Statistical Field theory Thierry Giamarchi November 8, 2003 Contents 1 Introduction 5 2 Equilibrium statistical mechanics 7 2.1 Basics; correlation functions . ............................... 7 2.2Externalfield;linearresponse................................ 8 2.3 Continuous systems: functional integral . ....................... 11 2.4Examples:elasticsystems.................................. 17 3 Calculation of Functional Integrals 29 3.1Perturbationtheory..................................... 29 3.2Variationalmethod...................................... 31 3.3Saddlepointmethod..................................... 32 4 Quantum problems; functional integral 35 4.1Singleparticle......................................... 35 4.2 Correlation functions; Matsubara frequencies ....................... 39 4.3 Many degrees of freedom: example of the elastic system . ............... 42 4.4Linkwithclassicalproblems................................. 44 5 Perturbation; Linear response 47 5.1Method............................................ 47 5.2Analyticalcontinuation................................... 50 5.3Fluctuationdissipationtheorem............................... 52 5.4Kuboformula......................................... 53 5.5 Scattering, Memory function . ............................... 57 6 Examples 59 6.1 Compressibility . ....................................... 59 6.2 Conductivity of quantum crystals; Hall effect ....................... 61 6.3Commensuratesystems................................... 66 7 Out of equilibrium classical systems 69 7.1Overdampeddynamics.................................... 69 7.2Linkwiththermodynamics................................. 70 7.3MSRFormalism....................................... 72 3 Chapter 1 Introduction The material discussed in these notes can also be found in more specialized books. Among them • Statistical mechanics and functional integrals [1, 2, 3, 4] • Many body physics [5, 3] These notes are still in a larval stage. This is specially true for chapters 3 and 7. Despite the at- tempts to unify the notations and eliminates mistakes, factors of 2 and other errors of signs, it is clear that many of those remain. I’ll try to update the notes from time to time. If you find these notes useful and feel like helping to improve them, do not hesitate to email to me ([email protected]) if you have found errors in those notes or simply to let me know if there are parts that you would like to see improved or expanded. 5 Chapter 2 Equilibrium statistical mechanics 2.1 Basics; correlation functions In statistical mechanics one is interested in getting basic quantities such as the partition function. Space is meshed with a network on which some variable describe the state of the system. To be more specific let us consider an Ising model, on a square lattice. On each site i exists a spin σi = ±. The state of the system is describe by all spin variables σ1,σ2,...σN , where N grows as the volume of the system. The set of all these variables is a configuration that we denote in the following as {σ}.To compute physical quantities one needs in addition an Hamiltonian describing the energy of the system for a given configuration. To be specific let us again consider a Ising type Hamiltonian H0 = − Ji,jσiσj (2.1) i,j A physical quantity such as the partition function results from the sum over all configurations of the system Z = e−βH0 = ··· e−βH0[{σ}] (2.2) {σ} σ1=± σ1=± σN =± The difficulty if of course to make the sum over the humongous (thermodynamics) number of variables. From Z or F = −T log(Z) many simple physical quantities can be directly extracted (specific heat, etc.). In addition to these rather global quantities, one is often interested in correlation functions, mea- suring thermodynamics averages of local observables or products of local observables. Some examples are σi1 (2.3) σi1σi2 (2.4) where the thermodynamics average stands for −βH O[σα,...,σβ]e 0 {σ} O[α,...,β] = −βH (2.5) {σ} e 0 Where O is any operator. Note that the correlation functions depends on the spatial positions α,...,β, the average having been made over the microscopic variables σ. 7 In fact the correlation functions can also be obtained from a partition function. If one adds to the Hamiltonian a source term − βHs = hiOi (2.6) i one introduces the partition function, that now depends on a thermodynamics number of external sources −β[H0+Hs] Z[h1,h2,...,hN ]= e (2.7) {σ} and the associated free energy F [h1,h2,...,hN ]. It is then easy to see that 1 ∂Z[h1,h2,...,hN ] [Oα ...Oβ] = (2.8) Z[h =0] ∂hα ...∂hβ {h}=0 Of course this is a rather formal relation since computing Z[h1,h2,...,hN ] is in general a formidable task. Similar formulas can be derived from the free energy. In particular one can obtain from the free energy ∂F[h1,h2,...,hN ] Oα = −β (2.9) ∂hα {h}=0 It is easy to see that differentiating the free energy leads to the so-called connected correlations ∂F[h1,h2,...,hN ] − β =[OαOβ−OαOβ] (2.10) ∂hα∂hβ {h}=0 Each time one differentiate an average one gets two terms one coming from the numerator one from the denominator. Thus ∂hγ ... =[...Oγ−...Oγ] (2.11) It is thus easy to obtain the higher derivatives. The ingredients that we have illustrated on this example of the Ising model are totally general. It is important to understand that what is generally needed is to be able to perform the sums over the microsocpic degrees of freedom. 2.2 External field; linear response Among the various correlation functions some are of special importance. Let us assume that a physical variable of the system can be coupled to an external field, in a way similar to (2.6). In our Ising example this is the case if one puts the system in a magnetic field. The natural variable associated with such a perturbation is the magnetization of the system. One would have two way to compute it. Adding the magnetic field to the Hamiltonian H0 leads to Hh = −h σi (2.12) i From H = H0 + Hh one can compute the free energy F [h]. Standard thermodynamics tells us that the total magnetization M of the system is simply given by dF [h] M = − (2.13) dh This relation comes immediately from (2.8) using O = − i σi. M describes the response of the system to the external field h. Even for this simple case (h is space independent) is is quite complicated to get the magnetization. To generalize this notion of response, let us consider an operator Oi (that can a priori depend on the position) that couples to an external field hi (that can also depend on space) trough the Hamiltonian Hp = − hiOi (2.14) i Let us furthermore assume that Oi is such that in the absence of perturbation Hp one has Oi0 =0, where 0 denotes averages performed with H0 alone. If it is not the case one can simply subtract the average value from Oi. In that case one can measure the response of the system, that is given by Oi[{h}]=Oi (2.15) The notation Oi[{h}] means that the value Oi at point i depends in fact on the value of the external field at all points, i.e. of the whole set of values of h at each spatial point. For even a simple H0 it is hopeless to expect to compute the full response Oi. A simpler case, but of considerable interest, is to compute the linear response of the system. Indeed if the external field is small one can expand (2.15), in powers of the external field. The lowest order term is linear in the external field (hence the term linear response). One has d Oi[{h}]=0+a χijhj (2.16) j which is the most general linear term that one can write. The factor ad is here for convenience (see below). At that stage this merely defines χ. If we assume in addition that H0 is invariant by translation, then χij can only depend on the distance between i and j (that we denote symbolically by i − j). A more transparent way to write the convolution (2.16) is to go to Fourier space. Let us recall the convention that we use for fourier transform on the lattice. d −iq·r hq = a e i hi (2.17) i 1 iq·r hi = e i hq (2.18) Ω q=2πn/L where we have introduced a lattice spacing a (a can be set to 1 at that stage since it is purely formal). The volume of the system is Ω = Nad and one introduces a distance r = ai. The sum over q runs from q = −π/a to q =+π/a, i.e. over N values of n. This corresponds to the standard Brillouin zone. In that case iq·ri e = Nδq,0 (2.19) i where δi,j is the discrete δ. For a continuous system ri → r, one has the continuous Fourier transform h(q)= ddre−iq·rh(r) (2.20) Ω 1 h(r)= eiq·rh(q) (2.21) Ω q=2πn/L where Ω = Ld is the volume of the system. The relation between the lattice variables and the continuous ones is thus hi → h(r) (2.22) hq → h(q) (2.23) For the continuous case iq·r dre =Ωδq,0 (2.24) still with a discrete δ. In the limit where the volume of the system goes to infinity, one can use 1 → 1 d d d q (2.25) Ω q (2π) d Ωδq,0 → (2π) δ(q) (2.26) where δ(q) is the Dirac δ. Using these definitions for the Fourier transform one can rewrite (2.16) as d 1 −iqri+iq rj Oq = a e χi−jhq (2.27) N ij q using ri = rj + rl one has d 1 i q−q r −iqr O ( ) j l q = a e e χrl hq (2.28) N jl q d −iqrl = a e χrl hq (2.29) l = χqhq (2.30) Thus in Fourier space the relation between the response and the external field is truly linear, each q mode being independent.