Winter School on Strongly Correlated Quantum Matter

TBA

TBA

TBA

TBA

Conventional magnets typically undergo a phase transition as temperature crosses the characteristic interaction energy scale, separating a disordered paramagnetic phase from an ordered one. The resulting behaviour has been the focus of attention and the setting of countless important discoveries over the past century in physics. Over the last few decades, an even richer plethora of phenomena has been uncovered in presence of frustration, when the competition between energy terms suppresses the transition temperature and opens the door to a new regime: A 'spin liquid' where the magnetic degrees of freedom remain disordered and yet 'strongly correlated', when the temperature becomes substantially smaller than the characteristic interactions. These new phases of matter can give rise to unusual and interesting phenomena, encompassing emergent symmetries and topological order, and excitations that can fractionalise the properties of the very constituent elements that make up the system. In this talk we will review some of the basic concepts relating to frustrated magnets and spin liquids. We will focus in particular on how one can model and understand their elementary excitations, both at the classical and quantum mechanical level.

Conventional magnets typically undergo a phase transition as temperature crosses the characteristic interaction energy scale, separating a disordered paramagnetic phase from an ordered one. The resulting behaviour has been the focus of attention and the setting of countless important discoveries over the past century in physics. Over the last few decades, an even richer plethora of phenomena has been uncovered in presence of frustration, when the competition between energy terms suppresses the transition temperature and opens the door to a new regime: A 'spin liquid' where the magnetic degrees of freedom remain disordered and yet 'strongly correlated', when the temperature becomes substantially smaller than the characteristic interactions. These new phases of matter can give rise to unusual and interesting phenomena, encompassing emergent symmetries and topological order, and excitations that can fractionalise the properties of the very constituent elements that make up the system. In this talk we will review some of the basic concepts relating to frustrated magnets and spin liquids. We will focus in particular on how one can model and understand their elementary excitations, both at the classical and quantum mechanical level.

Conventional magnets typically undergo a phase transition as temperature crosses the characteristic interaction energy scale, separating a disordered paramagnetic phase from an ordered one. The resulting behaviour has been the focus of attention and the setting of countless important discoveries over the past century in physics. Over the last few decades, an even richer plethora of phenomena has been uncovered in presence of frustration, when the competition between energy terms suppresses the transition temperature and opens the door to a new regime: A 'spin liquid' where the magnetic degrees of freedom remain disordered and yet 'strongly correlated', when the temperature becomes substantially smaller than the characteristic interactions. These new phases of matter can give rise to unusual and interesting phenomena, encompassing emergent symmetries and topological order, and excitations that can fractionalise the properties of the very constituent elements that make up the system. In this talk we will review some of the basic concepts relating to frustrated magnets and spin liquids. We will focus in particular on how one can model and understand their elementary excitations, both at the classical and quantum mechanical level.

TBA

TBA

The standard model of particle physics is a relativistic $SU(3) \times SU(2) \times U(1)$ gauge theory that describes the strong and electroweak interactions between quarks and leptons mediated by gauge bosons. The strong interactions between quarks and gluons are governed by Quantum Chromodynamics (QCD). The corresponding $SU(3)$ gauge interactions lead to the confinement of color and to the spontaneous breakdown of the chiral $SU(N_f)_L \times SU(N_f)_R$ symmetry of left- and right-handed quark fields with $N_f$ flavors. These non-perturbative phenomena can be addressed in the framework of Wilson's lattice QCD, in which ultraviolet divergences are regularized by a 4-d Euclidean space-time lattice. The naive lattice discretization of Dirac fermions leads to the notorious fermion doubling problem, which is intimately connected to the anomalies of chiral symmetry, i.e. to the explicit breaking of its axial $U(1)$ subgroup at the quantum level. The continuum limit of zero lattice spacing is mathematically equivalent to a second order phase transition in statistical mechanics, which is characterized by an infinite correlation length in lattice units. Monte Carlo simulations of lattice QCD have led to numerous qualitative insights and quantitative results for the structure of hadrons (3-quark baryons and quark-anti-quark mesons) as well as for the strongly interacting quark-gluon plasma at high temperature. At large baryon density, e.g. in the deep interior of neutron stars, the importance sampling that underlies Monte Carlo simulations fails due to a very severe sign problem. Quantum link models provide an alternative regularization of QCD in which the dimensional reduction of discrete quantum link variables gives rise to emergent gluon fields. This regularization can be used in quantum simulation experiments, e.g. with ultra-cold alkaline-earth atoms in an optical lattice, which do not suffer from the sign problem.

The standard model of particle physics is a relativistic $SU(3) \times SU(2) \times U(1)$ gauge theory that describes the strong and electroweak interactions between quarks and leptons mediated by gauge bosons. The strong interactions between quarks and gluons are governed by Quantum Chromodynamics (QCD). The corresponding $SU(3)$ gauge interactions lead to the confinement of color and to the spontaneous breakdown of the chiral $SU(N_f)_L \times SU(N_f)_R$ symmetry of left- and right-handed quark fields with $N_f$ flavors. These non-perturbative phenomena can be addressed in the framework of Wilson's lattice QCD, in which ultraviolet divergences are regularized by a 4-d Euclidean space-time lattice. The naive lattice discretization of Dirac fermions leads to the notorious fermion doubling problem, which is intimately connected to the anomalies of chiral symmetry, i.e. to the explicit breaking of its axial $U(1)$ subgroup at the quantum level. The continuum limit of zero lattice spacing is mathematically equivalent to a second order phase transition in statistical mechanics, which is characterized by an infinite correlation length in lattice units. Monte Carlo simulations of lattice QCD have led to numerous qualitative insights and quantitative results for the structure of hadrons (3-quark baryons and quark-anti-quark mesons) as well as for the strongly interacting quark-gluon plasma at high temperature. At large baryon density, e.g. in the deep interior of neutron stars, the importance sampling that underlies Monte Carlo simulations fails due to a very severe sign problem. Quantum link models provide an alternative regularization of QCD in which the dimensional reduction of discrete quantum link variables gives rise to emergent gluon fields. This regularization can be used in quantum simulation experiments, e.g. with ultra-cold alkaline-earth atoms in an optical lattice, which do not suffer from the sign problem.

These two lectures will give an overview of recent achievements in the quantum simulation of many-body problems using synthetic quantum systems based on atoms. The platforms consist of cold atoms atoms held in optical lattices or in arrays of tweezers, and where the interactions between atoms can be controlled. We will illustrate their potential by describing situations ranging from quantum magnetism to topological matter.

These two lectures will give an overview of recent achievements in the quantum simulation of many-body problems using synthetic quantum systems based on atoms. The platforms consist of cold atoms atoms held in optical lattices or in arrays of tweezers, and where the interactions between atoms can be controlled. We will illustrate their potential by describing situations ranging from quantum magnetism to topological matter.

TBA

The standard model of particle physics is a relativistic $SU(3) \times SU(2) \times U(1)$ gauge theory that describes the strong and electroweak interactions between quarks and leptons mediated by gauge bosons. The strong interactions between quarks and gluons are governed by Quantum Chromodynamics (QCD). The corresponding $SU(3)$ gauge interactions lead to the confinement of color and to the spontaneous breakdown of the chiral $SU(N_f)_L \times SU(N_f)_R$ symmetry of left- and right-handed quark fields with $N_f$ flavors. These non-perturbative phenomena can be addressed in the framework of Wilson's lattice QCD, in which ultraviolet divergences are regularized by a 4-d Euclidean space-time lattice. The naive lattice discretization of Dirac fermions leads to the notorious fermion doubling problem, which is intimately connected to the anomalies of chiral symmetry, i.e. to the explicit breaking of its axial $U(1)$ subgroup at the quantum level. The continuum limit of zero lattice spacing is mathematically equivalent to a second order phase transition in statistical mechanics, which is characterized by an infinite correlation length in lattice units. Monte Carlo simulations of lattice QCD have led to numerous qualitative insights and quantitative results for the structure of hadrons (3-quark baryons and quark-anti-quark mesons) as well as for the strongly interacting quark-gluon plasma at high temperature. At large baryon density, e.g. in the deep interior of neutron stars, the importance sampling that underlies Monte Carlo simulations fails due to a very severe sign problem. Quantum link models provide an alternative regularization of QCD in which the dimensional reduction of discrete quantum link variables gives rise to emergent gluon fields. This regularization can be used in quantum simulation experiments, e.g. with ultra-cold alkaline-earth atoms in an optical lattice, which do not suffer from the sign problem.

In this colloquium, I will review the computational strategies that have been employed recently to advance our understanding of the quantum many-body problem and strongly correlated quantum matter, both for models and materials. Some success stories will be presented and illustrated with physical examples, together with challenges that lie ahead.

TBA

Quantum Chromodynamics (QCD) is the fundamental theory for elementary particles (quarks and gluons) which constitutes the proton, the neutron and the atomic nuclei. Because the quarks and gluons are strongly interacting, their arise characteristic phases at finite temperature and baryon density: The hadronic phase at low temperature and density with the non-perturbative phenomena such as confinement and chiral symmetry breaking, the quark-gluon plasma phase at high temperature and low density, and the superconducting quark matter at low temperature and high density. In this colloquium, I discuss the characteristic properties of these phases on the basis of the many-body approach to QCD as well as their relevance to the physics of early universe, accelerator experiments, neutron stars and gravitational waves.

In these lectures, I will review Monte Carlo strategies with a focus on modern problems in condensed matter, quantum information, and machine learning. Topics will include Hamiltonian and data-driven approaches, the QMC sign problem, and contrasts between Markov Chain and autoregressive approaches.

In these lectures, I will review Monte Carlo strategies with a focus on modern problems in condensed matter, quantum information, and machine learning. Topics will include Hamiltonian and data-driven approaches, the QMC sign problem, and contrasts between Markov Chain and autoregressive approaches.

This lecture will describe how quantum magnets are investigated experimentally. Topics covered will start from the choice of materials designed to enhance quantum properties, to signatures of quantum fluctuations present in bulk physical properties and finally the investigations of ground states and excitations using muon spectroscopy and inelastic neutron scattering. The lecture will also cover how experimental data can be compared to theoretical calculations and how the investigation of real materials broadens our understanding of quantum phenomena.

These lectures will cover the main machine learning techniques adopted to study interacting many-body quantum systems. First the concept of neural-network quantum states, a representation of the many-body wave-function based on artificial neural networks will be introduced. Then, I will show how these states can be used in variational applications to theoretically study the physical properties of interacting many-body quantum systems both at equilibrium and out-of-equilibrium. The lectures will be followed by hands-on sessions, based on the open-source software NetKet.

These lectures will cover the main machine learning techniques adopted to study interacting many-body quantum systems. First the concept of neural-network quantum states, a representation of the many-body wave-function based on artificial neural networks will be introduced. Then, I will show how these states can be used in variational applications to theoretically study the physical properties of interacting many-body quantum systems both at equilibrium and out-of-equilibrium. The lectures will be followed by hands-on sessions, based on the open-source software NetKet.

These lectures will cover the main machine learning techniques adopted to study interacting many-body quantum systems. First the concept of neural-network quantum states, a representation of the many-body wave-function based on artificial neural networks will be introduced. Then, I will show how these states can be used in variational applications to theoretically study the physical properties of interacting many-body quantum systems both at equilibrium and out-of-equilibrium. The lectures will be followed by hands-on sessions, based on the open-source software NetKet.

During those two tutorials we will be showing how to practically apply the variational methods presented in the main lectures to investigate large, strongly correlated quantum systems in polynomial time. Tutorials will assume familiarity with python and will be based on the netket package. We will begin by briefly reviewing how to obtain the exact ground-state by diagonalising the hamiltonian. Then, we will show how to optimise variationally optimise several ansatze (Mean Field, Matrix-Product-States, Restricted-Bolztmann-Machine). We will also illustrate how to use Metropolis Sampling in order to circumvent the exponential size of the hilbert space. During the second session, we will show how to perform the real-time evolution of the same systems using the Time Dependent Variational Principle. The case of quenches of the ground state will be considered. Lastly, we will show how it is possible to study the dissipative dynamics of the same system coupled to a thermal bath using the Lindblad master equation.

During those two tutorials we will be showing how to practically apply the variational methods presented in the main lectures to investigate large, strongly correlated quantum systems in polynomial time. Tutorials will assume familiarity with python and will be based on the netket package. We will begin by briefly reviewing how to obtain the exact ground-state by diagonalising the hamiltonian. Then, we will show how to optimise variationally optimise several ansatze (Mean Field, Matrix-Product-States, Restricted-Bolztmann-Machine). We will also illustrate how to use Metropolis Sampling in order to circumvent the exponential size of the hilbert space. During the second session, we will show how to perform the real-time evolution of the same systems using the Time Dependent Variational Principle. The case of quenches of the ground state will be considered. Lastly, we will show how it is possible to study the dissipative dynamics of the same system coupled to a thermal bath using the Lindblad master equation.