TY - JOUR
AB - We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.
AU - Henheik, Sven Joscha
AU - Teufel, Stefan
ID - 10600
IS - 1
JF - Journal of Mathematical Physics
KW - mathematical physics
KW - statistical and nonlinear physics
SN - 0022-2488
TI - Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap
VL - 63
ER -
TY - JOUR
AB - We investigate the BCS critical temperature Tc in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of Tc at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory.
AU - Henheik, Sven Joscha
ID - 10623
IS - 1
JF - Mathematical Physics, Analysis and Geometry
KW - geometry and topology
KW - mathematical physics
SN - 1385-0172
TI - The BCS critical temperature at high density
VL - 25
ER -
TY - JOUR
AB - Based on a result by Yarotsky (J Stat Phys 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this a strong version of the local perturbations perturb locally (LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences.
AU - Henheik, Sven Joscha
AU - Teufel, Stefan
AU - Wessel, Tom
ID - 10642
IS - 1
JF - Letters in Mathematical Physics
KW - mathematical physics
KW - statistical and nonlinear physics
SN - 0377-9017
TI - Local stability of ground states in locally gapped and weakly interacting quantum spin systems
VL - 112
ER -
TY - JOUR
AB - We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap.
AU - Henheik, Sven Joscha
AU - Teufel, Stefan
ID - 10643
JF - Forum of Mathematics, Sigma
KW - computational mathematics
KW - discrete mathematics and combinatorics
KW - geometry and topology
KW - mathematical physics
KW - statistics and probability
KW - algebra and number theory
KW - theoretical computer science
KW - analysis
TI - Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk
VL - 10
ER -
TY - JOUR
AB - We review old and new results on the Fröhlich polaron model. The discussion includes the validity of the (classical) Pekar approximation in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass.
AU - Seiringer, Robert
ID - 10852
IS - 01
JF - Reviews in Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0129-055X
TI - The polaron at strong coupling
VL - 33
ER -
TY - JOUR
AB - We show that the energy gap for the BCS gap equation is
Ξ=μ(8e−2+o(1))exp(π2μ−−√a)
in the low density limit μ→0. Together with the similar result for the critical temperature by Hainzl and Seiringer (Lett Math Phys 84: 99–107, 2008), this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential V. The results hold for a class of potentials with negative scattering length a and no bound states.
AU - Lauritsen, Asbjørn Bækgaard
ID - 9121
JF - Letters in Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0377-9017
TI - The BCS energy gap at low density
VL - 111
ER -
TY - JOUR
AB - We first review the problem of a rigorous justification of Kubo’s formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo’s formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper, we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.
AU - Henheik, Sven Joscha
AU - Teufel, Stefan
ID - 9285
IS - 01
JF - Reviews in Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0129-055X
TI - Justifying Kubo’s formula for gapped systems at zero temperature: A brief review and some new results
VL - 33
ER -
TY - JOUR
AB - Extending on ideas of Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)], we present a modified “floating crystal” trial state for jellium (also known as the classical homogeneous electron gas) with density equal to a characteristic function. This allows us to show that three definitions of the jellium energy coincide in dimensions d ≥ 2, thus extending the result of Cotar and Petrache [“Equality of the Jellium and uniform electron gas next-order asymptotic terms for Coulomb and Riesz potentials,” arXiv: 1707.07664 (2019)] and Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)] that the three definitions coincide in dimension d ≥ 3. We show that the jellium energy is also equivalent to a “renormalized energy” studied in a series of papers by Serfaty and others, and thus, by the work of Bétermin and Sandier [Constr. Approximation 47, 39–74 (2018)], we relate the jellium energy to the order n term in the logarithmic energy of n points on the unit 2-sphere. We improve upon known lower bounds for this renormalized energy. Additionally, we derive formulas for the jellium energy of periodic configurations.
AU - Lauritsen, Asbjørn Bækgaard
ID - 9891
IS - 8
JF - Journal of Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0022-2488
TI - Floating Wigner crystal and periodic jellium configurations
VL - 62
ER -
TY - JOUR
AB - In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.
AU - Wirth, Melchior
AU - Zhang, Haonan
ID - 9973
JF - Communications in Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0010-3616
TI - Complete gradient estimates of quantum Markov semigroups
VL - 387
ER -
TY - JOUR
AB - We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit.
AU - Bálint, Péter
AU - De Simoi, Jacopo
AU - Kaloshin, Vadim
AU - Leguil, Martin
ID - 8415
IS - 3
JF - Communications in Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0010-3616
TI - Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards
VL - 374
ER -
TY - JOUR
AB - The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum. The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold P+ (resp. P−). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity P∞, which we call the manifold at parabolic infinity. On P∞ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside P∞ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.
AU - Delshams, Amadeu
AU - Kaloshin, Vadim
AU - de la Rosa, Abraham
AU - Seara, Tere M.
ID - 8417
IS - 3
JF - Communications in Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0010-3616
TI - Global instability in the restricted planar elliptic three body problem
VL - 366
ER -
TY - JOUR
AB - We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number 1/q is polynomially sense in the smooth case, and exponentially dense in the analytic case.
AU - Kaloshin, Vadim
AU - Zhang, Ke
ID - 8420
IS - 11
JF - Nonlinearity
KW - Mathematical Physics
KW - General Physics and Astronomy
KW - Applied Mathematics
KW - Statistical and Nonlinear Physics
SN - 0951-7715
TI - Density of convex billiards with rational caustics
VL - 31
ER -
TY - JOUR
AB - In the present note we announce a proof of a strong form of Arnold diffusion for smooth convex Hamiltonian systems. Let ${\mathbb T}^2$ be a 2-dimensional torus and B2 be the unit ball around the origin in ${\mathbb R}^2$ . Fix ρ > 0. Our main result says that for a 'generic' time-periodic perturbation of an integrable system of two degrees of freedom $H_0(p)+\varepsilon H_1(\theta,p,t),\quad \ \theta\in {\mathbb T}^2,\ p\in B^2,\ t\in {\mathbb T}={\mathbb R}/{\mathbb Z}$ , with a strictly convex H0, there exists a ρ-dense orbit (θε, pε, t)(t) in ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ , namely, a ρ-neighborhood of the orbit contains ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ .
Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are the usage of crumpled normally hyperbolic invariant cylinders from [9], flower and simple normally hyperbolic invariant manifolds from [36] as well as their kissing property at a strong double resonance. This allows us to build a 'connected' net of three-dimensional normally hyperbolic invariant manifolds. To construct diffusing orbits along this net we employ a version of the Mather variational method [41] equipped with weak KAM theory [28], proposed by Bernard in [7].
AU - Kaloshin, Vadim
AU - Zhang, K
ID - 8498
IS - 8
JF - Nonlinearity
KW - Mathematical Physics
KW - General Physics and Astronomy
KW - Applied Mathematics
KW - Statistical and Nonlinear Physics
SN - 0951-7715
TI - Arnold diffusion for smooth convex systems of two and a half degrees of freedom
VL - 28
ER -
TY - JOUR
AB - The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462–465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797–808, 1998) proposed to look for an example of a Hamiltonian near H0(I)=⟨I,I⟩2 with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian H0(I)+εH1(θ,I,ε) which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.
AU - Kaloshin, Vadim
AU - Saprykina, Maria
ID - 8502
IS - 3
JF - Communications in Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0010-3616
TI - An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension
VL - 315
ER -
TY - JOUR
AB - Let M be a smooth compact manifold of dimension at least 2 and Diffr(M) be the space of C r smooth diffeomorphisms of M. Associate to each diffeomorphism f;isin; Diffr(M) the sequence P n (f) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diffr(M) such for a Baire generic diffeomorphism f∈N the number of periodic points P n f grows with a period n faster than any following sequence of numbers {a n } n ∈ Z + along a subsequence, i.e. P n (f)>a ni for some n i →∞ with i→∞. In the cases of surface diffeomorphisms, i.e. dim M≡2, an open set N with a supergrowth of the number of periodic points is a Newhouse domain. A proof of the man result is based on the Gontchenko–Shilnikov–Turaev Theorem [GST]. A complete proof of that theorem is also presented.
AU - Kaloshin, Vadim
ID - 8525
JF - Communications in Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0010-3616
TI - Generic diffeomorphisms with superexponential growth of number of periodic orbits
VL - 211
ER -
TY - JOUR
AB - We introduce a new potential-theoretic definition of the dimension spectrum of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if and is a Borel probability measure with compact support in , then under almost every linear transformation from to , the q-dimension of the image of is ; in particular, the q-dimension of is preserved provided . We also present results on the preservation of information dimension and pointwise dimension. Finally, for and q > 2 we give examples for which is not preserved by any linear transformation into . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.
AU - Hunt, Brian R
AU - Kaloshin, Vadim
ID - 8527
IS - 5
JF - Nonlinearity
KW - Mathematical Physics
KW - General Physics and Astronomy
KW - Applied Mathematics
KW - Statistical and Nonlinear Physics
SN - 0951-7715
TI - How projections affect the dimension spectrum of fractal measures
VL - 10
ER -