Landau- 1 Chennai Mathematical Institute April 25, 2011 1 Final year project under Prof. K. Narayan, CMI
Interacting fermion system I Basic properties of metals (heat capacity, susceptibility,...) were explained by Sommerfeld, Pauli and others in 1928 They described metals as system of non-interacting Fermions, a Fermi gas This description is known to work over a large temperature range (several orders of magnitude) [Coleman, 2010] However, at sufficiently low T, electron repulsion comparable to its energy Why does the non-interacting picture work so well?
Interacting fermion system II In 1956, Lev Landau developed a of interacting spin-1/2 Fermions, to explain stability of Fermi gas against perturbations The Landau- has been successful in explaining systems such as metals, nuclear matter, liquid He-3 etc. [Schofield, 1999] The key ideas behind Landau s are the principle of adiabaticity and Pauli s exclusion prnciple. Generic Hamiltonian H = ɛ p c pσc pσ + V (q)c p 1 q c p 2 +q c p 1 c p2 p,σ p i,q,σ
Adiabaticity Start with ground state of non-interacting system and slowly turn on interactions Landau s argument If full Hamiltonian has the same symmetries of unperturbed Hamiltonian then its ground states can be obtained by perturbatively transforming the simple ground state. [Pines and Noziéres, 1966] Note: This means that we ignore the possibility of a phase transition Similarly, low-lying excitations of non-interacting are mapped to low lying excitations of interacting
Adiabaticity Start with ground state of non-interacting system and slowly turn on interactions Landau s argument If full Hamiltonian has the same symmetries of unperturbed Hamiltonian then its ground states can be obtained by perturbatively transforming the simple ground state. [Pines and Noziéres, 1966] Note: This means that we ignore the possibility of a phase transition Similarly, low-lying excitations of non-interacting are mapped to low lying excitations of interacting
Adiabaticity Start with ground state of non-interacting system and slowly turn on interactions Landau s argument If full Hamiltonian has the same symmetries of unperturbed Hamiltonian then its ground states can be obtained by perturbatively transforming the simple ground state. [Pines and Noziéres, 1966] Note: This means that we ignore the possibility of a phase transition Similarly, low-lying excitations of non-interacting are mapped to low lying excitations of interacting
Adiabaticity Start with ground state of non-interacting system and slowly turn on interactions Landau s argument If full Hamiltonian has the same symmetries of unperturbed Hamiltonian then its ground states can be obtained by perturbatively transforming the simple ground state. [Pines and Noziéres, 1966] Note: This means that we ignore the possibility of a phase transition Similarly, low-lying excitations of non-interacting are mapped to low lying excitations of interacting
Adiabaticity II Turning on interactions conserves dynamical variables such as spin and momentum of excitations (this is a consequence of Pauli s exclusion principle) The low lying excitations of the full systems are called Landau quasiparticles
Adiabaticity II Turning on interactions conserves dynamical variables such as spin and momentum of excitations (this is a consequence of Pauli s exclusion principle) The low lying excitations of the full systems are called Landau quasiparticles
Adiabaticity II Turning on interactions conserves dynamical variables such as spin and momentum of excitations (this is a consequence of Pauli s exclusion principle) The low lying excitations of the full systems are called Landau quasiparticles
Scattering rate I Quasiparticle excitations come from adiabatic continuation of noninteracting states Thus, they need not be eigenstates of the new Hamiltonian. As such they have finite lifetime τ, arising solely from scattering with other quasiparticles We consider only long-lived excitations, viz, those having lifetime τ, where ɛ is the excitation energy ɛ Estimate scattering rate for external particle of energy ɛ 1
Scattering rate II It can be shown [Phillips, 2003] that the scattering rate 1 τ ρ = (ɛ 1 ɛ F ) 2 At low temperature T, we can take the average excitation energy to be k B T and hence we have 1 τ T 2 for small T Thus, transport properties, such as resistivity, have T 2 temperature dependance. This quadratic dependence is a key property of Fermi
Two-point I Like any field, the two-point correlator gives important information about the. We have the two point [Abrikosov et al., 1975] G( r 2 r 1, t 2 t 1 ) = i 0 T ψ( r 2, t 2 )ψ ( r 1, t 1 ) 0 where T denotes the time-ordered product, defined (for Fermionic operators A, B) as { A(t 1 )B(t 2 ) for t 1 > t 2 T A(t 1 )B(t 2 ) = B(t 2 )A(t 1 ) for t 2 > t 1 We expand G( r, t) in basis m of momentum eigenstates
Two-point II Use notation: E m E 0 = ɛ m + µ, E m E 0 = ɛ m µ µ is chemical potential Taking Fourier transform we get G( p, ω) = i + i m m δ( p p m ) ω ɛ m µ + iδ ψ 0m 2 δ( p + p m ) ω + ɛ m µ iδ ψ 0m 2
Analytic structure Seperating the real and imaginary parts in the expression we get Re G( p, ω) = 1 π P Im G( p, ω ) sgn(ω µ) ω ω We can define s G R, G A such that Re G = Re G R = Re G A Im G R = Im G sgn(ω µ) Im G A = Im G sgn(ω µ) Observe that G R, G A are analytic in upper- and lower-half plans respectively
Poles and residues Recall the expression for G(p, ω), δ( p p m ) G( p, ω) = i ω ɛ m µ + iδ ψ 0m 2 + i m m δ( p + p m ) ω + ɛ m µ iδ ψ 0m 2 Note that this expression has poles at ω = ɛ m + µ and ω = ɛ m + µ for all m, m That is, every single-particle excitation (quasiparticle or hole) corresponds to pole of the. For given quasiparticle state m (hole m ), we have pole at ω = ɛ m + µ (or ω = ɛ m + µ) with residues Z m = ψ 0m 2 = m ψ 0 2
Poles and residues Recall the expression for G(p, ω), Note that this expression has poles at ω = ɛ m + µ and ω = ɛ m + µ for all m, m That is, every single-particle excitation (quasiparticle or hole) corresponds to pole of the. For given quasiparticle state m (hole m ), we have pole at ω = ɛ m + µ (or ω = ɛ m + µ) with residues Z m = ψ 0m 2 = m ψ 0 2 Z m = ψ 0m 2 = m ψ 0 2 This is physical interpretation for residues
Occupation number I Suppose we have a given momentum state Let us write the number operator expectation value (Schrödinger representation) as This can be written as N( p) = S 0 ψ p ψ p 0 S N( p) = 2i lim t 0 + G( p, ω)e iωt dω We split this integral as µ G( p, ω)e iωt dω + µ G( p, ω)e iωt dω
Occupation number II Recall that for ω < µ (ω > µ), G has poles in upper (lower) half plane It can be shown that two parts of the integral cancel out, except for residue from pole corresponding to p (only if it is above p F ) [Lifshitz and Pitaevskii, 1980] Thus, we have N(p > p F ) N(p < p F ) = Z p where Z p is the residue at Fermi surface
Occupation number III Figure: Jump in the occupation number at the Fermi surface Jump in occupation number is another important property of Fermi Z = 1, 0 correspond to Fermi-gas and non- respectively
Breakdown of behaviour Over the past thirty years, materials have been found which behave as Fermi with renormalization of up to 10 3 (heavy Fermion systems) This illustrates the robustness of Landau s. However, materials have also been found that violate low-energy description of quasiparticles (eg.: High-T c cuprates) [Senthil, 2008]
Vanishing residue Suppose we expand a quasiparticle state perturbatively, [Varma et al., 2002] m = Z 1/2 m ψ 0 +N k 1,k 2,k 3 V (q p)c k 1 c k2 c k 3 δ(q; p) 0 +... (first term is free particle creation, second one comes from interaction) We say behaviour breaks down as Z m 0 This is equivalent to diverging contribution from many-particle states
I Abrikosov, A., Gorkov, L., and Dzyaloshinski, I. (1975). Methods of quantum field in statistical physics. Dover books on physics. Dover Publications. Coleman, P. (2010). to many body physics. Rutgers University (draft version). Lifshitz, E. M. and Pitaevskii, L. P. (1980). Statistical Physics Part 2. Elsevier. Phillips, P. (2003). Advanced solid state physics. Frontiers in Physics Series. Westview Press.
II Pines, D. and Noziéres, P. (1966). The of quantum. W. A. Benjamin Inc. Schofield, A. J. (1999).. Contemporary Physics, 40:95 115. Senthil, T. (2008). Critical fermi surfaces and non-fermi liquid metals. Phys. Rev. B, 78(3):035103. Varma, C. M., Nussinov, Z., and van Saarloos, W. (2002). Singular or non-fermi. Physics Reports, 361(5-6):267 417.
Acknowledgements I would like to thank my advisor Prof. K. Narayan for his support I would also like to thank my colleague, Debangshu Mukherjee, for helpful comments I also thank Prof. Baskaran, IMSc., and Prof. Kedar Damle, TIFR, for useful discussions