Dr. Benjamin Hinrichs

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Mathematische Physik komplexer Quantensysteme

Nachwuchsgruppenleiter - Wissenschaftlicher Mitarbeiter

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+49 5251 60-2648

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benjamin.hinrichs(at)math.upb(dot)de

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J2.322

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33098 Paderborn

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2023

Non-Fock Ground States in the Translation-Invariant Nelson Model Revisited Non-Perturbatively

D. Hasler, B. Hinrichs, O. Siebert, 2023

Abstract

arXiv

The Nelson model, describing a quantum mechanical particle linearly coupled to a bosonic field, exhibits the infrared problem in the sense that no ground state exists at arbitrary total momentum. However, passing to a non-Fock representation, one can prove the existence of so-called dressed one-particle states. In this article, we give a simple non-perturbative proof for the existence of such one-particle states at arbitrary coupling strength and for almost all total momenta in a physically motivated momentum region. Our results hold both for the non- and the semi-relativistic Nelson model.

2022

Existence of Ground States in the Infrared-Critial Spin Boson Model

B. Hinrichs, in: Mathematical aspects of quantum fields and related topics, 2022, pp. 60-73

Abstract

arXiv

We review recent results on the existence of ground states for the infrared-critical spin boson model, which describes the interaction of a massless bosonic field with a two-state quantum system. Explicitly, we derive a critical coupling $\lambda_{\mathsf c}>0$ such that the spin boson model exhibits a ground state for coupling constants $\lambda$ with $|\lambda|<\lambda_{\mathsf c}$. The proof combines a Feynman-Kac-Nelson formula for the spin boson model with external magnetic field, a 1D-Ising model correlation bound and a compactness argument in Fock space. Elaborating on the connection to a long-range 1D-Ising model, we briefly discuss the conjecture that the spin boson model does not have a ground state at large coupling. This note is based on joint work with David Hasler and Oliver Siebert.

Absence of ground states in the renormalized massless translation-invariant Nelson model

T.N. Dam, B. Hinrichs, Reviews in Mathematical Physics (2022), 34(10)

Abstract

DOI

arXiv

We consider a model for a massive uncharged non-relativistic particle interacting with a massless bosonic field, widely referred to as the Nelson model. It is well known that an ultraviolet renormalized Hamilton operator exists in this case. Further, due to translation-invariance, it decomposes into fiber operators. In this paper, we treat the renormalized fiber operators. We give a description of the operator and form domains and prove that the fiber operators do not have a ground state. Our results hold for any non-zero coupling constant and arbitrary total momentum. Our proof for the absence of ground states is a new generalization of methods recently applied to related models. A major enhancement we provide is that the method can be applied to models with degenerate ground state eigenspaces.

FKN Formula and Ground State Energy for the Spin Boson Model with External Magnetic Field

D. Hasler, B. Hinrichs, O. Siebert, Annales Henri Poincaré (2022), 23(8), pp. 2819-2853

Abstract

DOI

arXiv

We consider the spin boson model with external magnetic field. We prove a path integral formula for the heat kernel, known as Feynman–Kac–Nelson (FKN) formula. We use this path integral representation to express the ground state energy as a stochastic integral. Based on this connection, we determine the expansion coefficients of the ground state energy with respect to the magnetic field strength and express them in terms of correlation functions of a continuous Ising model. From a recently proven correlation inequality, we can then deduce that the second order derivative is finite. As an application, we show existence of ground states in infrared-singular situations.

Feynman-Kac formula and asymptotic behavior of the minimal energy for the relativistic Nelson model in two spatial dimensions

B. Hinrichs, O. Matte, 2022

Abstract

arXiv

We consider the renormalized relativistic Nelson model in two spatial dimensions for a finite number of spinless, relativistic quantum mechanical matter particles in interaction with a massive scalar quantized radiation field. We find a Feynman-Kac formula for the corresponding semigroup and discuss some implications such as ergodicity and weighted $L^p$ to $L^q$ bounds, for external potentials that are Kato decomposable in the suitable relativistic sense. Furthermore, our analysis entails upper and lower bounds on the minimal energy for all values of the involved physical parameters when the Pauli principle for the matter particles is ignored. In the translation invariant case (no external potential) these bounds permit to compute the leading asymptotics of the minimal energy in the three regimes where the number of matter particles goes to infinity, the coupling constant for the matter-radiation interaction goes to infinity and the boson mass goes to zero.

Super-Gaussian Decay of Exponentials: A Sufficient Condition

B. Hinrichs, D.W. Janssen, J. Ziebell, 2022

Abstract

arXiv

In this article, we present a sufficient condition for the exponential $\exp({-f})$ to have a tail decay stronger than any Gaussian, where $f$ is defined on a locally convex space $X$ and grows faster than a squared seminorm on $X$. In particular, our result proves that $\exp({-p(x)^{2+\varepsilon}+\alpha q(x)^2})$ is integrable for all $\alpha,\varepsilon>0$ w.r.t. a Radon Gaussian measure on a nuclear space $X$, if $p$ and $q$ are continuous seminorms on $X$ with compatible kernels. This can be viewed as an adaptation of Fernique's theorem and, for example, has applications in quantum field theory.

Existence of Ground States for Infrared-Critical Models of Quantum Field Theory

B. Hinrichs, 2022

DOI

2021

Correlation bound for a one-dimensional continuous long-range Ising model

D. Hasler, B. Hinrichs, O. Siebert, Stochastic Processes and their Applications (2021), 146, pp. 60-79

Abstract

DOI

arXiv

We consider a measure given as the continuum limit of a one-dimensional Ising model with long-range translationally invariant interactions. Mathematically, the measure can be described by a self-interacting Poisson driven jump process. We prove a correlation inequality, estimating the magnetic susceptibility of this model, which holds for small norm of the interaction function. The bound on the magnetic susceptibility has applications in quantum field theory and can be used to prove existence of ground states for the spin boson model.

On Existence of Ground States in the Spin Boson Model

D. Hasler, B. Hinrichs, O. Siebert, Communications in Mathematical Physics (2021), 388(1), pp. 419-433

Abstract

DOI

arXiv

We show the existence of ground states in the massless spin boson model without any infrared regularization. Our proof is non-perturbative and relies on a compactness argument. It works for arbitrary values of the coupling constant under the hypothesis that the second derivative of the ground state energy as a function of a constant external magnetic field is bounded.

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