The nonequilibrium theory described in the previous chapters reduces to a particularly simple form for a system of fermions very close to zero temperature. To see this, let us define a “local occupation number” f ( p , ω ; R , T ) $ f(p, \omega ;R, T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math12_1.tif"/> by writing 12.1 g < ( p , ω ; R , T ) = a ( p , ω ; R , T ) f ( p , ω ; R , T ) $$ g^{ < } (p,~\omega ;R,~T) = a(p,~\omega ;R,~T)f(p,~\omega ;R,~T) $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math12_1.tif"/>
The Landau Fermi liquid theory of metals has been so successful that exceptions to it draw a fair amount of attention. The theory rests on the assumption that the low‐energy properties of interacting electrons in a metal in three dimensions may be described in terms of weakly repulsive charged spin 1 2 $ \frac{1}{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_1.tif"/> fermionic excitations— the quasiparticles. This is of course not generally expected to hold following a phase transition, for example, in the form of a Pomeranchuk instability or a superconducting transition. In Chapter 2 we showed that Pomeranchuk instabilities occur whenever some Landau F function satisfies the condition F l s ( a ) / ( 2 l + 1 ) = - 1 $ F_{l}^{s(a)} /(2l + 1) = - 1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_2.tif"/> , where l = 0, 1, 2, . . . denotes the relevant angular momentum channel where the instability occurs. The l = 0 instability in the spin‐symmetric channel is associated with a divergence of compressibility that leads to phase separation and consequent breakdown of the Fermi liquid. In the spin‐antisymmetric channel, the l = 0 instability corresponds to ferromagnetism. In the ferromagnetic quantum critical region (“classical” Gaussian regime in Hertz theory), we found in Sec. 9.6.2.3 that spin fluctuations lead to the destruction of the Landau Fermi liquid and this is evident, for example, in the ln ( 1 / T ) $ {\text{ln }}(1/T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_3.tif"/> divergence of the specific heat coefficient γ ( T ) . Proximity to an electronic instability is however not a necessary condition for the breakdown of the Fermi liquid. In Sec. 10.2 we found that transverse electromagnetic fluctuations are only weakly screened by conduction electrons and that results in a γ ( T ) ∼ ln ( 1 / T ) $ \gamma (T) \sim {\text{ln }}(1/T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_4.tif"/> divergence for the specific heat coefficient without needing to be in the vicinity of a ferromagnetic quantum critical point 1 . The lack of screening of such electromagnetic fluctuations is a feature shared with low energy spin fluctuations in the Hertz theory of quantum critical ferromagnets. Another route to non Fermi‐liquid behavior, not reliant on the existence of long‐range interactions, appeared in Sec. 11.4 where we analyzed non‐optimally screened Kondo models. It was found that singularities in properties such as the susceptibility and specific heat coefficient may be expected in the event that residual low‐energy degeneracies associated with spin‐flip scattering with conduction electrons persist down to the lowest temperatures. Finally, weak potential disorder was found to preserve the Fermi liquid (in three dimensions) down to the lowest temperatures (see Chapter 8 ).
The quantum revolution in the beginning of the 20th century changed our concepts of dynamics. The dynamic variables such as position r $ {\mathbf{r}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_1.tif"/> and momentum p $ {\mathbf{p}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_2.tif"/> are replaced by the corresponding position and momentum operators r ^ $ {\hat{\mathbf{r}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_3.tif"/> and p ^ $ {\hat{\mathbf{p}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_4.tif"/> , respectively When one would like to observe a dynamical variable, say ω , of a particle, one has to introduce a wavefunction ψ , which contains all the dynamical information of particles, and to apply the corresponding dynamical operator Ω ^ $ \hat{\Omega } $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_5.tif"/> to the wavefunction ψ . Then one may obtain the desired dynamical value of the particle as the eigenvalue of the operator, 1.1 Ω ^ ψ = ω ψ . Ω^ψ=ωψ. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math1_1.tif"/>
The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second‐quantized, Heisenberg representation, particle‐creation, and annihilation operators. The creation operator ψ ^ † ( r , t ) $ \hat{\psi }^{\dag } ({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_1.tif"/> , when acting to the right on a state of the system, adds a particle to the state at the space‐time point r , $ {\mathbf{r}}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_2.tif"/> t ; the annihilation operator ψ ^ ( r , t ) $ \hat{\psi }({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_3.tif"/> , the adjoint of the creation operator acting to the right, removes a particle from the state at the point r , t .
For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.