Landau s Fermi Liquid Theory


 Mervin Kristian Paul
 1 years ago
 Views:
Transcription
1 Thors Hans Hansson Stockholm University
2 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
3 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
4 Fermi liquids  Why do we want it? The free, i.e. noninteracting, 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
5 Fermi liquids  Why do we want it? The free, i.e. noninteracting, 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 noninteracting 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.
6 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:
7 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
8 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
9 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
10 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??
11 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
12 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:
13 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
14 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 1 2 and charge ±e.
15 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 1 2 and charge ±e. The state of the Fermi liquid is described as a collection of n quasiparticles, k 1, α 1... k N, α N
16 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 1 2 and charge ±e. The state of the Fermi liquid is described as a collection of n quasiparticles, k 1, α 1... k N, α N The interaction between the quasiparticles is described by a few Fermi liquid parameters f l.
17 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 1 2 and charge ±e. The state of the Fermi liquid is described as a collection of n quasiparticles, k 1, α 1... k N, α N The interaction between the quasiparticles is described by a few Fermi liquid parameters f l. 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.
18 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
19 Fermi Liquids How?
20 Fermi Liquids How? I will now describe Landaus intuitive approach to Fermi liquids, based on adiabatic evolution from the free Fermi gas.
21 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.
22 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.
23 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
24 The free Fermi gas For a gas of free fermions with mass m e, the zero temperature ground state is obtained by filling all single particle (plane wave) states up to the Fermi energy ɛ F. With the T = 0 distribution function n 0 ( p) = θ( p p F ) we get, Total energy ˆ E = V Tr ˆ Tr d 3 p (2π) 3 ɛ(p)n 0(p) dτ p2 n 0 (p) = V pf 5 2m e 5π 2 = Vp2 F 2m e 5π 2 ɛ F where V is the, ɛ(p) = p2 2m e, ɛ F = ɛ(p F ), and ˆ ˆ d 3 p dτ = V (2π) 3 = ki
25 The free Fermi gas Total number of particles ˆ N = dτ n 0 (p) = V p3 F 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 p F ) + 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:
26 The free Fermi gas Finite T FermiDirac distribution function n β (p) = 1 e β(ɛ(p) ɛ 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.
27 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
28 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
29 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 oneone 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, α free p, α int. 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
30 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
31 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)] = dτɛ 0 (p)δn(p) + 1 2V ˆ dτdτ f ( p, p )δn(p)δn(p ) +... where V is the volume of the system, and ɛ 0 (p) = δe[n] δn(p) n=n 0 f ( p, p ) = δ2 E[n] δn(p)δn(p ) n=n 0
32 The quasiparticle distribution function Since the entropy follows from a counting argument, and since there is an oneone correspondence between the states in the interacting and the noninteracting system, the calculation of the entropy will look precisely the same, except that the constraint δe = 0 now becomes, ˆ dτ ɛ[n, p]δn(p) = 0, Minimizing the entropy under this constraint gives, Functional equation for the quasiparticle distribution function n β (p) = 1 e β(ɛ[n,p] ɛ F ) + 1. which is a very complicated functional equation. From the previous arguments we however know that it will approach n 0 (p) = θ(p p F ) as T goes to zero.
33 The effective mass m I  definition Now we assume that the temperature is low enough that we can put n(p) = θ(p p F ) and expand around the Fermi momentum, ɛ[n(p ), p] ɛ[θ(p p F ), p] ɛ F + v F (p p F ) +... To understand the meaning of ɛ F and v F, recall that p F is the same as in the free theory, but the Fermi energy is not. But ɛ[n, p] is the energy cost for adding a single quasiparticle, so ɛ F = µ = chemical potential For the noninteracting gas, From this we conclude ɛ(p) = ɛ F + p F m e (p p F ) v F = Fermi velocity = p F m p F m e This relation defines the effective mass m.
34 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
35 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, c V. It is the same as for the free electron gas with the substitution m e m, c V = m p F 3 k 2 B T, The effective mass can vary a lot: 3 He m /m e 3 Heavy fermion compounds such as UPt 3 m /m e
36 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
37 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) = Ψ FL c pc p Ψ FL Z is the wave function renormalization constant, or the strength of the single particle pole.
38 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
39 When is FL theory applicable? A quasielectron at a momentum p 1 it can decay into two quasielectrons and one quasihole, p 1 p 2 + p 3 + p 2 p 1 p 3 + 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 Γ p p F 2 For FL theory to be applicable, we must have T T F T F for metallic sodium is 40, 000K but for liquid He 3 only 7K!!
40 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
41 Interactions  definition of the FL parameters Writing out the spin indices ɛ[n(p ), p] αβ = ɛ 0,αβ (p) + 1 V ˆ dτ f ( p, p ) αγ,βδ δn γδ (p ) f only depends on the angle θ given by p p = pp cos θ pf 2 cos θ Symmetries (rotation, TR) and Fermi statistics implies, f ( p, p ) αγ,βδ = f (cos θ)δ αβ δ γδ + g(cos θ) S αβ S γδ. For an isotropic liquid (no B fields), we have n αβ = nδ αβ, and ˆ ɛ[n(p ), p] = ɛ F + v F (p p F ) + dτ f (cos θ)δn(p ) Fparameters F (θ) = p F m π 2 f (cos θ) = (2L + 1)F L P L (cos θ) L=0
42 Important relations The effective mass m is not independent of the parameters F L. In fact follows from Gallilean invariance. The compressibility κ is given by, where ρ is the density. κ = m = 1 + F 1 m e 3 m p F π 2 ρ 2 (1 + F 0 ) The response to a magnetic field involves the function g(cos θ), and the susceptibility is determined by the parameter G 0, where G L is defined analogously to F L
43 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
44 Momentum cutoff From a microscopic eucledian action S = S Λ, construct a sequence of actions S Λn where Λ n is a cutoff and the action S Λn describes the physics for momenta p < Λ n. The first Λ can be thought of as a physical cutoff.
45 Momentum cutoff From a microscopic eucledian action S = S Λ, construct a sequence of actions S Λn where Λ n is a cutoff and the action S Λn describes the physics for momenta p < Λ n. The first Λ can be thought of as a physical cutoff. Study the Eucledian partition function of a φ 4 theory ˆ Z[T, µ,... ] Z = D[φ, φ ] e S[φ] (1) ˆ [ ] 1 S = d d x 2 φ ( 2 + g 2 )φ + g 4 φ 4 = S 0 + S int (2)
46 Momentum cutoff From a microscopic eucledian action S = S Λ, construct a sequence of actions S Λn where Λ n is a cutoff and the action S Λn describes the physics for momenta p < Λ n. The first Λ can be thought of as a physical cutoff. Study the Eucledian partition function of a φ 4 theory ˆ Z[T, µ,... ] Z = D[φ, φ ] e S[φ] (1) ˆ [ ] 1 S = d d x 2 φ ( 2 + g 2 )φ + g 4 φ 4 = S 0 + S int (2) In Fourier space, φ( x) = 0 p Λ e i p x φ( p) = φ < ( x) + φ > ( x) 0 p Λ 1 e i p x φ( p) + Λ 1 p Λ e i p x φ( p) (3)
47 The effective action The action becomes S = 1 2 (p2 + g 2 ) φ ( p) φ( p) + 0 p Λ 1 = S 0 [φ < ] + S 0 [φ > ] + S int [φ <, φ > ] Λ 1 p Λ 1 2 (p2 + g 2 ) φ ( p) φ( p) + S int
48 The effective action The action becomes S = 1 2 (p2 + g 2 ) φ ( p) φ( p) + 0 p Λ 1 = S 0 [φ < ] + S 0 [φ > ] + S int [φ <, φ > ] Λ 1 p Λ 1 2 (p2 + g 2 ) φ ( p) φ( p) + S int We define the effective action S eff at scale Λ 1 = Λ/s by ˆ e Seff [φ <] = e S 0[φ <] D[φ > ] e S 0[ p >] S int [φ <,φ >] Rescale : p < 1 s p φ < ζ φ which resets the cutoff to Λ and the kinetic term to p 2. We then get (suppressing the s), ˆ [ S eff = d d x φ ( 2 + g 2)φ + g 4 φ 4 + g 6 φ 6 + g 22 ] 2 φ
49 Dimensional analysis and flow equations Mass dimensions: [g 2 ] = 2 and [g 4 ] = 4 d (recall that [φ] = d 2 under the scale transformation, 1) so g 2 g 2 = s 2 g 2 and g 4 g 4 = s 4 d g 4 We expect g 2 to be relevant and g 4 relevant or irrelevant depending on whether d < 4 or d > 4. The RG equations are s dg i ds = dg dt = β i(g 1, g 2,... ) The d = 4 βfunctions can be obtained from perturbation theory, dg 2 dt = 2g 2 + ag 4 dg 4 dt = bg 2 4
50 The RG flow for φ 4 theory in d = 4
51 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
52 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 {g i }. For fermions the momenta are restricted to a smaller ands smaller shell around the Fermi momentum p F. An RG fixed point is a set of coupling functions {f i ( Ω)} defined on the Fermi surface, p = p F Ω.
53 Two body scattering Consider scattering between two particles in d = 3 close to the Fermi surface, Parametrize: k i = (p F + k i ) Ω i, i = 1,... 4 where Ω = (cos θ, sin θ). Ω 1 = Ω 3 and Ω 2 = Ω 4 (I ) Ω 1 = Ω 4 and Ω 2 = Ω 3 (II ) Ω 1 = Ω 2 and Ω 3 = Ω 4 (III )
54 Fixed point functions A four particle interaction in momentum space can be written as, ˆ 4 S int = dω i ψ p 3 ψ p 4 ψ p2 ψ p1 f (ω i, k i, θ i ) δ 2 ( p i )δ( i=1 p i i i ω i ) Processes I and II are allowed for arbitrary angles θ 1 and θ 2, and because of rotational invariance, it is characterized by a fixed point function: F (θ 1 θ 2 ) = f (θ 1, θ 2, θ 1, θ 2 ) = f (θ 1, θ 2, θ 2, θ 1 ) 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 θ 1 θ 3, V (θ 1 θ 3 ) = f (θ 1, θ 1, θ 3, θ 3 ).
55 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 dv (θ) = 1 ˆ dθ dt 4π 2π V (θ θ )V (θ ) which has the solution: V L (0) V L (t) = 1 + V L (0)t/(4π) where V L (0) are the starting values for the RG evolution.
56 Physical interpretation V L (t) = V L (0) 1 + V L (0)t/(4π) Take all V L (0) > 0, as would be the case for a reasonable repulsive potential, then
57 Physical interpretation V L (t) = V L (0) 1 + V L (0)t/(4π) Take all V L (0) > 0, as would be the case for a reasonable repulsive potential, then V (θ) will renormalize to zero.
58 Physical interpretation V L (t) = V L (0) 1 + V L (0)t/(4π) Take all V L (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 V L (0) < 0 is, as would be the case for any attractive interaction, the RG flow will hit a singularity for some t, and
59 Physical interpretation V L (t) = V L (0) 1 + V L (0)t/(4π) Take all V L (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 V L (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, V L grows out of control, and the perturbative treatment can no longer be trusted.
60 Physical interpretation V L (t) = V L (0) 1 + V L (0)t/(4π) Take all V L (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 V L (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, V L grows out of control, and the perturbative treatment can no longer be trusted. The renormalization of V (θ) comes from the BCS diagram c.
61 Physical interpretation V L (t) = V L (0) 1 + V L (0)t/(4π) Take all V L (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 V L (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, V L 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 L th partial wave of the particle  particle scattering amplitude, corresponding to the formation of a Cooper pair with angular momentum L.
Fermi Liquid and BCS Phase Transition
Fermi Liquid and BCS Phase Transition Yu, Zhenhua November 2, 25 Abstract Landau fermi liquid theory is introduced as a successful theory describing the low energy properties of most fermi systems. Besides
More informationEmergent Quantum Criticality
(Non)Fermi Liquids and Emergent Quantum Criticality from gravity Hong Liu Massachusetts setts Institute te of Technology HL, John McGreevy, David Vegh, 0903.2477 Tom Faulkner, HL, JM, DV, to appear SungSik
More informationr 2 dr h2 α = 8m2 q 4 Substituting we find that variational estimate for the energy is m e q 4 E G = 4
Variational calculations for Hydrogen and Helium Recall the variational principle See Chapter 16 of the textbook The variational theorem states that for a Hermitian operator H with the smallest eigenvalue
More informationCorrelations between spin accumulation and degree of timeinverse breaking for electron gas in solid
Correlations between spin accumulation and degree of timeinverse breaking for electron gas in solid V.Zayets * Spintronic Research Center, National Institute of Advanced Industrial Science and Technology
More informationPhys Midterm. March 17
Phys 7230 Midterm March 17 Consider a spin 1/2 particle fixed in space in the presence of magnetic field H he energy E of such a system can take one of the two values given by E s = µhs, where µ is the
More informationPart III: Impurities in Luttinger liquids
Functional RG for interacting fermions... Part III: Impurities in Luttinger liquids 1. Luttinger liquids 2. Impurity effects 3. Microscopic model 4. Flow equations 5. Results S. Andergassen, T. Enss (Stuttgart)
More informationLecture 3 (Part 1) Physics 4213/5213
September 8, 2000 1 FUNDAMENTAL QED FEYNMAN DIAGRAM Lecture 3 (Part 1) Physics 4213/5213 1 Fundamental QED Feynman Diagram The most fundamental process in QED, is give by the definition of how the field
More informationFermi liquids and fractional statistics in one dimension
UiO, 26. april 2017 Fermi liquids and fractional statistics in one dimension Jon Magne Leinaas Department of Physics University of Oslo JML Phys. Rev. B (April, 2017) Related publications: M Horsdal, M
More informationarxiv:condmat/ v2 [condmat.strel] 25 Jun 2003
Magnetic fieldinduced Landau Fermi Liquid in hight c metals M.Ya. Amusia a,b, V.R. Shaginyan a,c 1 arxiv:condmat/0304432v2 [condmat.strel] 25 Jun 2003 a The Racah Institute of Physics, the Hebrew
More informationNuclear structure III: Nuclear and neutron matter. National Nuclear Physics Summer School Massachusetts Institute of Technology (MIT) July 1829, 2016
Nuclear structure III: Nuclear and neutron matter Stefano Gandolfi Los Alamos National Laboratory (LANL) National Nuclear Physics Summer School Massachusetts Institute of Technology (MIT) July 1829, 2016
More informationIntermediate valence in Yb Intermetallic compounds
Intermediate valence in Yb Intermetallic compounds Jon Lawrence University of California, Irvine This talk concerns rare earth intermediate valence (IV) metals, with a primary focus on certain Ybbased
More informationC. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.
Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationc E If photon Mass particle 81
Nuclear Force, Structure and Models Readings: Nuclear and Radiochemistry: Chapter 10 (Nuclear Models) Modern Nuclear Chemistry: Chapter 5 (Nuclear Forces) and Chapter 6 (Nuclear Structure) Characterization
More informationRegularization Physics 230A, Spring 2007, Hitoshi Murayama
Regularization Physics 3A, Spring 7, Hitoshi Murayama Introduction In quantum field theories, we encounter many apparent divergences. Of course all physical quantities are finite, and therefore divergences
More informationChapter 2 Superconducting Gap Structure and Magnetic Penetration Depth
Chapter 2 Superconducting Gap Structure and Magnetic Penetration Depth Abstract The BCS theory proposed by J. Bardeen, L. N. Cooper, and J. R. Schrieffer in 1957 is the first microscopic theory of superconductivity.
More informationMicroscopic Properties of BCS Superconductors (cont.)
PHYS598 A.J.Leggett Lecture 8 Microscopic Properties of BCS Superconductors (cont.) 1 Microscopic Properties of BCS Superconductors (cont.) References: Tinkham, ch. 3, sections 7 9 In last lecture, examined
More information2 Feynman rules, decay widths and cross sections
2 Feynman rules, decay widths and cross sections 2.1 Feynman rules Normalization In nonrelativistic quantum mechanics, wave functions of free particles are normalized so that there is one particle in
More informationMODEL WITH SPIN; CHARGE AND SPIN EXCITATIONS 57
56 BOSONIZATION Note that there seems to be some arbitrariness in the above expressions in terms of the bosonic fields since by anticommuting two fermionic fields one can introduce a minus sine and thus
More informationFunctional RG for fewbody physics
Functional RG for fewbody physics Michael C Birse The University of Manchester Review of results from: Schmidt and Moroz, arxiv:0910.4586 Krippa, Walet and Birse, arxiv:0911.4608 Krippa, Walet and Birse,
More informationHolographic superconductors
Holographic superconductors Sean Hartnoll Harvard University Work in collaboration with Chris Herzog and Gary Horowitz : 0801.1693, 0810.1563. Frederik Denef : 0901.1160. Frederik Denef and Subir Sachdev
More informationNuclear structure Anatoli Afanasjev Mississippi State University
Nuclear structure Anatoli Afanasjev Mississippi State University 1. Nuclear theory selection of starting point 2. What can be done exactly (abinitio calculations) and why we cannot do that systematically?
More informationMesonic and nucleon fluctuation effects in nuclear medium
Mesonic and nucleon fluctuation effects in nuclear medium Research Center for Nuclear Physics Osaka University Workshop of Recent Developments in QCD and Quantum Field Theories National Taiwan University,
More informationAngular Momentum Quantization: Physical Manifestations and Chemical Consequences
Angular Momentum Quantization: Physical Manifestations and Chemical Consequences Michael Fowler, University of Virginia 7/7/07 The SternGerlach Experiment We ve established that for the hydrogen atom,
More informationLikewise, any operator, including the most generic Hamiltonian, can be written in this basis as H11 H
Finite Dimensional systems/ilbert space Finite dimensional systems form an important subclass of degrees of freedom in the physical world To begin with, they describe angular momenta with fixed modulus
More information7.4. Why we have two different types of materials: conductors and insulators?
Phys463.nb 55 7.3.5. Folding, Reduced Brillouin zone and extended Brillouin zone for free particles without lattices In the presence of a lattice, we can also unfold the extended Brillouin zone to get
More informationExotic phases of the Kondo lattice, and holography
Exotic phases of the Kondo lattice, and holography Stanford, July 15, 2010 Talk online: sachdev.physics.harvard.edu HARVARD Outline 1. The Anderson/Kondo lattice models Luttinger s theorem 2. Fractionalized
More informationChemistry 3502/4502. Exam III. All Hallows Eve/Samhain, ) This is a multiple choice exam. Circle the correct answer.
B Chemistry 3502/4502 Exam III All Hallows Eve/Samhain, 2003 1) This is a multiple choice exam. Circle the correct answer. 2) There is one correct answer to every problem. There is no partial credit. 3)
More informationShortRanged Central and Tensor Correlations. Nuclear ManyBody Systems. Reaction Theory for Nuclei far from INT Seattle
ShortRanged Central and Tensor Correlations in Nuclear ManyBody Systems Reaction Theory for Nuclei far from Stability @ INT Seattle September 6, Hans Feldmeier, Thomas Neff, Robert Roth Contents Motivation
More informationEffective Field Theory for Nuclear Physics! Akshay Vaghani! Mississippi State University!
Effective Field Theory for Nuclear Physics! Akshay Vaghani! Mississippi State University! Overview! Introduction! Basic ideas of EFT! Basic Examples of EFT! Algorithm of EFT! Review NN scattering! NN scattering
More informationINSTRUCTIONS PART I : SPRING 2006 PHYSICS DEPARTMENT EXAM
INSTRUCTIONS PART I : SPRING 2006 PHYSICS DEPARTMENT EXAM Please take a few minutes to read through all problems before starting the exam. Ask the proctor if you are uncertain about the meaning of any
More informationUnitarity, Dispersion Relations, Cutkosky s Cutting Rules
Unitarity, Dispersion Relations, Cutkosky s Cutting Rules 04.06.0 For more information about unitarity, dispersion relations, and Cutkosky s cutting rules, consult Peskin& Schröder, or rather Le Bellac.
More informationSeptember 6, 3 7:9 WSPC/Book Trim Size for 9in x 6in book96 7 Quantum Theory of ManyParticle Systems Eigenstates of Eq. (5.) are momentum eigentates.
September 6, 3 7:9 WSPC/Book Trim Size for 9in x 6in book96 Chapter 5 Noninteracting Fermi gas The consequences of the Pauli principle for an assembly of fermions that is localized in space has been discussed
More informationMesoscopic NanoElectroMechanics of Shuttle Systems
* Mesoscopic NanoElectroMechanics of Shuttle Systems Robert Shekhter University of Gothenburg, Sweden Lecture1: Mechanically assisted singleelectronics Lecture2: Quantum coherent nanoelectromechanics
More informationQuantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University
Quantum Field Theory and the Standard Model MATTHEW D. Harvard University SCHWARTZ!H Cambridge UNIVERSITY PRESS t Contents v Preface page xv Part I Field theory 1 1 Microscopic theory of radiation 3 1.1
More informationLecture: Scattering theory
Lecture: Scattering theory 30.05.2012 SS2012: Introduction to Nuclear and Particle Physics, Part 2 2 1 Part I: Scattering theory: Classical trajectoriest and crosssections Quantum Scattering 2 I. Scattering
More informationApplied Nuclear Physics (Fall 2006) Lecture 8 (10/4/06) NeutronProton Scattering
22.101 Applied Nuclear Physics (Fall 2006) Lecture 8 (10/4/06) NeutronProton Scattering References: M. A. Preston, Physics of the Nucleus (AddisonWesley, Reading, 1962). E. Segre, Nuclei and Particles
More informationThermal and Statistical Physics Department Exam Last updated November 4, L π
Thermal and Statistical Physics Department Exam Last updated November 4, 013 1. a. Define the chemical potential µ. Show that two systems are in diffusive equilibrium if µ 1 =µ. You may start with F =
More informationIdeas on nonfermi liquid metals and quantum criticality. T. Senthil (MIT).
Ideas on nonfermi liquid metals and quantum criticality T. Senthil (MIT). Plan Lecture 1: General discussion of heavy fermi liquids and their magnetism Review of some experiments Concrete `Kondo breakdown
More informationPoS(Confinement8)147. Universality in QCD and Halo Nuclei
HelmholtzInstitut für Strahlen und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, University of Bonn, Germany Email: hammer@itkp.unibonn.de Effective Field Theory (EFT) provides a powerful
More informationTopological Kondo Insulators!
Topological Kondo Insulators! Maxim Dzero, University of Maryland Collaborators: Kai Sun, University of Maryland Victor Galitski, University of Maryland Piers Coleman, Rutgers University Main idea Kondo
More informationApplied Statistical Mechanics Lecture Note  3 Quantum Mechanics Applications and Atomic Structures
Applied Statistical Mechanics Lecture Note  3 Quantum Mechanics Applications and Atomic Structures Jeong Won Kang Department of Chemical Engineering Korea University Subjects Three Basic Types of Motions
More informationLecture 3: Fermiliquid theory. 1 General considerations concerning condensed matter
Phys 769 Selected Topics in Condensed Matter Physics Summer 200 Lecture 3: Fermiliquid theory Lecturer: Anthony J. Leggett TA: Bill Coish General considerations concerning condensed matter (NB: Ultracold
More informationQuantum phase transitions
Quantum phase transitions Thomas Vojta Department of Physics, University of MissouriRolla Phase transitions and critical points Quantum phase transitions: How important is quantum mechanics? Quantum phase
More information3. Introductory Nuclear Physics 1; The Liquid Drop Model
3. Introductory Nuclear Physics 1; The Liquid Drop Model Each nucleus is a bound collection of N neutrons and Z protons. The mass number is A = N + Z, the atomic number is Z and the nucleus is written
More informationPH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5
PH575 Spring 2009 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 Spring 2009 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 Spring 2009 POP QUIZ
More informationPhase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality. HansHenning Klauss. Institut für Festkörperphysik TU Dresden
Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality HansHenning Klauss Institut für Festkörperphysik TU Dresden 1 References [1] Stephen Blundell, Magnetism in Condensed
More informationNTNU Trondheim, Institutt for fysikk
NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel. 99890701 Allowed tools: mathematical tables Some formulas can be found on p.2. 1. Concepts.
More informationHelium3, Phase diagram High temperatures the polycritical point. Logarithmic temperature scale
Helium3, Phase diagram High temperatures the polycritical point Logarithmic temperature scale Fermi liquid theory Start with a noninteracting Fermi gas and turn on interactions slowly, then you get a
More informationElectronpositron production in kinematic conditions of PrimEx
Electronpositron production in kinematic conditions of PrimEx Alexandr Korchin Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine 1 We consider photoproduction of e + e pairs on a nucleus
More informationRelativistic magnetotransport in graphene
Relativistic magnetotransport in graphene Markus Müller in collaboration with Lars Fritz (Harvard) Subir Sachdev (Harvard) Jörg Schmalian (Iowa) Landau Memorial Conference June 6, 008 Outline Relativistic
More informationHigh order corrections to density and temperature of fermions and bosons from quantum fluctuations and the CoMDα Model
High order corrections to density and temperature of fermions and bosons from quantum fluctuations and the CoMDα Model Hua Zheng, 1,2 Gianluca Giuliani, 1 Matteo Barbarino, 1 and Aldo Bonasera 1,3 1 Cyclotron
More informationTopological insulator part II: Berry Phase and Topological index
Phys60.nb 11 3 Topological insulator part II: Berry Phase and Topological index 3.1. Last chapter Topological insulator: an insulator in the bulk and a metal near the boundary (surface or edge) Quantum
More information221B Lecture Notes Scattering Theory II
22B Lecture Notes Scattering Theory II Born Approximation Lippmann Schwinger equation ψ = φ + V ψ, () E H 0 + iɛ is an exact equation for the scattering problem, but it still is an equation to be solved
More informationSuperfluid Helium3: From very low Temperatures to the Big Bang
Superfluid Helium3: From very low Temperatures to the Big Bang Universität Frankfurt; May 30, 2007 Dieter Vollhardt Contents: The quantum liquids 3 He and 4 He Superfluid phases of 3 He Broken symmetries
More informationEvaluation of Triangle Diagrams
Evaluation of Triangle Diagrams R. Abe, T. Fujita, N. Kanda, H. Kato, and H. Tsuda Department of Physics, Faculty of Science and Technology, Nihon University, Tokyo, Japan Email: csru11002@g.nihonu.ac.jp
More informationFermi s Golden Rule and Simple Feynman Rules
Fermi s Golden Rule and Simple Feynman Rules ; December 5, 2013 Outline Golden Rule 1 Golden Rule 2 Recipe For the Golden Rule For both decays rates and cross sections we need: The invariant amplitude
More informationLecture 9  Rotational Dynamics
Lecture 9  Rotational Dynamics A Puzzle... Angular momentum is a 3D vector, and changing its direction produces a torque τ = dl. An important application in our daily lives is that bicycles don t fall
More informationLecture 5. HartreeFock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 HartreeFock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particlenumber representation: General formalism The simplest starting point for a manybody state is a system of
More informationFinitetemperature Field Theory
Finitetemperature Field Theory Aleksi Vuorinen CERN Initial Conditions in Heavy Ion Collisions Goa, India, September 2008 Outline Further tools for equilibrium thermodynamics Gauge symmetry FaddeevPopov
More informationOrigin of the first Hund rule in Helike atoms and 2electron quantum dots
in Helike atoms and 2electron quantum dots T Sako 1, A Ichimura 2, J Paldus 3 and GHF Diercksen 4 1 Nihon University, College of Science and Technology, Funabashi, JAPAN 2 Institute of Space and Astronautical
More informationChiral Magnetic and Vortical Effects at Weak Coupling
Chiral Magnetic and Vortical Effects at Weak Coupling University of Illinois at Chicago and RIKENBNL Research Center June 19, 2014 XQCD 2014 Stonybrook University, June 1920, 2014 Chiral Magnetic and
More informationQuantum Cluster Methods (CPT/CDMFT)
Quantum Cluster Methods (CPT/CDMFT) David Sénéchal Département de physique Université de Sherbrooke Sherbrooke (Québec) Canada Autumn School on Correlated Electrons Forschungszentrum Jülich, Sept. 24,
More informationFundamental Interactions (Forces) of Nature
Chapter 14 Fundamental Interactions (Forces) of Nature Interaction Gauge Boson Gauge Boson Mass Interaction Range (Force carrier) Strong Gluon 0 shortrange (a few fm) Weak W ±, Z M W = 80.4 GeV/c 2 shortrange
More informationThermal & Statistical Physics Study Questions for the Spring 2018 Department Exam December 6, 2017
Thermal & Statistical Physics Study Questions for the Spring 018 Department Exam December 6, 017 1. a. Define the chemical potential. Show that two systems are in diffusive equilibrium if 1. You may start
More informationCounting Schwarzschild and Charged Black Holes
SLACPUB984 SUITP940 September 199 hepth/909075 Counting Schwarzschild and Charged Black Holes Edi Halyo 1, Barak Kol 1, Arvind Rajaraman 2 and Leonard Susskind 1 1 Department of Physics, Stanford
More informationHolography and Mottness: a Discrete Marriage
Holography and Mottness: a Discrete Marriage Thanks to: NSF, EFRC (DOE) Ka Wai Lo M. Edalati R. G. Leigh Mott Problem emergent gravity Mott Problem What interacting problems can we solve in quantum mechanics?
More informationThe Heisenberg uncertainty principle. The Pauli exclusion principle. Classical conductance.
John Carroll 1 The Heisenberg uncertainty principle. Some quick particle physics. Beta decay example. The Pauli exclusion principle. The existence of energy bands. Classical conductance. Why it is inaccurate.
More informationLecture 8 Feynman diagramms. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 8 Feynman diagramms SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Photon propagator Electronproton scattering by an exchange of virtual photons ( Diracphotons ) (1) e  virtual
More information+ 1. which gives the expected number of Fermions in energy state ɛ. The expected number of Fermions in energy range ɛ to ɛ + dɛ is then dn = n s g s
Chapter 8 Fermi Systems 8.1 The Perfect Fermi Gas In this chapter, we study a gas of noninteracting, elementary Fermi particles. Since the particles are noninteracting, the potential energy is zero,
More informationJoint ICTPIAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation August Introduction to Nuclear Physics  2
235820 Joint ICTPIAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation 617 August 2012 Introduction to Nuclear Physics  2 P. Van Isacker GANIL, Grand Accelerateur National d'ions Lourds
More information1 Multiplicity of the ideal gas
Reading assignment. Schroeder, section.6. 1 Multiplicity of the ideal gas Our evaluation of the numbers of microstates corresponding to each macrostate of the twostate paramagnet and the Einstein model
More informationIntro to Nuclear and Particle Physics (5110)
Intro to Nuclear and Particle Physics (5110) March 23, 2009 From Nuclear to Particle Physics 3/23/2009 1 Nuclear Physics Particle Physics Two fields divided by a common set of tools Theory: fundamental
More informationColor superconductivity in dense quark matter and Astrophysical implications of dense quark matter
Color superconductivity in dense quark matter and Astrophysical implications of dense quark matter Seminar WS 2009/2010 Relativistische Schwerionenphysik Interface of QuarkGluon Plasma and Cold Quantum
More informationProton Neutron Scattering
March 6, 205 Lecture XVII Proton Neutron Scattering Protons and neutrons are both spin /2. We will need to extend our scattering matrix to dimensions to include all the possible spin combinations fof the
More informationA Renormalization Group Primer
A Renormalization Group Primer Physics 295 2010. Independent Study. Topics in Quantum Field Theory Michael Dine Department of Physics University of California, Santa Cruz May 2010 Introduction: Some Simple
More informationSemiclassical Electron Transport
Semiclassical Electron Transport Branislav K. Niolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 64: Introduction to Solid State Physics http://www.physics.udel.edu/~bniolic/teaching/phys64/phys64.html
More informationIntermission: Let s review the essentials of the Helium Atom
PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent
More informationNuclear Spin and Stability. PHY 3101 D. Acosta
Nuclear Spin and Stability PHY 3101 D. Acosta Nuclear Spin neutrons and protons have s = ½ (m s = ± ½) so they are fermions and obey the Pauli Exclusion Principle The nuclear magneton is eh m µ e eh 1
More informationPhysics 607 Final Exam
Physics 607 Final Exam Please show all significant steps clearly in all problems. 1. Let E,S,V,T, and P be the internal energy, entropy, volume, temperature, and pressure of a system in thermodynamic equilibrium
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 12, 2011 1:00PM to 3:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of this
More informationTwoBody Problem. Central Potential. 1D Motion
TwoBody Problem. Central Potential. D Motion The simplest nontrivial dynamical problem is the problem of two particles. The equations of motion read. m r = F 2, () We already know that the center of
More informationThe Superfluid Phase s of Helium 3
The Superfluid Phase s of Helium 3 DIETER VOLLHARD T RheinischWestfälische Technische Hochschule Aachen, Federal Republic of German y PETER WÖLFL E Universität Karlsruhe Federal Republic of Germany PREFACE
More informationColor Superconductivity in High Density QCD
Color Superconductivity in High Density QCD Roberto Casalbuoni Department of Physics and INFN  Florence Bari,, September 9 October 1, 004 1 Introduction Motivations for the study of highdensity QCD:
More informationPhysics 342: Modern Physics
Physics 342: Modern Physics Final Exam (Practice) Relativity: 1) Two LEDs at each end of a meter stick oriented along the x axis flash simultaneously in their rest frame A. The meter stick is traveling
More informationPHYSICAL SCIENCES PART A
PHYSICAL SCIENCES PART A 1. The calculation of the probability of excitation of an atom originally in the ground state to an excited state, involves the contour integral iωt τ e dt ( t τ ) + Evaluate the
More informationPHZ 7427 SOLID STATE II: Electronelectron interaction and the Fermiliquid theory
PHZ 7427 SOLID STATE II: Electronelectron interaction and the Fermiliquid theory D. L. Maslov Department of Physics, University of Florida (Dated: February 2, 204) CONTENTS I.Notations 2 II.Electrostatic
More informationLecture 18: 3D Review, Examples
Lecture 18: 3D Review, Examples A real (2D) quantum dot http://pages.unibas.ch/physmeso/pictures/pictures.html Lecture 18, p 1 Lect. 16: Particle in a 3D Box (3) The energy eigenstates and energy values
More informationNuclear Level Density with Nonzero Angular Momentum
Commun. Theor. Phys. (Beijing, China) 46 (2006) pp. 514 520 c International Academic Publishers Vol. 46, No. 3, September 15, 2006 Nuclear Level Density with Nonzero Angular Momentum A.N. Behami, 1 M.
More informationRenormalization group methods in nuclear few and manybody problems
Renormalization group methods in nuclear few and manybody problems Lecture 1 S.K. Bogner (NSCL/MSU) 2011 National Nuclear Physics Summer School University of North Carolina at Chapel Hill Useful readings
More informationLecture 6 Scattering theory Partial Wave Analysis. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 6 Scattering theory Partial Wave Analysis SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 The Born approximation for the differential cross section is valid if the interaction
More informationString / gauge theory duality and ferromagnetic spin chains
String / gauge theory duality and ferromagnetic spin chains M. Kruczenski Princeton Univ. In collaboration w/ Rob Myers, David Mateos, David Winters Arkady Tseytlin, Anton Ryzhov Summary Introduction mesons,,...
More informationPHYS 5012 Radiation Physics and Dosimetry
PHYS 5012 Radiation Physics and Dosimetry Tuesday 12 March 2013 What are the dominant photon interactions? (cont.) Compton scattering, photoelectric absorption and pair production are the three main energy
More informationSound modes and the twostream instability in relativistic superfluids
Madrid, January 17, 21 1 Andreas Schmitt Institut für Theoretische Physik Technische Universität Wien 1 Vienna, Austria Sound modes and the twostream instability in relativistic superfluids M.G. Alford,
More informationCritical Phenomena in Gravitational Collapse
Critical Phenomena in Gravitational Collapse Yiruo Lin May 4, 2008 I briefly review the critical phenomena in gravitational collapse with emphases on connections to critical phase transitions. 1 Introduction
More informationStatistical Mechanics Notes. Ryan D. Reece
Statistical Mechanics Notes Ryan D. Reece August 11, 2006 Contents 1 Thermodynamics 3 1.1 State Variables.......................... 3 1.2 Inexact Differentials....................... 5 1.3 Work and Heat..........................
More informationPreliminary Quantum Questions
Preliminary Quantum Questions Thomas Ouldridge October 01 1. Certain quantities that appear in the theory of hydrogen have wider application in atomic physics: the Bohr radius a 0, the Rydberg constant
More informationPhonon II Thermal Properties
Phonon II Thermal Properties Physics, UCF OUTLINES Phonon heat capacity Planck distribution Normal mode enumeration Density of states in one dimension Density of states in three dimension Debye Model for
More informationThe interacting boson model
The interacting boson model P. Van Isacker, GANIL, France Dynamical symmetries of the IBM Neutrons, protons and Fspin (IBM2) T=0 and T=1 bosons: IBM3 and IBM4 The interacting boson model Nuclear collective
More information