Renormalization Theory in Condensed Matter Physics

Transcription

1 Renormalization Theory in Condensed Matter Physics Manfred Salmhofer Universität Heidelberg Wolfhart Zimmermann Memorial Symposium Max-Planck-Institut für Physik, München, May 23, 2017

2 Specification and Historical Note Renormalization, as discussed in this talk, is based on the Renormalization Group of Kadanoff and Wilson, and used by a group of people that work in constructive field theory, which aims at mathematically constructing quantum field theory, given a Lagrangian or Hamiltonian. There is a part of the history of renormalization theory that often goes unmentioned but that has led to very important non-perturbative results, both mathematical and in applications. A partial list is Aizenman, Fröhlich, Lüscher, Weisz. Lüscher, Osterwalder, Seiler, Borgs,... Triviality of φ 4 models Lattice gauge theories, in particular QCD Gawȩdzki, Kupiainen and, independently Feldman, Magnen, Rivasseau, Sénéor. Infrared φ 4 in four dimensions, Gross-Neveu model in 2, 2 + ε, 3 dimensions Ba laban. Stability of Yang-Mills-theory in d = 4, nonlinear sigma models in three dimensions Brydges, Kennedy; Abdesselam Rivasseau. Tree and forest formulas and their derivation from Wilsonian RG equations.

3 Functional integral of quantum statistical mechanics Partition function for inverse temperature β = 1 k B T Z = Tr e β(h µn) = D φ S( φ,φ) Dφ e and chemical potential µ S( φ, φ) = D φ Dφ = β β dτ ( φ(τ), ( + µ)φ(τ)) τ X dτ H( φ(τ), φ(τ)) 0 0 d φ(τ, x)dφ(τ, x), (φ(τ),ψ(τ)) X = φ(τ, x)ψ(τ, x)dx τ [0,β) x X Fields φ(τ, x) and φ(τ, x): complex for bosons, Grassmann for fermions. Boundary condition in time: periodic for bosons, antiperiodic for fermions. H = H(a, a ) is the Hamiltonian in second-quantized representation. X Correlation functions A = 1 Z D φ Dφ e S( φ,φ) A( φ, φ)

4 Strictly speaking To make the trace well-defined, X must have a finite volume X, and a shortdistance (UV) regularization in x is required, e.g. by making X a lattice or restricting to smooth functions. The thermodynamic limit X will be taken. The UV regularization is left in place here. The Euclidian time τ arises from a Trotter formula, so the continuum formulation given here is the limit of a discrete-time version. It is nontrivial to prove that the limit exists. For bosons, this was done recently in a series of papers by T. Ba laban, J. Feldman, H. Knörrer, and E. Trubowitz. (Decimation-type renormalization group argument, as well as analysis of oscillatory integrals: the φ τ φ term has no positivity properties it is not even real.) For fermions, the situation is simpler a proof avoiding a multiscale analysis was given by W. Pedra & M.S. The time continuum limit has also been shown to exist (in a form useful for the analysis of the thermodynamic limit) for a class of Bose-Fermi systems by M.A.Nitzschner & MS. These proofs also work for real time.

5 Generic example A gas of quantum particles interacting by a stable pair potential v(x y) H( φ, φ) = 1 2 ( φ, E φ) X + λ( φφ, v φφ) X λ is a coupling constant. The kinetic operator E in the quadratic part of the Hamiltonian can be or a suitable discretization thereof, or a hopping term for a lattice system, or come from a periodic Schrödinger operator. At positive temperature T = 1 k B β, the variable k 0 dual to τ is discrete because of the (anti-)periodic boundary conditions for the fields. In terms of the Matsubara frequencies ω n = nπt, k 0 M B = {ω n : n even } for bosons, and k 0 M F = {ω n : n odd } for fermions.

6 Small denominators From the point of view of analysis, the renormalization problem can be viewed as a small-denominator problem. If E has spectrum ε α, then the covariance has an eigenfunction expansion in which C(k 0, α) = 1 ik 0 ε α + µ appears. When T gets small, the denominator can get small, depending on µ. Positive temperature, i.e. β <, poses a natural infrared cutoff, but in the limit T 0, infrared divergences arise in Feynman graph expansions. Since temperature is a physical parameter, it is important to ask what properties are uniform in temperature, or how things depend on temperature. A simple regularization is to take Ω > 0, replace C by C Ω (k 0, α) = 1 ik 0 ε α + µ k 2 0 Ω 2 + k 2 0 = Q Ω (k 0, α) 1, and study Ω 0 (many other choices of regulator have been used as well)

7 Translation-invariant case In this case, α = (k, σ), where k is (quasi)momentum and σ is a spin index. For X, k becomes a continuous variable. The UV regulator implies that large k do not occur (either due to a cutoff, or a lattice spacing). The ε α is then replaced by a function (k, σ) h(k), where h is a matrix in the spin variables. In the simplest case, h σσ (k) = ε(k) δ σσ. The nature of the singularity (pointlike or extended) depends on µ. For free (λ = 0) bosons, µ < 0 is needed in the grand canonical ensemble. For λ > 0, µ 0 is possible, and required to increase the density. For fermions, µ R. If µ > inf α ε α, then one has a degenerate Fermi system with a Fermi surface, i.e. the singularity (for T = 0) is on a curve in d = 2 and on a surface in d = 3 dimensions.

8 Renormalization group flow e WΩ( η,η) = D ψdψ e ( ψ,q Ω ψ) V ( ψ,ψ) ( η,ψ) (η, ψ) e Ω Ω e W Ω( η,η) = = = ( D ψdψ e ( ψ,q Ω ψ) V ( ψ,ψ) ( ψ, Q ( η,ψ) (η, ψ) Ω ψ) e D ψdψ e ( ψ,q Ω ψ) V ( ψ,ψ) ) e W Ω( η,η) δ, δη Q Ω δ δ η ( δ, δη ) δ Q Ω δ η ( η,ψ) (η, ψ) e Ẇ Ω ( η, η) = e WΩ( η,η) Q Ω e WΩ( η,η), Q Ω = ( ) δw = Q Ω W Ω ( η, η) Ω, δw Q Ω δη Ω δ η ( δ δη, ) δ Q Ω δ η

9 Irreducible vertex functions γ Λ m Obtained by a Legendre transformation Γ Ω ( φ, φ) = W Ω ( η, η) ( η, φ) ( φ, η) φ = δw Ω, φ = δw Ω δ η δη ( ( ) ) Γ Ω = ( φ, Q δ Ω φ) Tr Q 2 1 Γ Ω δ φδφ Expansion in the fields: Γ Ω (Ψ) = m 0 Γ(m) Ω (Ψ) K = (k, α), ψ(k) = ψ α(k). Γ (m) Ω (Ψ) = dk 1...dK 2m δ( k i ) ψ(k 1 )...ψ(k 2m ) γm(k Ω 1,..., K 2m )

10 γ Ω 1 is the inverse of the full two-point function. Under the assumption that U(1) SU(2) invariance is unbroken, γ Ω 1 (K, K ) = δ α,α (ik 0 E(k) Σ Ω (k)χ Ω (k)) χ Ω (k) = k 2 0 Ω 2 + k 2 0 Similarly, U(1) SU(2) symmetry implies that γ Ω 2 (K 1,..., K 4 ) is determined by the vertex v Ω (k 1, k 2, k 3 ) that describes the interaction of particles with spin conservation. Truncations. Setting Γ (m) = 0 for m m 0 allows to solve the equations recursively in terms of Σ Ω and V Ω (in principle). Already the truncation Γ (3) = 0 (or slight modifications thereof) gives a nontrivial system of integro-differential equations. [MS,C.Honerkamp, 2001; Katanin, 2004] Its status in comparison to the full hierarchy is similar to that of the Boltzmann equation in relation to the quantum BBGKY hierarchy.

11 Mathematical physics problems attacked using this method A major open problem for the interacting Bose gas is a proof of long-range order ( Bose-Einstein condensation) in the thermodynamic limit, for µ > 0. In the above formulation, one has a sigma model with a particular kinetic term. A perturbation expansion around the symmetry-broken state appears non-renormalizable at first. It is only when the U(1) Ward identities are used that one can show power counting even perturbatively. [Benfatto, Grilli et al., Ba laban-feldman-knörrer-trubowitz,...] A similar, foundational, problem for interacting fermion systems is the proof of superconductivity at low enough temperatures. It is also important to distinguish Fermi liquid and non-fermi liquid phases, and estimate their boundary lines to broken-symmetry phases, in fermion systems. This problem is closest to perturbative renormalization, and most rigorous results have been obtained on this question. [Feldman-Knörrer-Trubowitz, Feldman- MS-Trubowitz, MS, Disertori-Rivasseau, Pedra, Benfatto-Giuliani-Mastropietro, Afchain-Magnen-Rivasseau,...]

12 Renormalization with momentum-dependent counterterms Renormalization can be done with counterterms, but symmetry does not, in general, restrict the form of the counterterms sufficiently to reduce to a finite number of parameters. The counterterm action is of the form d d k φ(0, k)f (k)φ(0, k) where F (k) is a function of k that is not just proportional to ε(k). Therefore, sufficient regularity needs to be proven. The physical meaning of F is that it fixes the Femri surface. It contains the information about the deformation of the Fermi surface under the influence of the interaction. Very early attempts to renormalize the perturbation expansion in many-electron systems failed to take this into account (the anomalous diagrams of Brueckner- Goldstone perturbation theory). The models were correctly renormalized, including all regularity proofs, only when methods of constructive QFT were applied [Feldman-MS-Trubowitz]. Based on this, fermionic models were constructed by an iteration of convergent perturbation expansions.

13 Regular Fermi Curves These results indicate: In two-dimensional Fermi systems with weak, short-range interactions, the self-energy is a regular function if the Fermi surface of the noninteracting Hamiltonian is nonsingular, i.e. it does not contain any points k with ε(k) = 0 has no perfect nesting, i.e. it does not contain any segments that are straight lines What happens when one of these conditions is violated? In the following, I present some results about Fermi surfaces with Van Hove singularities, i.e. points where ε(k) = 0.

14 Hubbard model The glossina morsitans of correlated fermion models, and also an effective one-band model for cuprate materials. (t, t, U)-Hubbard model: on-site repulsion U, and t t ɛ(k) = 2t(cos k 1 + cos k 2 ) 4t cos k 1 cos k 2 k B = R 2 /2πZ 2 = [ π, π) 2 t = 0 3 t t < For finite-range hopping, k ɛ(k) analytic in k. In general: hopping amplitude t x x with sufficient decay as x x.

15 Van Hove Singularities Van Hove (1953): k ɛ(k) always has saddle points k s : ɛ(k s ) = 0. Generically, ε i 0 ɛ(k s + Rp) = E VH ε 1 p ε 2 p O( p 3 ) R rotation, ε i 0 e.g. k s = (π, 0), E VH = 4t, ε 1 ε 2 = 1 θ 1+θ, 0 < θ = 2t t < 1 Singular Fermi surfaces. µ = E VH k s on the Fermi surface S. Density of states ρ(e) = d d k δ(e ɛ(k)) (2π) d has a Van Hove singularity (VHS) at E = E VH. d = 2 : ρ(e) log W E E VH

16 Effects on self-consistency equations e.g. BCS equation for superconductivity (SC) = g de ρ(e) 2 (E µ) 2 + tanh β (E µ) T c e ρ(µ)/g if ρ is regular for E µ. T c e K/ g if µ = E V H VHS also enhance tendencies towards ferromagnetism (FM) in d = 2, already at weak coupling, T F M e K/J if J U and g U 2 : FM and SC can compete but: simple gap equations or resummations are not sufficient to describe this competition a simple comparison of free energy densities would give a first order phase transition

17 Van Hove scenario in connection to high-tc The Fermi surface observed in angleresolved photoemission spectroscopy (ARPES) is in the vicinity of Van Hove points. Picture from Borisenko et al, 2006 I. Dzyaloshinskii, J. Phys. I France 6 (1996) 119: studied the possibility of an extended Van Hove singularity with flat bands, marginal- and non-fermi-liquid behaviour. N. Furukawa, T.M. Rice, MS, Phys. Rev. Lett. 81 (1998) 3195 argued for pinning of the Fermi surface at the Van Hove points V. Yu. Irkhin, A. A. Katanin, M. I. Katsnelson, Phys. Rev. Lett. 89 (2002) used a two-patch RG method to study robustness of extended Van Hove singularities.

18 Self-Energy at Van Hove points Theorem (J. Feldman & MS, 2008). Consider a two-dimensional Fermi system whose bare Fermi curve contains finitely many Van Hove singularities, and is non-nested (in a very weak sense) away from the Van Hove points, and consider the renormalized perturbation expansion to all orders, i.e. as a formal power series in the coupling. Renormalized means that the Fermi surface is fixed by a counter term. Then, to any order in this expansion, the self-energy Σ(ω, k) is C 1 in k, uniformly in ω and k, but the ω-derivative of Σ is singular at ω = 0 at the Van Hove point in the zero-temperature limit. The quasiparticle weight Z(k) = (1 + i ω Σ(0, k)) 1 vanishes at this singularity. In observables relevant for ARPES, this appears as a disappearance of the Fermi surface in the vicinity of the Van Hove points.

19 Remarks The asymmetry in the regularity can be motivated by looking at the two-loop selfenergy. In a nutshell, the vanishing of e at the Van Hove points cancels a singularity in the q-derivative, but not in the q 0 -derivative. The full proof uses the standard Fermi surface renormalization group technique. Generalization of the inversion theorem of [Feldman, MS, Trubowitz, 2000] to justify the counterterms may hold. I know of no way of justifying counterterms for the frequency derivative, in contrast to the situation with momentum-dependent counterterms. Might be proven non-perturbatively with the same technique and more careful bounds, but only nonuniformly in β because, in general, the effective interaction increases under the RG iteration, indicating onset of symmetry-breaking. Is there a situation where the renormalized expansion can be controlled down to zero temperature?

20 A quantum critical point in the Hubbard model Consider a curve of (t, t, U)-Hubbard models parametrized by θ = t, where t the chemical potential µ(θ) is chosen such that the Fermi curve contains a Van Hove point, and U is fixed. Then there is a point near θ 0 = 0.341, where v Ω stays small down to very low scales. At θ 0, the self-energy has small-frequency behaviour Σ(ω, (π, 0)) i sgn(ω) ω γ with γ < 1 (γ 0.74). [C. Honerkamp & MS; K. Giering & MS] Ω /t U = 3t t 2 /t 1 The momentum and frequency dependence of the vertex functions is studied as well, using a composite-field expansion.

21 Why is there a QCP at 0.341? This work was started in Leipzig, and Leipzig has area code There is no reason to assume that this number is universal. But a simple consideration shows that θ 0 is close to the place where two second-order contributions to the effective action cancel. The location of the downturn will change when the interaction is changed, but it is not special to an on-site interaction.

22 Mean-field calculation with the effective action PhD thesis of [K. Veschgini] It can be understood in a simple way why mean-field theory is able to describe a downturn of scale instead of a first-order transition.

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24 Conclusion and Outlook Renormalization is necessary and natural in models of quantum statistical mechanics that are directly relevant for condensed-matter physics. The renormalization group allows to do a rigorous mathematical analysis of gapless systems, e.g. a mathematical characterization of Fermi liquids. Work is in progress on fundamental phenomena, such as superconductivity and Bose-Einstein condensation. The same renormalization group technique has also become the method of choice to determine phase diagrams and calculate order parameters of correlated-fermion materials, cold gases of fermionic and bosonic particles, Bose-Fermi mixtures, and analyzing topological phases of correlated fermion systems, e.g. Rashba models. Singular Fermi curves in two dimensions are specific candidates for non-fermi-liquids. There is evidence, both numerical and analytical, of robustness and even extension of Van Hove singularities, and of a deconfined quantum critical point in the twodimensional Hubbard model T / t approximate nesting regime saddle point regime d wave regime µ / t

25 References J. Feldman, M. Salmhofer, E. Trubowitz. An inversion theorem in Fermi surface theory, Comm. Pure Appl. Math. 53 (2000) J. Feldman, M. Salmhofer. Singular Fermi surfaces I and II, Rev. Math. Phys. 20 (2008) , C. Honerkamp, M. Salmhofer, Magnetic and superconducting instabilities of the Hubbard model at the van Hove filling, Phys. Rev. Lett. 87 (2001) C. Husemann, K-U. Giering, M. Salmhofer, Frequency Dependent Vertex Functions of the (t,t )-Hubbard Model at Weak Coupling, Phys. Rev. B 85 (2012) K-U. Giering, M. Salmhofer, Self-energy flows in the two-dimensional repulsive Hubbard model, Phys. Rev. B 86 (2012) G. A. H. Schober, K.-U. Giering, M. M. Scherer, C. Honerkamp, M. Salmhofer Functional renormalization and mean-field approach to multiband systems with spin-orbit coupling. Phys. Rev. B 93 (2016) M. Salmhofer, C. Honerkamp, Fermionic renormalization group flows technique and theory, Prog. Theor. Phys. 105 (2001) 1 W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, K. Schönhammer, Functional renormalization group approach to correlated fermion systems, Rev. Mod. Phys. 84 (2012) 299