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space-time coordinate (0, 0) followed by particle creation at (τ, r) and (b) the creation of a particle at (0, 0) followed by its annihilation at (τ, r).65 The annihilation process initiating (a) may be interpreted alternatively as the creation of a hole, i.e. the composite process D describes the joint propagation of ...
= ωn, i.e. q eiq·xDq, whereD q = T Ld limR→0 R−1 p e−ip·xψa ¯ψa p2+qψa p2 ¯ψa p1 p1+q p1+q ψa p1p2 Dq = T Ld lim R→0 1 R ωn p1p2 ¯ψa p1,ωn ψa p1+q,ωn+m ¯ψa p2+q,ωn+m ψa p2,ωn ψ. (6.54) A3: Only diagrams that contain one free summation over the Fermi surface per impurity line contribute to leading order in (pF)−1. (Form...
C is deﬁned as the solution of the equation (\|ωm\| + D(−i∇r + 2A)2)Γ(r, ωm) = (2πντ Ld)−1δd(r) (a formal solution of which is given by the right-hand side of the deﬁnition above). Crucially, the presence of the vector potential spoils the singularity of the cooperon mode in the limit q, ωm → 0. In other words, the magne...
nate representation) wherein time-reversal amounts to transposition. 75 Note that, under time-reversal, the sign of the time-derivative is reversed: dτ ( ¯ψ∂τ ψ)T = − dτ (∂τ ψT ) ¯ψT = − dτ ψT (−∂τ ) ¯ψT . 76 Notice that the temporal minus sign in the ﬁelds ψT and ¯ψT is introduced to remove the unwanted sign multiplyi...
d conductors at low temperatures. Our target is a low-energy action of the elements Q = T ΛT −1 ∈ Sp(4RM )/(Sp(2RM ) × Sp(2RM )) of the Goldstone mode manifold. The action of this nonlinear σ-model can be constructed by substituting the slowly ﬂuctuating matrix conﬁguration Q = {Qα,α nn (r)} into the action (6.59) and ...
essing a spatially homogeneous mean-ﬁeld conﬁguration may host other solutions of physical signiﬁcance. Indeed, we have encountered a scenario of this type already in Chapter 3 above. Exploring quantum double-well tunneling, we observed that the imaginary time Euler–Lagrange equations of a particle in a doubly degenera...
than Tc. To this end, add to and subtract from the right-hand side of . Then expand to leading order in the small the gap equation the integral parameters δT /Tc and Δ/Tc, whereδT = Tc − T > 0. dx tanh x x Answer: (a) For Δ = 0, λ(ξ) = (ξ2 + Δ2)1/2 = \|ξ\| and Eq. (6.28) assumes the form 1 gν = 0 ωD/2Tc dx tanh x x , wh...
dependent free energy by Z [ ˜V0] = exp(−βF [ ˜V0]), noting that ˜V0 shifts the chemical potential, F [ ˜V0](μ) =F (μ − i ˜V0), and neglecting the ˜V0-dependence of G[ ˜V0] G, we obtain the saddle-point equation 0 = ∂ ∂ ˜V0 1 4EC ˜V 2 0 − iN0 ˜V0 − F (μ − i ˜V0) = 1 2EC ˜V0 − iN0 + i ˆN μ−i ˜V0 , F (μ − i ˜V0) =− i∂μF ...
g to the superconducting case, show that, at an intermediate stage, the action is given by Seﬀ [V, φ] = Sc[V ] − tr ln ∂τ + ( ˆξ1 + i( ˙φ1 + V )/2)σ3 + Δσ1 T †ei(φ1−φ2)σ3/2 e−i(φ1−φ2)σ3/2T ∂τ + ( ˆξ2 + i( ˙φ2 − V )/2)σ3 + Δσ1 , where ˆξi, i = 1, 2, comprise the single-particle energies of the system, and Sc[V ] = dτ V ...
Leggett 1938– Co-recipient of the 2003 Nobel Prize in Physics (with Alexei A. Abrikosov and Vitaly L. Ginzburg) “for pioneering contributions to the theory of superconductors and superﬂuids.” He has made important contributions to the theory of normal and superﬂuid helium liquids and other strongly coupled superﬂuids,...
ctron–hole condensate. As a result experimentalists have used “Feshbach resonance” phenomena to tune the atomic pair interaction from weak to strong coupling, whence the atoms exist as tightly bound pairs. By monitoring the dynamics of BEC formation, attempts have been made to infer the properties of the ephemeral BCS-...
ternatively, if the transition to the magnetic phase is of second order (i.e. the expectation value of the magnetization ﬁeld grows continously from zero as the interaction is increased through Uc), a ﬁeld theory of the system near the critical point can be developed as a perturbative expansion 6.7 Problems 353 of the ...
s point, see J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford University Press, 1993). 6.7 Problems 357 Notice that both the lack of gauge invariance of the measure and the UV problems manifest themselves in an integral over (quantum) ﬂuctuations, i.e. while the symmetry is preserved on the classical...
th the other two categories of experiment the situation is diﬀerent. Transport and spectroscopic measurements can be used to probe both static and dynamical features of a system; further, fully angle/frequency-resolved spectroscopic data contain detailed information on the spatio-temporal structure of the dominant exci...
sion (Ω(K), K)) of the states inside the solid. This is where the detective work of spectroscopy begins. What we know is that the dispersions of the scattered particles and conof the (k, ω(k)) stituents, and (K, Ω(k)), respecrelated are tively, sample Sir Chandrasekhara V. Raman 1888– 1970 (left), Lord (John Rayleigh S...
ction, we argued that condensed matter experiments typically probe the . Such linear (linear) response of a system to the application of weak perturbations response can be cast in terms of a generalized susceptibility χ: Eq. (7.2). In the following we try to give the formal expression (7.2) a concrete meaning. Speciﬁca...
ditional operator representation, i.e. expressions with circumﬂexes, ˆX, represent canonically quantized operators and · · · = Z −1tr(· · ·exp{− β[ ˆH − μ ˆN ]}) represents the quantumthermal expectation value. Restricting ourselves to correlation functions of operators taken at two diﬀerent times,11 the general deﬁnit...
epresentations above apply to C T,+,−(z = ω) whereω is restricted to the real axis.) This extended interpretation allows us to view C T,+,− as complex functions with singularities in the immediate vicinity of the real axis. More speciﬁcally: The retarded correlation function C + has singularities for z = −Ξαβ − iη slig...
the eigenvalues of this operator – which are still functions of z – are given by the correlation function Ga(z) above. Numerous physical observables can be compactly represented in terms of the operator Green function. For example, using Eq. (7.19), it is straightforward to verify that the single-particle density of s...
ion of A as a probability measure describing in what way the spectral weight carried by the state c† a\|α is spread out over the 18 Strictly speaking, we can integrate A only if the variance of Σ(ω) over the interval [ξa + Σ − Σ, ξa + Σ + Σ] in which the Lorentzian is peaked is negligible (but see below). 7.3 Analytic s...
n, Many Particle Physics (Plenum Press, 1981). 21 We assume that we are dealing with a Fermi liquid, i.e. that we can think of the constituents of the system as fermionic quasi-particles, in the spirit of Landau’s theory. 7.3 Analytic structure of correlation functions 387 a linear approximation, the induced charge den...
dx ∂μ this condition can only be generally valid if → ∂μ Kμν = 0. Summarizing, Gauge invariance and particle number conservation demand the identity → ∂μ Kμν = Kμν ← ∂ν= 0. We next turn to the derivation of the linear response kernel. Our starting point is an observation made already in Chapter 1 (and several times th...
the Info block above shows that any meaningful analysis of the conductivity must take account of the presence of static disorder; without disorder, the ﬁeld would make the electrons freely accelerate, i.e. there would be no such thing as a steady current ﬂow in metals. Thus, what we should have in mind when we think ab...
es the (ﬁctitious) propagation of an electron backwards in time. However, an electron propagating in a chronologically reversed direction can be interpreted as a hole moving forward in time. Thus, the advanced electronic Green function eﬀectively represents a descriptor of hole dynamics. That the product of two Green f...
in turn implies that the overlap \|Ψ\|Ψ\|2 vanishes as some negative power of N . sin2 δm (n−m+ δm n<nF,m>nF π )2 7.6.2 RPA dielectric function Much of the response of a system of charged fermions to an external electromagnetic perturbation is encoded in the dielectric function q. While the dielectric function cannot be c...
B ˆN . (7.61) (d) In Chapter 9, we apply the linear response formulae derived above to a ﬁeld-theoretical analysis of the quantum Hall eﬀect, the physics of a two-dimensional electron gas subject to a strong perpendicular ﬁeld. Our starting point will be the replicated partition sum Z = D(ψ, ¯ψ) exp(−S[ψ, ¯ψ]), where S...
entary particle theory and condensed matter physics (critical phenomena and the Kondo problem) to quantum chemistry and computer science. (Image c The Nobel Foundation.) The general line of reasoning above summarizes much of the thinking behind the renormalization group. Of course, the approach would be quite useless h...
nality? Consider, for example, a two-dimensional variant of the model. Here the formation of a large connected region of M mismatched spins incurs an energy cost U ∼ M 1/2J. To understand why, one may note that the energy cost is proportional to the length of the one-dimensional boundary that encloses a (circular) doma...
f the conceptual foundations of the RG on a model application. According to the general scheme outlined at the beginning of the chapter, our aim is to devise an algorithm to recursively trace out parts of the short-scale ﬂuctuations of the system and assess their inﬂuence on the remaining degrees of freedom. 418 The re...
. Indeed, we expect on general grounds that (see the discussion of the previous section) the reduced free energy should scale with the inverse of the correlation length ξ, which in turn diverges upon approaching the zero-temperature ﬁxed point. Comparing with Eq. (8.10), and noting that there are no reasons for the res...
carries the dimension [dτ ] = [frequency]−1 and, therefore, changes by a factor b. We are now in a position to compare the eﬀective actions S[θ] andS [θ] before and after the integration over the fast modes. Obviously, the principal eﬀect of the integration over fast modes is that the coupling constant of the periodic ...
t that the integration area is proportional to b, and we have used the fact that, for those narrow time windows, the ﬁeld integration will be oblivious to the diﬀerence between θ(τ ) andθ (τ ). The second proportionality is obtained by averaging the integrand along the lines of our previous calculations. After the resc...
ese operators in the action from the very beginning (with an a priori undetermined coupling constant). One then veriﬁes whether the augmented action represents a complete system, i.e. one that does not lead to the generation of operators beyond those that are already present. If necessary, one has to repeat this step u...
epts of RG theory (such as the scaling laws to be discussed below). 17 A marginal scaling ﬁeld corresponds to a direction in coupling constant space with vanishing partial derivative, ∂φα R\|g∗=0 = 0. In this case, to obtain a reﬁned picture, one sometimes considers the second-order derivative, ∂2 α. For x > 0 φα (x <0)...
th diverges as ξ ∼ \|t\|−ν . η: This implies that the correlation function, * C(r) ∼ 1 \|r\|d−2+η , exp[−\|r\|/ξ], \|r\| ξ, \|r\| ξ, crosses over from exponential to a power law scaling behaviour at the length scale ξ. To motivate the power, one may notice that C ∼ φφ carries twice the dimension of the ﬁeld 20 Unfortunately, the...
gα>1 are irrelevant (or, for that matter, marginal). We can then write C(pi, g1, gα) = bndφ C(pib, g1bλ1 , gαbλα ) = g −ndφ/λ1 1 g11≈ g −1/λ1 1 C(pig , 1, 0) ≡ g −ndφ/λ1 1 C(pig −1/λ1 1 , 1, gαg −λα/λ1 1 ) −ndφ/λ1 1 F (pig −1/λ1 1 ). Here, we have used the freedom of arbitrarily choosing the parameter b to set g1bλ1 = ...
e it is straightforward to attribute engineering dimensions to all other operators: φ2 = L2, φ4 = L−d+4, φn = Ld+(2−d)n/2, (∇mφ)2 = L2(1−m). These relations convey much about the potential signiﬁcance of all structurally allowed operators: The engineering dimension of the non-gradient operator ∼ φ2 is positive in all d...
that we are expanding not around the “true” mean-ﬁeld, i.e. the exact solution of (8.24), but rather around the solution ¯φ = 0 of the ﬁeld-free system. However, in view of the fact that h has the status of an external perturbation, this choice of the reference conﬁguration is quite natural. 8.4 RG analysis of the ferr...
action31 is multiplied by a large parameter (which, in the case of a quantum theory, might be −1). The expansion in the number of loops is then equivalent to an expansion in the inverse of that parameter (for a quantum theory, an expansion away from the classical limit). 30 L. H. Ryder, Quantum Field Theory, (Cambridg...
sion. Of course, a more qualiﬁed approach to the question is to explore what happens at higher order in the -expansion. Needless to say, the price to be paid for this ambition is that, at orders O(n>1), the analysis indeed becomes laborious. Nonetheless, the success of the ﬁrst-order expansion prompted researchers to d...
= J(π) encapsulates both the geometry of the Haar measure and the Jacobian associated with the transformation g → a πaTa. Referring for a more detailed discussion of the function J to text books on group theory, we here merely note that its Taylor expansion in π starts as J(π) = 1 + O(π4) where theO (π4) term will not ...
rehensive discussion of the role of geometry and topology in quantum ﬁeld theory will be developed in the next chapter.) Consider then a two-dimensional square lattice with a phase-like variable exp(iθi) ∈ S1 deﬁned on each of its sites i. Demanding that the Hamiltonian or action of the system be periodic in all θi and...
ormation at low temperatures and protects the integrity of the quasi-long-range ordered phase. To explore the range over which vortices are suppressed, one may explore the partition function for a conﬁguration with just a single vortex of unit 41 Notice that, if the spin degrees of freedom have three components or more...
ression into Eq. (8.45), and changing integration variables, one ﬁnds that the term linear in x integrates to zero while the angular average of (x · ∇XC)2 leads to x2(∇XC)2/2. Thus, to O(r4), one obtains e−Seff (r−r) e−4π2JC(r−r) × 1 + y2 0 (dx 2π x)e−4π2JC(x)8πJ 2 x2 2 d2X (∇X(C(r − X) − C(r − X)))2 . d2X [∇X(C(r−X)−C...
cussion was substantial enough to convince the reader of the enormous power and versatility RG methods have in disclosing “deep” physical information and to motivate further exploration of the subject. INFO While, for the most part, previous chapters were guided by symmetry principles, little has been said so far about...
critical theory. The aim of the present problem is to explore the nature of the critical phenomena when the critical point is driven to zero temperature – a quantum critical point. Motivated by the coherent state formulation of the quantum partition function of an interacting system, it is tempting to associate direct...
δ(b) = 1) while v(b) remains small, the perturbative expansion remains valid. However, (ii) if v(b) grows to order unity while δ(b) remains small, the scaling behavior moves into a region potentially controlled by non-Gaussian ﬂuctuations. From the equations above, the condition for Gaussian behavior, i.e. δ(b) = 1 and...
ariant (up to a phase) and can be omitted. In Section 3.3 we used this representation to identify the phase space of spin as the coset space SU(2)/U(1), the 2 sphere. In the Q language, the same sphere is represented as Q = gσ3g−1, where ˆSi = σi/2 are identiﬁed with the Pauli matrices. Again, the ψ factor, commutative...
es of the conductance, the scaling function approaches the asymptote β(g) → d − 2 corresponding to ohmic behavior. For very small values of conductance, the scaling function approaches β(g) → ln g, which is characteristic of insulating behavior. According to the weak localization expansion, a localization transition is...
lems 493 is associated with eight possible contributions, which can be grouped in pairs. (You may ﬁnd it helpful to enumerate these possibilities diagrammatically.) In particular, show that there exists a contribution J+J− k skskf ˆS−c † k s ↑ckf ↓ 1 E − ˆH22 † ˆS+c kf ↓cks↑\|ψ1, where the wavevectors kf (ks) index stat...
in a single chapter.2 Consequently, our discussion is example–oriented and often regrettably superﬁcial (with regard to both physical depth and, especially, mathematical structures). In fact, the aim of the present text is to demystify the subject of topology in ﬁeld theory, to arouse the interest of readers and to mo...
φ) removes A from the Hamiltonian while changing the boundary conditions to ψ(0) = e2πiAψ(2π). In the gauge-transformed picture, the presence of the magnetic ﬁeld thus amounts to a twist in the boundary conditions of the wavefunctions, and the persistent current is a measure of the sensitivity of the spectrum to this t...
ce yourself of the veracity of this statement) π1(T d . Turning to higher dimensions, it becomes more and more diﬃcult to identify homotopy classes simply by invoking one’s imagination. One of the last intuitively accessible examples is π2(S2) = Z: maps of the 2-sphere into itself can be classiﬁed according to how ofte...
. theories where the base manifold represents space-time) are called “instantons.” By contrast, “solitons” are topological solutions of classical equations of motions. However, this rule is also sometimes broken. The ubiquitous presence of the suﬃx “-on” reﬂects a widespread tendency in physics to associate excitations...
agnetic spin chains. Surprisingly, though, this expectation does not conform with experimental observation. Neutron scattering experiments on one-dimensional spin 1/2 antiferromagnets have indeed shown that, in the vicinity of the N´eel ordering wavevector q = π/a, the dispersion is linear. However, spin S = 1 chains s...
topological term modify this behavior? To get some idea of what might happen, let us reformulate the partition function as a sum over disjoint topological sectors, Z = DnW e2πiSW e−S0[nW ], (9.16) W ∈Z where nW denotes ﬁeld conﬁgurations of winding number W . Equation (9.16) provides a preliminary explanation for the ...
same time, the longitudinal resistivity/conductivity drops by as much as 13 orders of magnitude. In passing we note that the rapid oscillations visible in the ﬁgure at small ﬁeld strengths represent the familiar Shubnikov–de Haas oscillations. 21 One may recall that the ﬂux quantum is deﬁned through the relation Φ0 = ...
y a gauge transformation that does not alter the boundary conditions.) That, however, does not necessarily imply that individual basis states map onto themselves upon the completion of the path φ = 0 → φ = 2π. I.e., while the set of eigenstates as a whole gets reproduced, permutations of individual states are consisten...
any other characteristics. 27 J. E. Avron and R. Seiler, Quantization of the Hall conductance for general, multiparticle Schr¨odinger Hamil- tonians, Phys. Rev. Lett. 54 (1985), 259–62. 28 D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two- dimensional periodic potential...
discussed in Section 9.4, this is not accidental but rather reﬂects a fundamental property of topological terms. For future reference, we also anticipate that the boundary descendant of the bulk θ-term Eq. (9.23) is an example of a Wess–Zumino term. 3 T −1 → T hσar We next show that, under the conditions stated above,...
f an Onsager relation. Consistency with these relations requires that the topological series be of the form σ11 = σ0 11 + δσ11 + ∞ W =1 cos(2πW σ0 12) a(W )(σ0 11), σ12 = σ0 12 + ∞ W =1 sin(2πW σ0 12) b(W )(σ0 11), (9.27) 534 Topology where a(W ) and b(W ) are expansion coeﬃcients. The notation emphasizes that, in the ...
φ (x) T U′ z z′ z–1 z (φ (x)) V z′ z′(φ (x)) V′ Figure 9.10 On the construction of coordinate representations of ﬁeld manifolds. For a discussion, see the main text. then explain the general ideas behind WZ ﬁeld theory before we turn to the discussion of a number of interesting applications. 9.4.1 A crash-course in diﬀ...
vince yourself that, for manifolds T ⊂ Rn, this reduces to the standard deﬁnition of the diﬀerential with which you are familiar.) However, not every 1-form is a diﬀerential of a function. Consider, for example, the tangent basis of T = S1 constructed in the exercise on page 539. Let us deﬁne a 1-form by setting ω(e) =...
with an atlas of identically oriented charts is said to be orientable. (A prominent counterexample is presented by the M¨obius strip.) The deﬁnition of the integral above implies that we have chosen a deﬁnite orientation. For future reference, we remark that, if φ : T1 ⊃ U1 → U2 ⊂ T2 is a diﬀeomorphism 9.4 Wess–Zumino ...
ot familiar with Stokes’ theorem may be helpful. First note that ∂C is a manifold by itself, with dim ∂C = n−1. Thus, ω is top-dimensional on ∂C and can be integrated. To gain some insight into the substance of Eq. (9.39) let us consider a few special cases. For example, let M be a three-dimensional manifold, n = 3. Co...
quantized according to Eq. (9.43). Having discussed the structure and geometry of WZ theories, we now return to physics. Using a simple prototype system as an example, we begin by exploring how WZ terms enter low-energy theories of many-body quantum systems. 9.4.3 Example: magnetic moment coupled to fermions Consider a...
erits its time dependence from the parameters x(t). We may thus write t γ(t) = i t 0 dt 0(x(t))\|∂xi(t)\|0(x(t))∂txi(t) = i c dx 0(x)\|∂x\|0(x) = i 0\|d0. c Here, the second integral has to be interpreted as a line integral in parameter space. It is taken along a curve c which starts at x(0), follows the evolution of the pa...
ystem. In fact, we shall begin by considering the totally non-interacting case.64 Switching from a lattice to a continuum description and linearizing around the two Fermi points (for details, see Section 2.2), we describe this prototypical system in terms of the action S[ψ†, ψ] = s=±1 dx dτ ψ†r s (−isvF∂x + ∂τ ) ψr s =...
ze. This tells us that S[g] alone does not suﬃce to establish the boson–fermion correspondence. But let us now inspect the second term in the action proposed by Witten, the two-dimensional WZ functional. The ﬁrst thing we have to understand is why the second term of the action indeed represents a WZ functional in the s...
its invariance under spin transformations, the bosonic representation of the umklapp operator reads ∼ λuk cos 4φ. ∂zφ and js = 1√ 8π 2π EXERCISE To obtain the bosonic representation of the umklapp operator, substitute Eq. (9.63) into its deﬁnition and obtain ∼ λukei4φtr(gσ2gT σ2) + h.c. ∼ λuk cos(4φ). Here the last equ...
system is described by the O(3)-model with topological angle 2πS = π. On the other hand, we might have approached the problem via the WZW route discussed above. Within that context, the ferromagnetic perturbation turns out to be irrelevant (see Aﬄeck and Haldane62), i.e. the long-range physics of the system is describ...
f vanishing strength. This nicely conforms with the experimental observation of Fermi-liquid-like behavior (no QHE) close to half ﬁlling. We have seen in Section 9.3.4 that, when a ﬂux quantum is adiabatically pushed through an annular quantum Hall geometry, an electron charge ﬂows from the inner to the outer perimeter...
In the following we will investigate what information can be obtained from the functional integral Eq. (9.76) about the physical behavior of the FQH system. Particle exchange in two dimensions Before embarking on this program, it may be of interest to discuss a few general aspects of Chern–Simons ﬁeld theory and partic...
iτ ∈ −i[0, β] is that the initial and ﬁnal points of our two space-time trajectories now have to be identiﬁed (the usual temporally periodic boundary conditions of the imaginary time formulation101). A glance at the ﬁgure shows that this leads to a pair of two closed world line curves which, for a winding angle φ = 2πn...
(a, σ) → (¯a, 0) in the action. Using the fact that aext = 2sνAext, we arrive at the last line, where the coupling constant qeﬀ = 1 − 2sν = 1 1 + 2sp , (9.82) is identiﬁed as the eﬀective charge of the CF. The line of reasoning above tells us that CFs eﬀectively carry fractional charge. The partial “screening” of the b...
eates an accumulation of statistical ﬂux that in turn acts as a scattering center of CFs, etc. Thus, we readily wind up with a full-blown problem “interaction + disorder + strong magnetic ﬁeld” whose rigorous microscopic solution seems to be elusive. Nonetheless, all evidence suggests that eventually the CFs will be lo...
on around the ring), many modes contribute to the integral and a continuum approximation is valid. Thus, the dominant contribution to 590 Topology the correlation function comes from frequencies \|ω\| < Ec, for which the discreteness of the sum really matters.114 Carry out the sum over momenta by the methods otherwise em...
e WZW action reduces to the θ-term for the unit-modular ﬁeld n ∈ S2 : Γ[in · σ] = πStop[n], where Stop[n] = 1 4π S2 d2x n · (∂1n × ∂2n). Answer: (a) Using the fact that g−1dg g−1 = −dg−1 (why?) the form ω can be rewritten as ω = tr(dg−1 ∧ dg ∧ dg−1 g), i.e. dω = −tr(dg−1 ∧ dg ∧ dg−1 ∧ dg). To show that this expression ...
φ ≡ ∂iφdxi, where we use the symbol d to denote the real space contribution to the exterior derivative. Plug this ansatz into the residual contribution to the action (after a0 has been integrated out) to reduce the ﬁeld strength tensor to dτ ∂xφ∂τ φ. Recalling that the CS action enters the boundary action S[φ] = the th...
nonequilibrium statistical (ﬁeld) theory. In the following, we introduce a spectrum of concepts central to the description of many particle systems out of statistical equilibrium. We will see that key elements of the theory – Langevin theory and the formalism of the Fokker–Planck equation – can be developed from the co...
Eq. (10.3), with ﬁrst and second cumulant μ1 = N ˜μ1 and μ2 = N ˜μ2, respectively. (All higher cumulants of the Gaussian distribution vanish. Exercise: verify this statement by showing that the generating function of the Gaussian distribution is again a Gaussian.) The ubiquity of additive random variables in nature ex...
2) = f (x 2). Assuming that f (x) =f (H(x)) ≡ f () is a function of energy, we conclude that probability conservation is compatible with energy conservation, 1 + 2 = 1 + 2, if ln f () = a + b is linear in energy. In this case, f (x1)f (x2) = exp(a(1 + 2) + 2b) = exp(a( 2) is indeed satisﬁed. To ﬁx the constants a and b...
dissipation or friction are intimately linked to the presence of ﬂuctuating forces. However, unlike the everyday phenomenon of friction, the corresponding ﬂuctuation forces are usually less noticeable. The reason is that friction acts in a directed way (cf. the action of a brake), while the response caused by ﬂuctuati...
bability to ﬁnd v at t for initial data (v0, t0). Weighing by the conditional probability to move on to v and summing over all intermediate conﬁgurations, v, we obtain the total probability (v0, t0) → (v, t). Equation (10.20) is the deﬁning property of a Markovian process, i.e. a process which is fully determined by on...
y now think of them as amoebae or another not too complex animal) with a mechanism of self-propulsion. To this end, we imagine the particles endowed with an energy storage. The time-dependent energy level, (t), will (a) increase due to food intake at some rate, q, (b) decrease at rate −c(t) due to metabolic activity (n...
obabilities. Identifying (1, 2) ↔ \|i as a classical approximation to a (coherent) state in two particle Hilbert space,16 and (1, 2) ↔ \|f as a ﬁnal state, we have the identiﬁcation, w(1, 2; 1, 2) ↔ \|Sﬁ\|2, where Sﬁ is the scattering matrix. Unitarity means that \|Sﬁ\|2 = i \|Sif \|2 = 1, or i d1d2 w(1, 2; 1, 2) = d1d2 w(1, 2...
reservoirs kept at diﬀerent temperature), we may then expect a scenario wherein the collision integral aims to establish local thermal equilibrium, while the left hand side will try to compromise between the “cost” of spatial variations and the need to adjust to the conditions imposed by the external perturbation. The ...
state (an+1, tn+1) depends on the current state (an, tn), but not on the history that got us there, (an−1, tn−1; . . .). Put diﬀerently, a Markov process lacks memory. Markov processes owe their popularity to the fact that they optimally compromise between descriptive power and analytical tractability; if possible, on...
2q)t and variance ∼ √ t. Random processes whose distributions are Gaussian are generally called Gaussian processes. The example illustrates that Gaussian processes are typically the results of the addition of a large number of elementary random variables. Example: Poisson process n Now consider a sequence of elementar...
he distribution, the Fokker–Planck prediction is by many orders of magnitude oﬀ the true result. 0 -10 -20 -30 -40 10 20 30 n 40 The origins of the this error can be traced to the second order expansion of the operator ˆE1 = e−∂x/N . Doing so, and replacing this operator by an eﬀective long-range approximation wherein ...
ni−1} is shorthand for the lattice time derivative. INFO The discretization leading to a unit functional determinant goes under the name Ito discretization. More general discretization schemes call for a treatment of the corresponding functional determinants. It is customary to represent these determinants in terms of...
. By generalization of the ideas exempliﬁed in Section 10.4.4 on the Poisson process, we can derive a path integral representation for the master equations describing Markovian stochastic processes. Unsurprisingly, this path integral turns out to be a close ally of the MSRJD functional integral above. For simplicity, ...
Hamiltonian operator. The price to be paid for this simplicity is that the integration variables ψ and ¯ψ do not have a direct physical interpretation. The network of diﬀerent theories constructed in the previous sections is summarized in Fig. 10.9. So far, we have been focusing on the case of low-dimensional problems...
function of the ﬁeld and its spatial derivatives. Representing the δ-constraint in terms of a Fourier integral over a “momentum ﬁeld” ψ and integrating over the noise ﬁeld we arrive at O[φ]ξ = D(φ, ψ) O[φ] ei dx ψ(x)(∂tφ(x)+D(−Δ)n δH[φ] δφ(x) )− dxdx ψ(x)K(x−x)ψ(x), 35 Typically, the dominant contribution (in the sense...
hat χ(ω) = C+(ω) = −iΘ(t)[ ˆX(t), ˆX(0)], where C + was the retarded response function. We then established a connection between C + and another correlation function, the time ordered correlation function, C T (t) ≡ −iTt ˆX(t) ˆX(0), 36 Chapter 7 was formulated for quantum theories, but that aspect will not be of relev...
xt allow the particles to hop between nearest neighbour lattice sites (subject to the condition of no more than single occupancy). This will render the system dynamical and allow it to settle in a conﬁguration compromising between single particle dynamics and interactions. To see this more clearly, imagine the dynamics...
of Eq. (10.93) will generate the bare version of a generalized Langevin equation. However, we must keep in mind that the strong anisotropy introduced by the ﬁeld will, by way of renormalization, render the eﬀective parameters of the Langevin anisotropic as well. This motivates the more general anisotropic representati...
the dimension below which nonlinear terms become qualitatively important. To this end, let us use the freedom to independently scale q, ˆq, t, φ, ψ in a manner to leave the dominant quadratic contributions to the action invariant. Deﬁning the scale transformation ˆq → b ˆq, φ → bdφ φ, q → bσq, t → b−zt, ψ → bdψ ψ, (10...
1; in inﬁnite dimensions, the unlimited number of options to go 43 H. Hinrichsen, Nonequilibrium critical phenomena and phase-transitions into absorbing states, Adv. Phys. 49, 815-958 (2000). 678 Nonequilibrium (classical) Figure 10.13 The two phases of a directed percolation network. Left, dry phase, wherein individua...
hree distinct phase portraits of directed percolation. Left: phase supporting a ﬁnite concentration of active sites, center: phase transition, right: empty state. and κ ≡ (Aλ)1/2.(In deriving Eq. (10.112), we made use of the freedom to rescale ﬁelds so as to make the coeﬃcients of the two nonlinear terms ∼ φψ2 and ∼ φ2...

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