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Manfred Salmhofer

Publications and source records attributed to Manfred Salmhofer.

At least 37 records · Page 2Linked to original sources
arXiv2018

Renormalization in Condensed Matter: Fermionic Systems - from Mathematics to Materials

Renormalization plays an important role in the theoretically and mathematically careful analysis of models in condensed-matter physics. I review selected results about correlated-fermion systems, ranging from mathematical theorems to applications in models relevant for materials science, such as the prediction of equilibrium phases of systems with competing ordering tendencies, and quantum criticality.

cond-mat.str-el↗
arXiv2018

Persistence of exponential decay and spectral gaps for interacting fermions

We consider systems of weakly interacting fermions on a lattice. The corresponding free fermionic system is assumed to have a ground state separated by a gap from the rest of the spectrum. We prove that, if both the interaction and the free Hamiltonian are sums of sufficiently rapidly decaying terms, and if the interaction is sufficiently weak, then the interacting system has a spectral gap as well, uniformly in the lattice size. Our approach relies on convergent fermionic perturbation theory, thus providing an alternative method to the one used recently in [MB Hastings. arXiv:1706.02270], and extending the result to include non-selfadjoint interaction terms.

math-ph↗
arXiv2013

Schwinger-Dyson Renormalization Group

We use the Schwinger-Dyson equations as a starting point to derive renormalization flow equations. We show that Katanin's scheme arises as a simple truncation of these equations. We then give the full renormalization group equations up to third order in the irreducible vertex. Furthermore, we show that to the fifth order, there exists a functional of the self-energy and the irreducible four-point vertex whose saddle point is the solution of Schwinger-Dyson equations.

cond-mat.str-el↗
arXiv2012

Self-energy flows in the two-dimensional repulsive Hubbard model

We study the two-dimensional repulsive Hubbard model by functional RG methods, using our recently proposed channel decomposition of the interaction vertex. The main technical advance of this work is that we calculate the full Matsubara frequency dependence of the self-energy and the interaction vertex in the whole frequency range without simplifying assumptions on its functional form, and that the effects of the self-energy are fully taken into account in the equations for the flow of the two-body vertex function. At Van Hove filling, we find that the Fermi surface deformations remain small at fixed particle density and have a minor impact on the structure of the interaction vertex. The frequency dependence of the self-energy, however, turns out to be important, especially at a transition from ferromagnetism to d-wave superconductivity. We determine non-Fermi-liquid exponents at this transition point.

cond-mat.str-el↗
arXiv2012

Functional renormalization group approach to correlated fermion systems

Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems, (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals, (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and non-equilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. We present a self-contained derivation of the exact flow equations and describe frequently used truncation schemes. Reviewing selected applications we then show how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems.

cond-mat.str-el↗
arXiv2011

Frequency Dependent Vertex Functions of the (t,t')-Hubbard Model at Weak Coupling

We present a functional renormalization group calculation for the two-dimensional (t,t')-Hubbard model at Van Hove filling. Using a channel decomposition we describe the momentum and frequency dependence of the vertex function in the normal phase. Compared to previous studies that neglect frequency dependences we find higher pseudo-critical scales and a smaller region of d-wave superconductivity. A large contribution to the effective interaction is given by a forward scattering process with finite frequency exchange. We test different frequency parameterizations and in a second calculation include the frequency dependence of the imaginary self-energy. We also generalize the channel decomposition to frequency-dependent fermion-boson vertex functions.

cond-mat.str-el↗
arXiv2009

Smoothing of Singular Legendre Transforms in Renormalization Group Flows

We consider O(N)-symmetric potentials with a logarithmic singularity in the second field derivative. This class includes BCS and Gross Neveu potentials. Formally, the exact renormalization group equation for the Legendre transform of these potentials seems to have ill-defined initial conditions. We show that the renormalization group equation for the local potential has well-defined initial conditions and that the logarithmic singularity is smoothed rapidly in the flow. Our analysis also provides an efficient method for numerical studies.

cond-mat.str-el↗
arXiv2009

Efficient Parametrization of the Vertex Function, $Ω$-Scheme, and the (t,t')-Hubbard Model at Van Hove Filling

We propose a new parametrization of the four-point vertex function in the one-loop one-particle irreducible renormalization group (RG) scheme for fermions. It is based on a decomposition of the effective two-fermion interaction into fermion bilinears that interact via exchange bosons. The numerical computation of the RG flow of the boson propagators reproduces the leading weak coupling instabilities of the two-dimensional Hubbard model at Van Hove filling, as they were previously obtained by a temperature RG flow. Instead of regularizing with temperature, we here use a soft frequency $Ω$-regularization that likewise does not artificially suppress ferromagnetism. Besides being more efficient than previous N-patch schemes, this parametrization also reduces the ambiguities in introducing boson fields.

cond-mat.str-el↗
arXiv2009

Efficient Fermionic One-Loop RG for the 2D Hubbard Model at Van Hove Filling

We propose a novel parametrization of the four-point vertex function in the one-loop one-particle irreducible renormalization group (RG) scheme for fermions. It is based on a decomposition of the effective two-fermion interaction into fermion bilinears that interact via exchange bosons. Besides being more efficient than previous N-patch schemes, this parametrization also reduces the ambiguity of introducing boson fields. We apply this parametrization to the two-dimensional (t,t')-Hubbard model using a novel $Ω$-frequency regularization.

cond-mat.str-el↗
arXiv2008

Feynman graphs and renormalization in quantum diffusion

We review our proof that in a scaling limit, the time evolution of a quantum particle in a static random environment leads to a diffusion equation. In particular, we discuss the role of Feynman graph expansions and of renormalization.

math-ph↗
arXiv2007

Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems

It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the covariance admits a Gram representation with a finite Gram constant. The covariances of the standard many--fermion systems do not fall into this class due to the slow decay of the covariance at large Matsubara frequency, giving rise to a UV problem in the integration over degrees of freedom with Matsubara frequencies larger than some Omega (usually the first step in a multiscale analysis). We show that these covariances do not have Gram representations on any separable Hilbert space. We then prove a general bound for determinants associated to chronological products which is stronger than the usual Gram bound and which applies to the many--fermion case. This allows us to prove convergence of the first integration step in a rather easy way, for a short--range interaction which can be arbitrarily strong, provided Omega is chosen large enough. Moreover, we give - for the first time - nonperturbative bounds on all scales for the case of scale decompositions of the propagator which do not impose cutoffs on the Matsubara frequency.

math-ph↗
arXiv2007

Singular Fermi Surfaces I. General Power Counting and Higher Dimensional Cases

We prove regularity properties of the self-energy, to all orders in perturbation theory, for systems with singular Fermi surfaces which contain Van Hove points where the gradient of the dispersion relation vanishes. In this paper, we show for spatial dimensions $d \ge 3$ that despite the Van Hove singularity, the overlapping loop bounds we proved together with E. Trubowitz for regular non--nested Fermi surfaces [J. Stat. Phys. 84 (1996) 1209] still hold, provided that the Fermi surface satisfies a no-nesting condition. This implies that for a fixed interacting Fermi surface, the self-energy is a continuously differentiable function of frequency and momentum, so that the quasiparticle weight and the Fermi velocity remain close to their values in the noninteracting system to all orders in perturbation theory. In a companion paper, we treat the more singular two-dimensional case.

math-ph↗
arXiv2007

Singular Fermi Surfaces II. The Two--Dimensional Case

We consider many--fermion systems with singular Fermi surfaces, which contain Van Hove points where the gradient of the band function $k \mapsto e(k)$ vanishes. In a previous paper, we have treated the case of spatial dimension $d \ge 3$. In this paper, we focus on the more singular case $d=2$ and establish properties of the fermionic self--energy to all orders in perturbation theory. We show that there is an asymmetry between the spatial and frequency derivatives of the self--energy. The derivative with respect to the Matsubara frequency diverges at the Van Hove points, but, surprisingly, the self--energy is $C^1$ in the spatial momentum to all orders in perturbation theory, provided the Fermi surface is curved away from the Van Hove points. In a prototypical example, the second spatial derivative behaves similarly to the first frequency derivative. We discuss the physical significance of these findings.

math-ph↗
arXiv2007

Quantum diffusion of the random Schrodinger evolution in the scaling limit

We consider random Schrödinger equations on $\bR^d$ for $d\ge 3$ with a homogeneous Anderson-Poisson type random potential. Denote by $λ$ the coupling constant and $ψ_t$ the solution with initial data $ψ_0$. The space and time variables scale as $x\sim λ^{-2 -κ/2}, t \sim λ^{-2 -κ}$ with $0< κ< κ_0(d)$. We prove that, in the limit $λ\to 0$, the expectation of the Wigner distribution of $ψ_t$ converges weakly to the solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data. The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the {\it non-ladder} diagrams is smaller than their "naive size" by an extra $λ^c$ factor {\em per non-(anti)ladder vertex} for some $c > 0$. This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series.

math-ph↗
arXiv2007

Quantum diffusion for the Anderson model in the scaling limit

We consider random Schrödinger equations on $\bZ^d$ for $d\ge 3$ with identically distributed random potential. Denote by $λ$ the coupling constant and $ψ_t$ the solution with initial data $ψ_0$. The space and time variables scale as $x\sim λ^{-2 -κ/2}, t \sim λ^{-2 -κ}$ with $0< κ< κ_0(d)$. We prove that, in the limit $λ\to 0$, the expectation of the Wigner distribution of $ψ_t$ converges weakly to a solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum $v$. This work is an extension to the lattice case of our previous result in the continuum \cite{ESYI}, \cite{ESYII}. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved.

math-ph↗
arXiv2006

Dynamical Adjustment of Propagators in Renormalization Group Flows

A class of continuous renormalization group flows with a dynamical adjustment of the propagator is introduced and studied theoretically for fermionic and bosonic quantum field theories. The adjustment allows to include self--energy effects nontrivially in the denominator of the propagator and to adapt the scale decomposition to a moving singularity, and hence to define flows of Fermi surfaces in a natural way. These flows require no counterterms, but the counterterms used in earlier treatments can be constructed using them. The influence of propagator adjustment on the strong--coupling behaviour of flows is examined for a simple example, and some conclusions about the strong coupling behaviour of renormalization group flows are drawn.

cond-mat.stat-mech↗
arXiv2006

Quantum diffusion of the random Schrodinger evolution in the scaling limit II. The recollision diagrams

We consider random Schrödinger equations on $\bR^d$ for $d\ge 3$ with a homogeneous Anderson-Poisson type random potential. Denote by $λ$ the coupling constant and $ψ_t$ the solution with initial data $ψ_0$. The space and time variables scale as $x\sim λ^{-2 -κ/2}, t \sim λ^{-2 -κ}$ with $0< κ< κ_0(d)$. We prove that, in the limit $λ\to 0$, the expectation of the Wigner distribution of $ψ_t$ converges weakly to the solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper the analysis of the non-repetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation.

math-ph↗