of non-trivial one-band AI topological insulators in every spatial dimension
by inspecting the second equivariant cohomology group which classifies "Real" line bundles. We show also that the classification of "Real" line bundles suffices for the complete classification of AI topological insulators in dimension d >= 3. In dimension d=4 the classification of different topological phases is given by the second "Real" Chern class which provides an even labeling identifiable with the degree of a suitable map. Finally, we provide an explicit construction for the realization of non trivial four-dimensional models of each topological degree.
Phan Tranh Nam
Collective excitations of Bose gases
In the mean-field regime, Bogoliubov's theory suggests that the spectrum of a large system of interacting bosons can be described asymptotically using an effective quadratic Hamiltonian acting on Fock space. Recently, Bogoliubov's theory has been justified rigorously for the low-energy eigenvalues when the Hartree minimizer is unique and non-degenerate. I will discuss how to extend the existing result for the low-enegy eigenvalues to the case of non-uniqueness and/or degeneracy of the Hartree minimizer. For example, we can consider rotating Bose gases when the rotation speed is large enough to break the symmetry and to produce multiple quantized vortices. I will also discuss the high-energy eigenvalues which correspond to the Hartree stationary states. This is joint work with Robert Seiringer.
Jan Philip Solovej
Afternoon session: Effective behavior of mean-field Bose gases, with Mathieu Lewin
In his 1932 paper, Eugene Wigner introduced the Wigner function in order to compute quantum corrections to classical equilibrium distributions. We show how to extend this program and compute semiclassical approximations to quantum mechanical equilibrium distributions for slow, semiclassical degrees of freedom coupled to fast, quantum mechanical degrees of freedom. The main examples are molecules and electrons in crystalline solids. The semiclassical formulas contain, in addition to quantum corrections similar to those already found by Wigner, also modifications of the classical Hamiltonian system used in the approximation: The classical energy and the Liouville measure on classical phase space have non-trivial-expansions in the semiclassical parameter. This talk is based on joint work with Wolfgang Gaim and Hans Stiepan.
When quantum particles are submitted to a sufficiently strong magnetic field the motion becomes two-dimensional and restricted to the Lowest Landau Level. Strong repulsive interactions can then lead to highly correlated states, descending from the Laughlin wave function. We investigate the response of such strongly correlated ground states to variations of an external potential. This leads to a family of variational problems of a new type. Our main results are rigorous energy estimates demonstrating strong rigidity of the response. In particular we obtain estimates indicating that there is a universal bound on the maximum local density of these states in the limit of large particle number. We refer to these as incompressibility estimates. This is joint work with Nicolas Rougerie.