Papers by Juerg Froehlich
Journal of Mathematical Physics, Jul 1, 2022
In this paper the local iterative Lie-Schwinger block-diagonalization method, introduced in [FP],... more In this paper the local iterative Lie-Schwinger block-diagonalization method, introduced in [FP], [DFPR1], and [DFPR2] for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation consisting of bounded interaction potentials of short range mutltiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of the size of the lattice. New ideas and concepts are necessary to extend our method to systems in dimension d > 1: As in our earlier work, a sequence of local block-diagonalization steps based on judiciously chosen unitary conjugations of the original Hamiltonian is introduced. The supports of effective interaction potentials generated in the course of these block-diagonalization steps can be identified with what we call minimal rectangles contained in the lattice, a concept that serves to tackle combinatorial problems that arise in the course of iterating the block-diagonalization steps. For a given minimal rectangle, control of the effective interaction potentials generated in each block-diagonalization step with support in the given rectangle is achieved by exploiting a variety of rather subtle mechanisms which include, for example, the use of weighted sums of paths consisting of overlapping rectangles and of large denominators, expressed in terms of sums of orthogonal projections, that serve to control analogous sums of projections in the numerators resulting from the unitary conjugations of the interaction potential terms involved in the local block-diagonalization step. In this paper we introduce and study a family of quantum lattice systems describing insulating materials in two or more dimensions. We are interested in analyzing the low-energy
Journal of Functional Analysis, 2023
In this paper we extend the local iterative Lie-Schwinger block-diagonalization methodintroduced ... more In this paper we extend the local iterative Lie-Schwinger block-diagonalization methodintroduced in [DFPR3] for quantum lattice systems with bounded interactions in arbitrary dimension-to systems with unbounded interactions, i.e., systems of bosons. We study Hamiltonians that can be written as the sum of a gapped operator consisting of a sum of on-site terms and a perturbation given by relatively bounded (but unbounded) interaction potentials of short range multiplied by a real coupling constant t. For sufficiently small values of |t| independent of the size of the lattice, we prove that the spectral gap above the ground-state energy of such Hamiltonians remains strictly positive. As in [DFPR3], we iteratively construct a sequence of local block-diagonalization steps based on unitary conjugations of the original Hamiltonian and inspired by the Lie-Schwinger procedure. To control the ranges and supports of the effective potentials generated in the course of our block-diagonalization steps, we use methods introduced in [DFPR3] for Hamiltonians with bounded interactions potentials. However, due to the unboundedness of the interaction potentails, weighted operator norms must be introduced, and some of the steps of the inductive proof by which we control the weighted norms of the effective potentials require special care to cope with matrix elements of unbounded operators. We stress that no "large-field problems" appear in our construction. In this respect our operator methods turn out to be an efficient tool to separate the low-energy spectral region of the Hamiltonian from other spectral regions, where the unbounded nature of the interaction potentials would become manifest.
Communications in Mathematical Physics, Aug 30, 2016
NATO advanced study institutes series, 1994
Physical Review Letters, Jan 18, 2012
We show that the evolution of magnetic fields in a primordial plasma, filled with Standard Model ... more We show that the evolution of magnetic fields in a primordial plasma, filled with Standard Model particles, at temperatures T 10MeV is strongly affected by the quantum chiral anomaly -an effect that has been neglected previously. Although reactions equilibrating left and right-chiral electrons are in deep thermal equilibrium for T 80 TeV, an asymmetry between these particle develops in the presence of strong magnetic fields. This results in magnetic helicity transfer from shorter to longer scales. This also leads to an effective generation of lepton asymmetry that may survive in the plasma down to temperatures T ∼ 10 MeV, which may strongly affect many processes in the early Universe. Although we report our results for the Standard Model, they are likely to play an important role also in its extensions.
WORLD SCIENTIFIC eBooks, Jul 1, 2003
This note is dedicated to H. Ezawa on the occasion of his 70 th birthday, with respect and affect... more This note is dedicated to H. Ezawa on the occasion of his 70 th birthday, with respect and affection.
Communications in Mathematical Physics, Mar 20, 2020
We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short ran... more We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.
Letters in Mathematical Physics, May 1, 2005
Isothermal processes of a finitely extended, driven quantum system in contact with an infinite he... more Isothermal processes of a finitely extended, driven quantum system in contact with an infinite heat bath are studied from the point of view of quantum statistical mechanics. Notions like heat flux, work and entropy are defined for trajectories of states close to, but distinct from states of joint thermal equilibrium. A theorem characterizing reversible isothermal processes as quasi-static processes ("isothermal theorem") is described. Corollaries concerning the changes of entropy and free energy in reversible isothermal processes and on the 0th law of thermodynamics are outlined.
Annales Henri Poincaré, Apr 8, 2017
We study a large class of models of two-dimensional quantum lattice systems with continuous symme... more We study a large class of models of two-dimensional quantum lattice systems with continuous symmetries, and we prove a general McBryan-Spencer-Koma-Tasaki theorem concerning algebraic decay of correlations. We present applications of our main result to the Heisenberg, Hubbard, and t-J models, and to certain models of random loops.
Nuclear physics, Mar 1, 2002
We describe a new order parameter for the confinement-deconfinement transition in lattice SU(2) Y... more We describe a new order parameter for the confinement-deconfinement transition in lattice SU(2) Yang-Mills theory. It is expressed in terms of magnetic monopole field correlators represented as sums over sheets of center vortices. Our construction establishes a link between "abelian" and "center dominance". It avoids an inconsistency in the treatment of small scales present in earlier definitions of monopole fields by respecting Dirac's quantization condition for magnetic fluxes.
Communications in Mathematical Physics, Nov 27, 2019
We present a systematic analysis of quantum Heisenberg-, xyand interchange models on the complete... more We present a systematic analysis of quantum Heisenberg-, xyand interchange models on the complete graph. These models exhibit phase transitions accompanied by spontaneous symmetry breaking, which we study by calculating the generating function of expectations of powers of the averaged spin density. Various critical exponents are determined. Certain objects of the associated loop models are shown to have properties of Poisson-Dirichlet distributions.
Journal of Functional Analysis, Nov 1, 2020
We consider quantum chains whose Hamiltonians are perturbations by interactions of short range of... more We consider quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we have proven that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant; see [DFPR]. The main result of this paper is that, under the same hypotheses, the ground-state energy is analytic for values of the coupling constant belonging to a fixed interval, uniformly in the length of the chain. Furthermore, assuming that the interaction potentials are invariant under translations, we prove that, in the thermodynamic limit, the energy per site is analytic for values of the coupling constant in the same fixed interval. In our proof we use a new method introduced in [FP], which is based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain. We prove a rather strong result concerning complex Hamiltonians corresponding to complex values of the coupling constant.
Advances in Mathematics, Apr 1, 2016
We consider a non-relativistic quantum gas of N bosonic atoms confined to a box of volume Λ in ph... more We consider a non-relativistic quantum gas of N bosonic atoms confined to a box of volume Λ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, ρ = N Λ , of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume Λ and small ratio Λ ρ . The initial state of the gas is assumed to be close to a product state of one-particle wave functions that are approximately constant throughout the box. The initial one-particle wave function of an excitation is assumed to have a compact support independent of Λ. We derive an effective non-linear equation for the time evolution of the one-particle wave function of an excitation and establish an explicit error bound tracking the accuracy of the effective non-linear dynamics in terms of the ratio Λ ρ . We conclude with a discussion of the dispersion law of low-energy excitations, recovering Bogolyubov's well-known formula for the speed of sound in the gas, and a dynamical instability for attractive two-body potentials.
Journal of Mathematical Physics, May 1, 2015
Simple proofs of uniqueness of the thermodynamic limit of KMS states and of the decay of equilibr... more Simple proofs of uniqueness of the thermodynamic limit of KMS states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures. New quantum correlation inequalities for general Heisenberg models are described. Finally, a simplified derivation of a general result on power-law decay of correlations in 2D quantum lattice systems with continuous symmetries is given, extending results of Mc Bryan and Spencer for the 2D classical XY model.
Nuclear Physics B, Feb 1, 2000
Communications on Pure and Applied Mathematics, 2007
We consider the nonlinear wave equation modelling the dynamics of (pseudo-relativistic) boson sta... more We consider the nonlinear wave equation modelling the dynamics of (pseudo-relativistic) boson stars. For spherically symmetric initial data, u0(x) ∈ C ∞ c (R 3 ), with negative energy, we prove blowup of u(t, x) in H 1/2 -norm within a finite time. Physically, this phenomenon describes the onset of "gravitational collapse" of a boson star. We also study blow-up in external, spherically symmetric potentials and we consider more general Hartree-type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0.
arXiv (Cornell University), Aug 7, 2006
We study finite-time blow-up for pseudo-relativistic Hartree-and Hartree-Fock equations, which ar... more We study finite-time blow-up for pseudo-relativistic Hartree-and Hartree-Fock equations, which are model equations for the dynamical evolution of white dwarfs. In particular, we prove that radially symmetric initial configurations with negative energy lead to finite-time blow-up of solutions. Furthermore, we derive a mass concentration estimate for radial blow-up solutions. Both results are mathematically rigorous and are in accordance with Chandrasekhar's physical theory of white dwarfs, stating that stellar configurations beyond a certain limiting mass lead to "gravitational collapse" of these objects. Apart from studying blow-up, we also prove local well-posedness of the initial-value problem for the Hartree-and Hartree-Fock equations underlying our analysis, as well as global-in-time existence of solutions with sufficiently small initial data, corresponding to white dwarfs whose stellar mass is below the Chandrasekhar limit.
Communications in Mathematical Physics, Jun 1, 2003
We study the classical decay of unstable scalar solitons in noncommutative field theory in 2 + 1 ... more We study the classical decay of unstable scalar solitons in noncommutative field theory in 2 + 1 dimensions. This can, but does not have to, be viewed as a toy model for the decay of D-branes in string theory. In the limit that the noncommutativity parameter θ is infinite, the gradient term is absent, there are no propagating modes and the soliton does not decay at all. If θ is large, but finite, the rotationally symmetric decay channel can be described as a highly excited nonlinear oscillator weakly coupled to a continuum of linear modes. This system is closely akin to those studied in the context of discrete breathers. We here diagonalize the linear problem and compute the decay rate to first order using a version of Fermi's Golden Rule, leaving a more rigorous treatment for future work.
Communications in Mathematical Physics, 2020
We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short ran... more We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.
Journal of Mathematical Physics, 2015
Simple proofs of uniqueness of the thermodynamic limit of KMS states and of the decay of equilibr... more Simple proofs of uniqueness of the thermodynamic limit of KMS states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures. New quantum correlation inequalities for general Heisenberg models are described. Finally, a simplified derivation of a general result on power-law decay of correlations in 2D quantum lattice systems with continuous symmetries is given, extending results of Mc Bryan and Spencer for the 2D classical XY model.
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Papers by Juerg Froehlich