The Hamiltonian for the many-electron system in finite volume [0, L]d is given by H = H0 + Hint where H0 = 1Ld ∑ eka + kσakσ (6.1) ek = k 2 2m − µ, and Hint = 1L3d ∑ ∑ V (k− p)a+kσa+q−k,τaq−p,τapσ (6.2) As usual, the interacting part may be represented by the following diagram: Since there is conservation of momentum, there are three independent momenta. One may consider three natural limiting cases with two independent momenta: forward exchange BCS In chapter 8 where we prove rigorous bounds on Feynman diagrams we show that the most singular contributions to the perturbation series come from the the third diagram in the above figure. To retain only this term, the q = 0 term of Hint given by (6.2), is the basic approximation of BCS theory. Furthermore V (k−p) is substituted by its value on the Fermi surface, V (k − p) ≈ V (kˆ − pˆ) where kˆ = kF k‖k‖ , kF = √ 2mµ, and the momenta k,p are restricted to values close to the Fermi surface, |ek|, |ep| ≤ ωD, where the cutoff ωD is referred to as the Debye frequency. Thus the BCS approximation reads Hint ≈ HBCS (6.3) where HBCS = 1L3d ∑ ∑ V (kˆ− pˆ) a+kσa+−k,τa−p,τapσ (6.4) We first make a comment on the volume factors. In (6.4) we retained only the q = 0 term of (6.2), but we did not cancel a volume factor. It is not obvious that (6.4) is still proportional to the volume. However, in the case of the forward approximation which is defined by putting k = p in (6.2) without canceling a volume factor, Hint ≈ Hforw := 1L3d ∑ ∑ V (0)a+kσa + p,τap,τakσ (6.5) there is an easy argument which shows that (6.5) is still proportional to the volume (for constant density). Namely, on a fixed N -particle space FN the interacting part Hint is a multiplication operator given by N∑ V (xi − xj) (6.6) Let δy(x) = δ(x − y). Then ϕ(x1, · · · ,xN ) := δy1 ∧ · · · ∧ δyN (x1, · · · ,xN ) is an eigenfunction of Hint with eigenvalue E = 12 N∑ V (yi − yj) = 12Ld N∑ ∑ e−i(yi−yj)qV (q) (6.7) which is, for V (x) ∈ L1, proportional to N or to the volume for constant density. One finds that ϕ is also an eigenvector of the forward term, Hforwϕ = Eforwϕ where Eforw is obtained from (6.7) by putting q = 0 without canceling a volume factor, V (q = 0) (6.8) Although the model defined by the Hamiltonian HBCS in (6.4) is still quartic in the annihilation and creation operators, it can be solved explicitly, without making a quadratic mean field approximation as it is usually done. The important point is that the approximation ‘putting q = 0 without canceling a volume factor’ has the effect that in the bosonic functional integral representation the volume factors enter the formulae in such a way that in the thermodynamic limit the integration variables are forced to take values at the global minimum of the effective potential. That is, fluctuations around the minimum configuration are suppressed for Ld →∞.
The Hamiltonian for the many-electron system in finite volume [0, L]d is given by H = H0 + Hint where H0 = 1Ld ∑ eka + kσakσ (6.1) ek = k 2 2m − µ, and Hint = 1L3d ∑ ∑ V (k− p)a+kσa+q−k,τaq−p,τapσ (6.2) As usual, the interacting part may be represented by the following diagram: Since there is conservation of momentum, there are three independent momenta. One may consider three natural limiting cases with two independent momenta: forward exchange BCS In chapter 8 where we prove rigorous bounds on Feynman diagrams we show that the most singular contributions to the perturbation series come from the the third diagram in the above figure. To retain only this term, the q = 0 term of Hint given by (6.2), is the basic approximation of BCS theory. Furthermore V (k−p) is substituted by its value on the Fermi surface, V (k − p) ≈ V (kˆ − pˆ) where kˆ = kF k‖k‖ , kF = √ 2mµ, and the momenta k,p are restricted to values close to the Fermi surface, |ek|, |ep| ≤ ωD, where the cutoff ωD is referred to as the Debye frequency. Thus the BCS approximation reads Hint ≈ HBCS (6.3) where HBCS = 1L3d ∑ ∑ V (kˆ− pˆ) a+kσa+−k,τa−p,τapσ (6.4) We first make a comment on the volume factors. In (6.4) we retained only the q = 0 term of (6.2), but we did not cancel a volume factor. It is not obvious that (6.4) is still proportional to the volume. However, in the case of the forward approximation which is defined by putting k = p in (6.2) without canceling a volume factor, Hint ≈ Hforw := 1L3d ∑ ∑ V (0)a+kσa + p,τap,τakσ (6.5) there is an easy argument which shows that (6.5) is still proportional to the volume (for constant density). Namely, on a fixed N -particle space FN the interacting part Hint is a multiplication operator given by N∑ V (xi − xj) (6.6) Let δy(x) = δ(x − y). Then ϕ(x1, · · · ,xN ) := δy1 ∧ · · · ∧ δyN (x1, · · · ,xN ) is an eigenfunction of Hint with eigenvalue E = 12 N∑ V (yi − yj) = 12Ld N∑ ∑ e−i(yi−yj)qV (q) (6.7) which is, for V (x) ∈ L1, proportional to N or to the volume for constant density. One finds that ϕ is also an eigenvector of the forward term, Hforwϕ = Eforwϕ where Eforw is obtained from (6.7) by putting q = 0 without canceling a volume factor, V (q = 0) (6.8) Although the model defined by the Hamiltonian HBCS in (6.4) is still quartic in the annihilation and creation operators, it can be solved explicitly, without making a quadratic mean field approximation as it is usually done. The important point is that the approximation ‘putting q = 0 without canceling a volume factor’ has the effect that in the bosonic functional integral representation the volume factors enter the formulae in such a way that in the thermodynamic limit the integration variables are forced to take values at the global minimum of the effective potential. That is, fluctuations around the minimum configuration are suppressed for Ld →∞.
If we consider ( ) 2,kr tΨ G to be the probability density for finding, at time t, particle 1 at particle k at and particle N at ,,1 … Gr …G ,kr ,Nr G a normalization condition analogous to Eq.(29.2) holds as (35.2) 1V* =ΨΨ∫ d The integral is defined in a space with 3N dimensions called the configuration space of the system, where a configuration point is given by 3N position coordinates 1 1 1, , , , , , , , , ,k k k N N N ,x y z x y z x y z… … and the volume element is NNNkkkNk dzdydxdzdydxdzdydxdddd ………… 1111 VVVV == Physical observables of the system will be represented by Hermitian operators, obtained from the classical representation in terms of the position and momentum variables as indicated by Postulate 2. Thus the many-particle Hamiltonian results from Eq.(2.32) as ( ) (∑ ∑ = ≠ ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ++∇−= k rrUtrU m H ,, 2 ˆ GGG= ) ) (35.3) where only acts on the coordinates of the kth particle, of mass The one-particle potential energy is due to external fields where this particle moves and k∇ .km( trU kk ,G( )jkkj rrU GG , gives the mutual interaction between the particles k and j. It is assumed that the position and momentum operators of different particles commute, and so do all the operators describing single-particle observables if they refer to different particles. The evolution of a many-particle system will be then formulated as a generalization of Eq.(30.6) in terms of given by Eqs.(35.1) and (35.3) HˆandΨ Ψ=∂ Ψ∂ H t i ˆ= (35.4) If the Hamiltonian (35.3) does not depend explicitly on the time, it is usual to express the time dependence of the many-particle state function by means of the evolution operator defined as ( )tUˆ ( ) ( ) ( )NkNk rrrtUtrrr G…G…GG…G…G ,,,,ˆ,,,,, 11 αΨ=Ψ (35.5) which is formally derived from Eq.(35.4) as ( ) ( ) ( )ˆˆ ˆU ti U t i HU t t t ˆα α α ∂∂ ⎡ ⎤Ψ = Ψ =⎣ ⎦∂ ∂= = Ψ It follows that (35.6) ( ) =/ˆˆ tHietU −= which is the analogue of Eq.(30.12). The evolution operator is unitary, Eq.(II.22), since ( ) ( )†ˆ ˆU t U t Iˆ= (35.7) Substituting Eq.(35.5), one obtains from Eq.(35.4) that the stationary states of the Nparticle systems are given by the energy eigenvalue equation ( ) ( )1 1ˆ , , , , , , , ,k N k NH r r r E r r rα α αΨ = ΨG G G G G G… … … … (35.8) The solution to the N-particle problem is facilitated if we take into account the symmetries of many systems of physical interest, which allow us to define constants of motion and criteria for simplifying the form (35.3) of the Hamiltonian operator. An alternative to complex explicit calculations is to associate with specific symmetries time independent unitary operators which transform the state functions into so that Eq.(35.4) gives ˆ ,U Ψ ˆ ,U Ψ ( ) ( )†ˆ ˆ ˆ ˆ ˆ ˆ ˆi U i U UH UHU Ut t∂ ∂ΨΨ = = Ψ = Ψ∂ ∂= = ˆ This is again a Schrödinger equation for the state Uˆ Ψ with the new Hamiltonian †ˆ ˆ ˆ ˆH UHU′ = If the Hamiltonian remains unchanged by the transformation, we have (35.9) †ˆ ˆ ˆ ˆ ˆ ˆor , 0H UHU U H⎡ ⎤= ⎣ ⎦ = According to the evolution equation (29.43), this implies that so that U is a constant of motion. If the unitary transformation which leaves the Hamiltonian invariant, is a function of an observable, its constancy implies that the given observable is a constant in any state of the system.
In many exactly solvable models of quantum statistical mechanics, an interaction Hamiltonian is equal to an integral of a product of operators of annihilation and creation and a potential divided by a volume of a system raised to a certain power. For example, the interaction Hamiltonian of the BCS model of superconductivity has the form HI,M=2Vg∬∬Φ(x1−x2,x′1−x′2)ψ+(x2)ψ(x′1)ψ(x′2)dx1dx2dx′1dx′2, (1) where integration is carried out over the whole three-dimensional Euclidean space ℝ3, V is the volume of ℝ3, ψ+ and ψ- are operators of creation and annihilation of Fermi particles, and Φ is a potential that satisfies the conditions Φ(x1−x2,x1′−x2′)=Φ(x1−x2,x′1−x′2)=Φ(x2′−x1′,x2−x2),Φ(x2−x1,x1′−x2′)=Φ(x2−x1,x2′−x1′) (2) and is a test function with respect to x 1-x 2 and x1′−x2′. The dependence on spin is omitted here.
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.
At the end of the last chapter, we considered the electron-electron interaction. Here, we found two terms: the Hartree energy and the exchange energy. These work fine if we can solve for the energy exactly. However, as was remarked, this can only be done for a few electron systems. For more meaningful systems, we need to develop an approximation technique for the interacting Green’s functions. The standard approach to this is via the interaction representation, which gives us a prescription for developing the perturbation series. As one might infer from our previous discussion, this will lead to multiple, messy integrals. Fortunately, the Feynman diagrams give us an elegant and pictorial method of looking at the various terms in the perturbation series. As a result, they also give us a much more intuitive method of examining the various terms.
The structural properties of all diagrammatic expansions developed so far can be analyzed systematically with the help of functional equations. 5.1 Generating Functionals In Chapter 3 we have seen that the correlation functions obtained from the functional derivatives of Z[j] via relation (2.14), and the generating functional itself, contain many disconnected parts. Ultimately, however, we shall be interested only in the connected parts of Z[j]. Remember that a meaningful description of a very large thermodynamic system can only be given in terms of the free energy which is directly proportional to the total volume. In the limit of an infinite volume, also called thermodynamic limit, one has then a well-defined free energy density. The partition function, on the other hand, has no proper infinite-volume limit. We can observe this property directly in the diagrammatic expansion of Z[j]. Each component of a disconnected diagram is integrated over the entire space, thus contributing a volume factor. The expansion of Z[j] therefore diverges at an infinite volume. In thermodynamics, we form the free energy from the logarithm of the partition function, which carries only a single overall volume factor and contains only connected diagrams. Therefore we expect the logarithm of Z[j] to provide us with the desired generating functional W[j]: W[j] = log Z[j]. (5.1) In this chapter we shall see that the functional derivatives of W[j] produce, indeed, precisely the connected parts of the Feynman diagrams in each correlation function. Consider the connected correlation functions G (n) c (x1,...,xn) defined by the functional derivatives G (n) c (x1,...,xn) = δ δ
Interactions in many-body systems bring about collective phenomena such as superconductivity and magnetism. In many cases, simple mean-field theory provides a basic understanding of these phenomena. In fermion systems in one dimension, however, neither the mean-field theory nor perturbation theory works if it starts from the non-interacting fermions. This is because the interaction effects in one dimension are much stronger than those in higher dimensions. Intuitively speaking, two particles cannot avoid collision in a single-way track in contrast with two and three dimensions. Thus the interaction effects appear in a drastic way in one dimension. Another aspect in one dimension, which overcompensates the difficulty of perturbation and mean-field theories, is that a complete account of interaction effects is possible under certain conditions. The class of systems satisfying such conditions is referred to as exactly solvable. Soon after the establishment of quantum mechanics, Bethe solved exactly the Heisenberg spin model in one dimension [28]. The basic idea of the solution is now called the Bethe ansatz. Since then, theoretical physics in one dimension has developed into a magnificent edifice, including sophisticated mathematical techniques. In many cases, the eigenfunctions derived by the Bethe ansatz consist of plane waves that are defined stepwise for each spatial configuration of particles. Since the coefficients of plane waves depend on the configuration, the property of the wave function cannot be made explicit without detailed knowledge of these coefficients. We mention some of the recent monographs on the Bethe ansatz and its extensions [54, 118, 179]. A comprehensive account on exactly solvable models has recently been given by Sutherland [178].