* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Quasi-set theory wikipedia, lookup

Standard Model wikipedia, lookup

Quantum logic wikipedia, lookup

Quantum chaos wikipedia, lookup

Interpretations of quantum mechanics wikipedia, lookup

Quantum chromodynamics wikipedia, lookup

Mathematical formulation of the Standard Model wikipedia, lookup

Quantum field theory wikipedia, lookup

Nuclear structure wikipedia, lookup

Old quantum theory wikipedia, lookup

Quantum electrodynamics wikipedia, lookup

Scale invariance wikipedia, lookup

Relational approach to quantum physics wikipedia, lookup

Canonical quantization wikipedia, lookup

Quantum gravity wikipedia, lookup

AdS/CFT correspondence wikipedia, lookup

Theory of everything wikipedia, lookup

Renormalization group wikipedia, lookup

Yang–Mills theory wikipedia, lookup

Topological quantum field theory wikipedia, lookup

Renormalization wikipedia, lookup

Scalar field theory wikipedia, lookup

Transcript

Landau’s Fermi Liquid Theory Thors Hans Hansson Stockholm University Quantum Field Theory for Condensed Matter Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Outline 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - Why do we want it? The free, i.e. non-interacting, Fermi gas give basic understanding of both cold Fermi systems. In particular, adding neutralizing or confining potentials it gives a qualitative understanding of Specific heat of (many) metals at low temperature The formation of neutron stars Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - Why do we want it? The free, i.e. non-interacting, Fermi gas give basic understanding of both cold Fermi systems. In particular, adding neutralizing or confining potentials it gives a qualitative understanding of Specific heat of (many) metals at low temperature The formation of neutron stars Adding a periodic potential, the band theory of non-interacting electrons also explains Conductors – semiconductors – metals Shape of Fermi surface and related spectroscopy The success of this ridiculously oversimplified model is not a coincidence. The theory of Fermi Liquids provides an explanation. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - Why do we want it? The concept of a Fermi Liquid describes (strongly) interacting fermions using concepts that naively would only be applicable to a very weakly interacting gas. That this is at all possible makes Fermi Liquid theory a very versatile, but at the same time its very success is puzzling. Fermi Liquid theory is: Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - Why do we want it? The concept of a Fermi Liquid describes (strongly) interacting fermions using concepts that naively would only be applicable to a very weakly interacting gas. That this is at all possible makes Fermi Liquid theory a very versatile, but at the same time its very success is puzzling. Fermi Liquid theory is: Is simple to use Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - Why do we want it? The concept of a Fermi Liquid describes (strongly) interacting fermions using concepts that naively would only be applicable to a very weakly interacting gas. That this is at all possible makes Fermi Liquid theory a very versatile, but at the same time its very success is puzzling. Fermi Liquid theory is: Is simple to use Is a good approximation to weakly interacting (dilute) Fermi systems Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - Why do we want it? The concept of a Fermi Liquid describes (strongly) interacting fermions using concepts that naively would only be applicable to a very weakly interacting gas. That this is at all possible makes Fermi Liquid theory a very versatile, but at the same time its very success is puzzling. Fermi Liquid theory is: Is simple to use Is a good approximation to weakly interacting (dilute) Fermi systems Gives a remarkably good description of real metals Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - Why do we want it? The concept of a Fermi Liquid describes (strongly) interacting fermions using concepts that naively would only be applicable to a very weakly interacting gas. That this is at all possible makes Fermi Liquid theory a very versatile, but at the same time its very success is puzzling. Fermi Liquid theory is: Is simple to use Is a good approximation to weakly interacting (dilute) Fermi systems Gives a remarkably good description of real metals But how can that be?? Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - What is it? Fermi liquid theory describe a strongly interacting Fermi system, such as nuclear matter, liquid Helium III, or electrons interacting via Coulomb forces. The basic concepts/ideas in Fermi liquid theory (of electrons) are: Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - What is it? Fermi liquid theory describe a strongly interacting Fermi system, such as nuclear matter, liquid Helium III, or electrons interacting via Coulomb forces. The basic concepts/ideas in Fermi liquid theory (of electrons) are: Weakly interacting quasielectrons and quasiholes Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - What is it? Fermi liquid theory describe a strongly interacting Fermi system, such as nuclear matter, liquid Helium III, or electrons interacting via Coulomb forces. The basic concepts/ideas in Fermi liquid theory (of electrons) are: Weakly interacting quasielectrons and quasiholes The quasiparticles have the same quantum numbers as electrons and holes, i.e. momentum ~k, spin 12 and charge ±e. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - What is it? Fermi liquid theory describe a strongly interacting Fermi system, such as nuclear matter, liquid Helium III, or electrons interacting via Coulomb forces. The basic concepts/ideas in Fermi liquid theory (of electrons) are: Weakly interacting quasielectrons and quasiholes The quasiparticles have the same quantum numbers as electrons and holes, i.e. momentum ~k, spin 12 and charge ±e. The state of the Fermi liquid is described as a collection of n quasiparticles, |~k1 , α1 . . . ~kN , αN i Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - What is it? Fermi liquid theory describe a strongly interacting Fermi system, such as nuclear matter, liquid Helium III, or electrons interacting via Coulomb forces. The basic concepts/ideas in Fermi liquid theory (of electrons) are: Weakly interacting quasielectrons and quasiholes The quasiparticles have the same quantum numbers as electrons and holes, i.e. momentum ~k, spin 12 and charge ±e. The state of the Fermi liquid is described as a collection of n quasiparticles, |~k1 , α1 . . . ~kN , αN i The interaction between the quasiparticles is described by a few Fermi liquid parameters fl . Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi liquids - What is it? Fermi liquid theory describe a strongly interacting Fermi system, such as nuclear matter, liquid Helium III, or electrons interacting via Coulomb forces. The basic concepts/ideas in Fermi liquid theory (of electrons) are: Weakly interacting quasielectrons and quasiholes The quasiparticles have the same quantum numbers as electrons and holes, i.e. momentum ~k, spin 12 and charge ±e. The state of the Fermi liquid is described as a collection of n quasiparticles, |~k1 , α1 . . . ~kN , αN i The interaction between the quasiparticles is described by a few Fermi liquid parameters fl . At finite temperature and chemical potential, the state of a Fermi Liquid is given by a density matrix for the quasiparticles, and there is a corresponding kinetic theory based on the distribution function f . Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi Liquids – How? Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi Liquids – How? I will now describe Landaus intuitive approach to Fermi liquids, based on adiabatic evolution from the free Fermi gas. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi Liquids – How? I will now describe Landaus intuitive approach to Fermi liquids, based on adiabatic evolution from the free Fermi gas. Later a diagrammatic approach was used to derive Fermi Liquid theory, but this is rather complicated and is based on resuming infinite classes of diagrams. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Fermi Liquids – How? I will now describe Landaus intuitive approach to Fermi liquids, based on adiabatic evolution from the free Fermi gas. Later a diagrammatic approach was used to derive Fermi Liquid theory, but this is rather complicated and is based on resuming infinite classes of diagrams. The modern Renormalization Group approach to Fermi Liquid theory, due primarily to Shankar and Polchinski, defines the Fermi Liquid as a fixed point of the RG flow. This roughly means that the FL description is what is left when all the high energy degrees of freedom are integrated out. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The free Fermi gas For a gas of free fermions with mass me , the zero temperature ground state is obtained by filling all single particle (plane wave) states up to the Fermi energy F . With the T = 0 distribution function n0 (~ p ) = θ(|~ p | − pF ) we get, Total energy ˆ d 3p E = V Tr (p)n0 (p) (2π)3 ˆ VpF2 p2 V p5 ≡ Tr dτ n0 (p) = 2 F = F 2me 5π 2me 5π 2 where V is the, (p) = p2 2me , F = (pF ), and ˆ ˆ X d 3p dτ = V = (2π)3 ~ ki Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The free Fermi gas Total number of particles ˆ N= dτ n0 (p) = V pF3 3π 2 Excitations from the ground state is obtained by changing the occupation numbers of the single particle levels, i.e. by changing the distribution function, n(p) = θ(p − pF ) + δn(p) For discrete momentum states, δn can only take the values ±1 because of the Pauli principle. The distribution function at finite temperature T is determined by maximizing the entropy, ˆ S = − dτ [n(p) ln n(p) + (1 − n(p)) ln(1 − n(p))] under the constraints δE = δN = 0 to get: Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The free Fermi gas Finite T Fermi-Dirac distribution function nβ (p) = 1 e β((p)−F ) +1 , β= 1 kT Note that no dynamical information was needed - only that the particles are fermions. These considerations can be generalized to include an external potential, but in those cases the integrals can in general not be calculated on a closed form and there are no analytical expression for energy, entropy etc. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Landaus basic idea Landau’s basic idea was that the interacting system can be thought of as ”connected” to the free fermi gas by an adiabatic switching process. In particular, this means Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Landaus basic idea Landau’s basic idea was that the interacting system can be thought of as ”connected” to the free fermi gas by an adiabatic switching process. In particular, this means there is an one-one correspondence between the excitations in the free and the interacting system. For a state with one fermion added to the ground state, this means |~ p , αifree → |~ p , αiint. The quantum numbers charge q, momentum p~, and spin α of the quasiparticle remain the same, but the energy is changed There is a Fermi surface also in the interacting system Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The Energy Functional E [n(p)] When interactions are present the total energy still depends on the distribution function, n(p) but this functional dependence is very complicated. We are however only interested in the change in energy as we vary the occupation numbers: Variation of E [n] ˆ δE [n(p)] = 1 dτ 0 (p)δn(p) + 2V ˆ dτ dτ 0 f (~ p , p~0 )δn(p)δn(p 0 ) + . . . where V is the volume of the system, 0 (p) = and f (~ p , p~0 ) = Thors Hans Hansson (Stockholm University) δE [n] |n=n0 δn(p) δ 2 E [n] |n=n0 δn(p)δn(p 0 ) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The quasiparticle distribution function Since the entropy follows from a counting argument, and since there is an one-one correspondence between the states in the interacting and the non-interacting system, the calculation of the entropy will look precisely the same, except that the constraint δE = 0 now becomes, ˆ dτ 0 [n, p]δn(p) = 0 , Minimizing the entropy under this constraint gives, Functional equation for the quasiparticle distribution function nβ (p) = 1 e β([n,p]−F ) +1 . which is a very complicated functional equation. From the previous arguments we however know that it will approach n0 (p) = θ(p − pF ) as T goes to zero. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The effective mass m? I - definition Now we assume that the temperature is low enough that we can put n(p) = θ(p − pF ) and expand around the Fermi momentum, [n(p 0 ), p] ≈ [θ(p − pF ), p] ≈ F + vF (p − pF ) + . . . To understand the meaning of F and vF , recall that pF is the same as in the free theory, but the Fermi energy is not. But [n, p] is the energy cost for adding a single quasiparticle, so F = µ = chemical potential For the non-interacting gas, (p) = F + pF (p − pF ) + . . . . me From this we conclude vF = Fermi velocity = pF pF 6= m? me This relation defines the effective mass m? . Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The effective mass m? II - how to measure One of the simplest observables that can be measured for a Fermi liquid is its specific heat, cV . It is the same as for the free electron gas with the substitution me → m? , cV = m? pF 2 kB T , 3 The effective mass can vary a lot: 3 He Heavy fermion compounds such as Thors Hans Hansson (Stockholm University) UPt3 m? /me ≈ 3 m? /me ≈ 100 − 1000 Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Meaning of the quasiparticle Fermi surface At zero temperature, the quasiparticle distribution function n(p) is sharp (dotted line). The momentum distribution of the electrons, N(p) = hΨFL |cp† cp |ΨFL i Z is the wave function renormalization constant, or the strength of the single particle pole. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 When is FL theory applicable? A quasielectron at a momentum p~1 it can decay into two quasielectrons and one quasihole, |~ p1 i− → |~ p2 i− + |~ p3 i− + |~ p2 − p~1 − p~3 i+ This makes the quasiparticles unstable !!! Landau’s fundamental observation Close to the Fermi surface, the phase space for decay vanish ∼ |~ p − p~F |2 and the quasiparticles become almost stable!! In fact, the life time τ becomes τ= 1 ∼ |~ p − p~F |−2 Γ For FL theory to be applicable, we must have T TF TF for metallic sodium is ≈ 40, 000K but for liquid He3 only ≈ 7K !! Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Interactions - definition of the FL parameters Writing out the spin indices 0 [n(p ), p]αβ 1 = 0,αβ (p) + V ˆ dτ 0 f (~ p , p~0 )αγ,βδ δnγδ (p 0 ) + . . . . f only depends on the angle θ given by p~ · p~0 = pp 0 cos θ ≈ pF2 cos θ Symmetries (rotation, TR) and Fermi statistics implies, f (~ p , p~0 )αγ,βδ = f (cos θ)δαβ δγδ + g (cos θ)S~αβ · S~γδ . ~ fields), we have nαβ = nδαβ , and For an isotropic liquid (no B ˆ 0 [n(p ), p] = F + vF (p − pF ) + dτ 0 f (cos θ)δn(p 0 ) F-parameters ∞ F (θ) = X pF m ? f (cos θ) = (2L + 1)FL PL (cos θ) π2 L=0 Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Important relations The effective mass m? is not independent of the parameters FL . In fact m? F1 =1+ me 3 follows from Gallilean invariance. The compressibility κ is given by, κ= m? pF π 2 ρ2 (1 + F0 ) where ρ is the density. The response to a magnetic field involves the function g (cos θ), and the susceptibility is determined by the parameter G0 , where GL is defined analogously to FL Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Momentum cutoff From a microscopic eucledian action S = SΛ , construct a sequence of actions SΛn where Λn is a cutoff and the action SΛn describes the physics for momenta p < Λn . The first Λ can be thought of as a physical cutoff. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Momentum cutoff From a microscopic eucledian action S = SΛ , construct a sequence of actions SΛn where Λn is a cutoff and the action SΛn describes the physics for momenta p < Λn . The first Λ can be thought of as a physical cutoff. Study the Eucledian partition function of a φ4 theory ˆ Z [T , µ, . . . ] ≡ Z = D[φ, φ? ] e −S[φ] ˆ 1 ? ~2 4 d φ (−∇ + g2 )φ + g4 |φ| = S0 + Sint S= d x 2 Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory (1) (2) Quantum Field Theory for Condensed Matter / 39 Momentum cutoff From a microscopic eucledian action S = SΛ , construct a sequence of actions SΛn where Λn is a cutoff and the action SΛn describes the physics for momenta p < Λn . The first Λ can be thought of as a physical cutoff. Study the Eucledian partition function of a φ4 theory ˆ Z [T , µ, . . . ] ≡ Z = D[φ, φ? ] e −S[φ] ˆ 1 ? ~2 4 d φ (−∇ + g2 )φ + g4 |φ| = S0 + Sint S= d x 2 In Fourier space, X φ(~x ) = e i p~·~x φ(~ p) = 0≤|~ p |≤Λ X e i p~·~x φ(~ p) + 0≤|~ p |≤Λ1 X e i p~·~x φ(~ p) (1) (2) (3) Λ1 ≤|~ p |≤Λ ≡ φ< (~x ) + φ> (~x ) Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The effective action The action becomes X 1 S= (p 2 + g2 ) φ? (~ p ) φ(~ p) + 2 0≤|~ p |≤Λ1 X Λ1 ≤|~ p |≤Λ 1 2 (p + g2 ) φ? (~ p ) φ(~ p ) + Sint 2 = S0 [φ< ] + S0 [φ> ] + Sint [φ< , φ> ] Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The effective action The action becomes X 1 S= (p 2 + g2 ) φ? (~ p ) φ(~ p) + 2 0≤|~ p |≤Λ1 X Λ1 ≤|~ p |≤Λ 1 2 (p + g2 ) φ? (~ p ) φ(~ p ) + Sint 2 = S0 [φ< ] + S0 [φ> ] + Sint [φ< , φ> ] We define the effective action S eff at scale Λ1 = Λ/s by ˆ −S eff [φ< ] S0 [φ< ] =e e D[φ> ] e −S0 [~p> ]−Sint [φ< ,φ> ] 1 p~ φ< → ζ φ 0 s which resets the cutoff to Λ and the kinetic term to p 2 . We then get (suppressing the ’ s), ˆ h i eff ~ 2 + g 0 )φ + g 0 |φ|4 + g6 |φ|6 + g22 |∇ ~ 2 φ|2 + ...... . S = d d x φ? (−∇ 2 4 Rescale : Thors Hans Hansson (Stockholm University) p~< → Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Dimensional analysis and flow equations Mass dimensions: [g2 ] = 2 and [g4 ] = 4 − d (recall that [φ] = under the scale transformation, g2 → g20 = s 2 g2 and d 2 − 1) so g4 → g40 = s4−d g4 We expect g2 to be relevant and g4 relevant or irrelevant depending on whether d < 4 or d > 4. The RG equations are s dg dgi = = βi (g1 , g2 , . . . ) ds dt The d = 4 β-functions can be obtained from perturbation theory, dg2 = 2g2 + ag4 dt dg4 = −bg42 dt Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 The RG flow for φ4 theory in d = 4 Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Next subject 1 Fermi Liquids – Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids – How? 2 Landau’s Phenomenological Approach The free Fermi gas Landaus basic idea The Energy Functional E [n(p)] The effective mass The quasiparticle Fermi surface When can Fermi Liquid theory be used? Quasiparticle interactions 3 Renormalization Group approach to FL theory An RG primer Fermi Liquids as a RG fixed point Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Why fermions are different For bosons the RG transformation restricts to momenta to a smaller and smaller ball around p = 0 . An RG fixed point is a set of coupling constants {gi }. For fermions the momenta are restricted to a smaller ands smaller shell around the Fermi momentum pF . An RG fixed point is a set of coupling ~ ~ functions {fi (Ω)} defined on the Fermi surface, p~ = pF Ω. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Two body scattering Consider scattering between two particles in d = 3 close to the Fermi surface, 1+2→3+4 ~ i , i = 1, . . . 4 where Ω ~ = (cos θ, sin θ). Parametrize: ~ki = (pF + ki )Ω ~1 = Ω ~ 3 and Ω ~2 = Ω ~4 Ω ~1 = Ω ~ 4 and Ω ~2 = Ω ~3 Ω ~ 1 = −Ω ~ 2 and Ω ~ 3 = −Ω ~4 Ω Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory (I ) (II ) (III ) Quantum Field Theory for Condensed Matter / 39 Fixed point functions A four particle interaction in momentum space can be written as, Sint = ˆ Y 4 i=1 dωi X ψp~†3 ψp~†4 ψp~2 ψp~1 f (ωi , ki , θi ) δ 2 ( X p~i )δ( i ~i p X ωi ) i Processes I and II are allowed for arbitrary angles θ1 and θ2 , and because of rotational invariance, it is characterized by a fixed point function: F (θ1 − θ2 ) = f (θ1 , θ2 , θ1 , θ2 ) = −f (θ1 , θ2 , θ2 , θ1 ) where the sign is due to fermi statistics. The Process III is scattering between particles at opposite positions on the Fermi circle. Here the coupling function can depend only on the angle θ1 − θ3 , V (θ1 − θ3 ) = f (θ1 , −θ1 , θ3 , −θ3 ) . Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Flow equations Here we just used geometric intuition to put in the constraints on the angles ”by hand”, but a carful evaluation of the diagrams taking the cutoff into account, will give the same result, and the flow eqs. dF (θ) =0 dt ˆ dV (θ) 1 dθ0 =− V (θ − θ0 )V (θ0 ) dt 4π 2π which has the solution: VL (0) VL (t) = 1 + VL (0)t/(4π) where VL (0) are the starting values for the RG evolution. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Physical interpretation VL (t) = VL (0) 1 + VL (0)t/(4π) Take all VL (0) > 0, as would be the case for a reasonable repulsive potential, then Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Physical interpretation VL (t) = VL (0) 1 + VL (0)t/(4π) Take all VL (0) > 0, as would be the case for a reasonable repulsive potential, then V (θ) will renormalize to zero. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Physical interpretation VL (t) = VL (0) 1 + VL (0)t/(4π) Take all VL (0) > 0, as would be the case for a reasonable repulsive potential, then V (θ) will renormalize to zero. F (θ) remains marginal and defines a fixed point theory which is a Fermi liquid!! If (at least) one VL (0) < 0 is , as would be the case for any attractive interaction, the RG flow will hit a singularity for some t, and Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Physical interpretation VL (t) = VL (0) 1 + VL (0)t/(4π) Take all VL (0) > 0, as would be the case for a reasonable repulsive potential, then V (θ) will renormalize to zero. F (θ) remains marginal and defines a fixed point theory which is a Fermi liquid!! If (at least) one VL (0) < 0 is , as would be the case for any attractive interaction, the RG flow will hit a singularity for some t, and The coupling constant, VL grows out of control, and the perturbative treatment can no longer be trusted. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Physical interpretation VL (t) = VL (0) 1 + VL (0)t/(4π) Take all VL (0) > 0, as would be the case for a reasonable repulsive potential, then V (θ) will renormalize to zero. F (θ) remains marginal and defines a fixed point theory which is a Fermi liquid!! If (at least) one VL (0) < 0 is , as would be the case for any attractive interaction, the RG flow will hit a singularity for some t, and The coupling constant, VL grows out of control, and the perturbative treatment can no longer be trusted. The renormalization of V (θ) comes from the ”BCS” diagram c. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39 Physical interpretation VL (t) = VL (0) 1 + VL (0)t/(4π) Take all VL (0) > 0, as would be the case for a reasonable repulsive potential, then V (θ) will renormalize to zero. F (θ) remains marginal and defines a fixed point theory which is a Fermi liquid!! If (at least) one VL (0) < 0 is , as would be the case for any attractive interaction, the RG flow will hit a singularity for some t, and The coupling constant, VL grows out of control, and the perturbative treatment can no longer be trusted. The renormalization of V (θ) comes from the ”BCS” diagram c. Results in a pole in the Lth partial wave of the particle - particle scattering amplitude, corresponding to the formation of a Cooper pair with angular momentum L. Thors Hans Hansson (Stockholm University) Landau’s Fermi Liquid Theory Quantum Field Theory for Condensed Matter / 39