IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Annual review of condensed matter physics

We give a general introduction to quantum phase transitions in strongly-correlated electron systems. These transitions which occur at zero temperature when a non-thermal parameter $g$ like pressure, chemical composition or magnetic field is tuned to a critical value are characterized by a dynamic exponent $z$ related to the energy and length scales $\Delta$ and $\xi$. Simple arguments based on an expansion to first order in the effective interaction allow to define an upper-critical dimension $D_{C}=4$ (where $D=d+z$ and $d$ is the spatial dimension) below which mean-field description is no longer valid. We emphasize the role of pertubative renormalization group (RG) approaches and self-consistent renormalized spin fluctuation (SCR-SF) theories to understand the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems. Finally we quote some recent inelastic neutron scattering experiments performed on heavy-fermions which lead to unusual scaling law in $\omega /T$ for the dynamical spin susceptibility revealing critical local modes beyond the itinerant magnetism scheme and mention new attempts to describe this local quantum critical point.

https://arxiv.org/pdf/cond-mat/0102119.pdf

Quantum Phase Transitions

A wide range of materials, like d-wave superconductors, graphene, and topological insulators, share a fundamental similarity: their low-energy fermionic excitations behave as massless Dirac particles rather than fermions obeying the usual Schrodinger Hamiltonian This emergent behavior of Dirac fermions in condensed matter systems defines the unifying framework for a class of materials we call "Dirac materials'' In order to establish this class of materials, we illustrate how Dirac fermions emerge in multiple entirely different condensed matter systems and we discuss how Dirac fermions have been identified experimentally using electron spectroscopy techniques (angle-resolved photoemission spectroscopy and scanning tunneling spectroscopy) As a consequence of their common low-energy excitations, this diverse set of materials shares a significant number of universal properties in the low-energy (infrared) limit We review these common properties including nodal points in the excitation spectrum, density of states, specific heat, transport, thermodynamic properties, impurity resonances, and magnetic field responses, as well as discuss many-body interaction effects We further review how the emergence of Dirac excitations is controlled by specific symmetries of the material, such as time-reversal, gauge, and spin-orbit symmetries, and how by breaking these symmetries a finite Dirac mass is generated We give examples of how the interaction of Dirac fermions with their distinct real material background leads to rich novel physics with common fingerprints such as the suppression of back scattering and impurity-induced resonant states

Dirac materials

Advances in physics

We study the chiral Ising, the chiral XY, and the chiral Heisenberg models at four-loop order with the perturbative renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and compute critical exponents for the Gross-Neveu-Yukawa fixed points to order $\mathcal{O}({\ensuremath{\epsilon}}^{4})$. Further, we provide Pad\'e estimates for the correlation length exponent, the boson and fermion anomalous dimension, as well as the leading correction to scaling exponent in $2+1$ dimensions. We also confirm the emergence of supersymmetric field theories at four loops for the chiral Ising and the chiral XY models with $N=1/4$ and $N=1/2$ fermions, respectively. Furthermore, applications of our results relevant to various quantum transitions in the context of Dirac and Weyl semimetals are discussed, including interaction-induced transitions in graphene and surface states of topological insulators.

/pdf/four-loop-critical-exponents-for-the-gross-neveu-yukawa-5nu7vmgznf.pdf

Four-loop critical exponents for the Gross-Neveu-Yukawa models

We consider the asymptotic-safety scenario for quantum gravity which constructs a nonperturbatively renormalizable quantum gravity theory with the help of the functional renormalization group (RG). We verify the existence of a non-Gaussian fixed point and include a running curvature-ghost coupling as a first step towards the flow of the ghost sector of the theory. We find that the scalar curvature-ghost coupling is asymptotically free and RG relevant in the ultraviolet. Most importantly, the property of asymptotic safety discovered so far within the Einstein-Hilbert truncation and beyond remains stable under the inclusion of the ghost flow.

Asymptotically free scalar curvature-ghost coupling in quantum Einstein gravity

In view of the measured Higgs mass of 125 GeV, the perturbative renormalization group evolution of the Standard Model suggests that our Higgs vacuum might not be stable. We connect the usual perturbative approach and the functional renormalization group which allows for a straightforward inclusion of higher-dimensional operators in the presence of an ultraviolet cutoff. In the latter framework we study vacuum stability in the presence of higher-dimensional operators. We find that their presence can have a sizable influence on the maximum ultraviolet scale of the Standard Model and the existence of instabilities. Finally, we discuss how such operators can be generated in specific models and study the relation between the instability scale of the potential and the scale of new physics required to avoid instabilities.

/pdf/the-higgs-mass-and-the-scale-of-new-physics-16j48npre4.pdf

The Higgs mass and the scale of new physics

Dirac and Weyl fermions appear as quasiparticle excitations in many different condensed-matter systems. They display various quantum transitions which represent unconventional universality classes related to the variants of the Gross-Neveu model. In this paper we study the bosonized version of the standard Gross-Neveu model---the Gross-Neveu-Yukawa theory---at three-loop order, and compute critical exponents in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions for a general number of fermion flavors. Our results fully encompass the previously known two-loop calculations, and agree with the known three-loop results in the purely bosonic limit of the theory. We also find the exponents to satisfy the emergent superscaling relations in the limit of a single-component fermion, order by order up to three loops. Finally, we apply the computed series for the exponents and their Pad\'e approximants to several phase transitions of current interest: metal-insulator transitions of spin-1/2 and spinless fermions on the honeycomb lattice, emergent supersymmetric surface field theory in topological phases, as well as the disorder-induced quantum transition in Weyl semimetals. Comparison with the results of other analytical and numerical methods is discussed.

Gross-Neveu-Yukawa model at three loops and Ising critical behavior of Dirac systems

The effect of particle-hole fluctuations for the BCS-BEC crossover is investigated by use of functional renormalization. We compute the Gorkov effect and the critical temperature for the whole range in the scattering length $a$. On the BCS side for small negative $a$ we recover the Gorkov approximation, while on the BEC side of small positive $a$ the particle-hole fluctuations play no important role, and we find a system of interacting bosons. In the unitarity limit of infinite scattering length our quantitative estimate yields ${T}_{c}/{T}_{F}=0.264$. We also investigate the crossover from broad to narrow Feshbach resonances---for the latter we obtain ${T}_{c}/{T}_{F}=0.204$ for ${a}^{\ensuremath{-}1}=0$. A key ingredient for our treatment is the computation of the momentum dependent four-fermion vertex and its bosonization in terms of an effective bound-state exchange.

Michael M. Scherer

Papers

Four-loop critical exponents for the Gross-Neveu-Yukawa models

Asymptotically free scalar curvature-ghost coupling in quantum Einstein gravity

The Higgs mass and the scale of new physics

Gross-Neveu-Yukawa model at three loops and Ising critical behavior of Dirac systems

Particle-hole fluctuations in BCS-BEC crossover