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ppreciably) we may write x(t) = a cos[ω(t)t θ], − as in (2.20). The energy of the oscillation is E = 1 2 m( ˙x2 + ω2x2) 1 2 mω2a2, ≡ essentially, and this now changes with time at the slow rate ˙E = m( ˙x¨x + ω2x ˙x + ω ˙ωx2) = mω ˙ωx2. By averaging over a period of the oscillation, so that e.g. E (above) we obtain E =...
re exact and involve no approximations for small-amplitude motions. Examples have been given of simple/important Hamiltonian systems which have regimes which exhibit chaos and a resulting breakdown of predictability. The action variables are eﬀectively constant when system control parameters are changed very slowly. Th...
lowly. Assuming that the kinetic energy of the molecules gives a measure of temperature and that (volume) the gas is ‘ideal’, so satisfying the ideal gas law: (pressure) (volume)5/3 for (temperature), obtain the relationship (pressure) × ∝ adiabatic change of a monatomic gas. ∝ ≥ 15. A particle of mass m moves smoothly...
a b) ∧ ∧ c = (a (b a ∧ ∧ c) = (a c)b (b · − c)a. c)b (a · − b)c. · · (A.15) (A.16) Note that these vectors are unequal, so that we cannot omit the brackets in a vector triple product. It is useful to note that in both of these formulae the term with the positive sign is the middle vector times the scalar product of the...
Stokes’ theorem. However, it is easy to verify for a small rectangular surface. (The proof proceeds by splitting up the surface into small sub-regions.) Suppose S is a rectangle in the xy-plane, of area dx dy. Then dS = k dx dy, so the surface integral is k (∇ A) dx dy = · ∧ ∂Ay ∂x − ∂Ax ∂y dx dy. (A.34) The line inte...
· whence ii + jj + kk = 1, (A.63) as may easily be veriﬁed by writing out the components. Similarly, we may write the relation (9.16) between angular momentum and angular velocity in the form J = m(r2ω rr · − ω) = I ω, · where the inertia tensor I is given explicitly by I = m(r21 rr). − × Vectors 403 Note the diﬀerence...
t may be useful to gather together the relevant mathematical information. B.1 Cartesian Form The most general conic would have an equation of the form Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0, where A, B, . . . , F are real constants, but by choosing the axes appropriately we can reduce this to a simpler form. First, we lo...
way that (C.6) § . × 2 matrix S = [ξ01 ... ξ02] and the In this similarity transformation the 2 1M S. It should diagonal matrix in (C.6) above then takes the form S − be noted here that λi, ξ0i, ci (i = 1, 2) could be complex, but even then (C.5) is the formal expression of the solution for ξ. In the case when M is sym...
f the ﬁxed point X, we may write xn + n = X, for each n, and, when n is small, we can expand F (xn) in (D.1) in the form F (xn) = F (X n) = F (X) − − nF (X) + 1 2 2 nF (X) + . . . , (D.3) 425 426 Classical Mechanics where F (X) = [dF (x)/dx]x=X , etc. A ﬁxed point X is asymptotically stable (and therefore an attractor)...
tribution attractors (via an equation for the distribution function — Perron–Frobenius), but no slick result like that above for r = 4. The distribution attractors are also characterized by an exponential divergence of iterates, leading to the sensitivity to initial conditions characteristic of chaos. If we choose to e...
e’ among the real numbers (see Problem 12) and these α correspond to resonances in the system. As in the 14.6, the sensitivity to discussion of the problem of small denominators in initial conditions is strongest for the rational α = k/s with small values of s. As the perturbation grows with , the breakdown associated ...
2a/v) 11. (a) no turning points, (b) 1 turning point, (c) 1 or 2 turning points. (2/ω) arctan(ωa/v). − 445 c/ma, (b) v < 2c/ma. g/γ. c/ma or − g/l t). − l/g; θ = θ0 cos( 1. k/m; 16 mm. 446 Classical Mechanics k/g u), (1/2k) ln(1 + ku2/g). √e2kh \| − − − < − ± 1). v \| e− gt/γ; ˙z 12. x = 2ma3/c; (a) 2(g/l)(cos θ → − 1/2 ...
= ω2 0, ω2 2 = ω2 0 + 2ω2 s , 0.0399 sin 63.0t) mm, (t in s). 10. (0.401 sin 6.27t 11. φ = (q/4π0)[14/a + (4x2 + y2)/2a3]; ω2 12. ω2 = g/l, g/l, 3g/l. 13. A1,2 = (F/√2m)(ω2 − ω2 + 2iγ1,2ω), with ω2 1,2 − γ1 = α/2m, γ2 = (α + 2β)/2m; α > √3k/ω0. 14. q2r = 0, q2r+1 = ( − 1)r4√2la/π2(2r + 1)2. CHAPTER 12 1. ω2 = g cos α/r...
4. Ar = r− 2 sin θ). (0, 0, 2r− 0 cos θ 1 0 sin θ 0 sin θ 0 cos θ ⎦ ρ(r)(3rr − 17. Q = ⎣ . ⎤ 15. ⎡ r 21)d3r. − APPENDIX B 1. Distances r1,2 are given by r2 1,2 = (a cos ψ 2. Tangent vector is t = ( ∓ ae)2 + (b sin ψ)2 = a2(1 e cos ψ)2. a sin ψ, b cos ψ), unit vectors from two foci are e, (b/a) sin ψ), scalar products a...
on, 334, 426 beam, colliding, 169 Bernoulli, 70 Bertrand’s theorem, 351 bicycle wheel, 124 bifurcation, 309, 428 billiard balls, 41 billiard systems, 363–366, 378 circular, 364 elliptical, 365, 378 oval, 366, 441 body cone, 224 Bohr orbit, 84, 121, 126 boost, 295, 298–300 bounce map, 364, 366 bouncing ball, 345 boundar...
nomic system, 232, 235 homoclinic intersection, 440 Hopf bifurcation, 326–328, 343 Huygens, 377 hydrogen atom, xv, 83–84, 126 hyperbola, 79, 82, 88, 410 in polar form, 413 hyperbolic point, 418, 440 ignorable co-ordinate, 282–285 impact parameter, 79, 89 improper node, 418 improper transformations, 300 impulse, 37 velo...
5 relaxation time, 28 renormalization, 429 repeller, 310 resonance, 32–34, 148, 187, 373 half-width of, 33 Richardson, 324 right-hand rule, 105, 385 rigid body, 197–225 and Euler angles, 221 angular momentum of, 197, 199, 203–208, 218 angular velocity of, 216–218 energy of, 198 free motion of, 223–225, 228 generalized ...
-order contributions to the Taylor expansion are immaterial in the long-range continuum limit. 6 From particles to ﬁelds T φ M S S[φ] Figure 1.3 Schematic visualization of a ﬁeld: a mapping φ from a base manifold M into a target space T (in this case, T are the real numbers; but, in general, T can be more complicated)....
the continuum generalization of the lattice momentum PI of Eq. (1.2). (Applied to PI , a continuum approximation like φI → φ(x) would produce π(x).) The Hamiltonian density is then deﬁned as usual through the Legendre transformation, H(φ, ∂xφ, π) = π ˙φ − L(φ, ∂xφ, ˙φ) ˙φ= ˙φ(φ,π) , (1.10) 9 In ﬁeld theory literature ...
l scalar product translates to an integral, f , g = N n=1 fngn → f, g = dx f (x)g(x). The analog of the nth unit vector is a δ-distribution, en → δx, where δx(x) ≡ δ(x − x), as can be seen from the following correspondence: fn !=f , en = m fm(en)m → f (x) !=f, δx = dx f (x)δx(x). Here (en)m = δnm denotes the mth compon...
ons and (b) gauge transformations, and (c) it should be simple! The most elementary choice compatible with these conditions is S[A] = d4x (c1 FμνF μν + c2 Aμjμ) , (1.22) μ dxμ = dt dx1 dx2 dx3 denotes the measure, jμ = (ρ, −j) the 4-current, where d4x = and c1,2 are undetermined constants. Up to quadratic order in A, E...
nfuse the atomic constituents, also oscillators (albeit coupled), with the independent collective oscillator modes described by ˆH. The description above, albeit perfectly valid, still suﬀers from a deﬁciency: our analysis amounts to explicitly describing the eﬀective low-energy excitations of the system (the waves) in...
ay verify that the Lagrangian assumes the form % L = 1 2 d3x (∂tA)2 − (∇ × A)2 . (1.35) By analogy with our discussion of the atomic chain, we would now proceed to “decouple” the theory by expanding the action in terms of eigenfunctions of the Laplace operator. The diﬀerence with our previous discussion is that we are ...
menon produced results for the force that, in conﬂict with experimental ﬁndings, were strongly temperature dependent. It was considered a major breakthrough of the new quantum mechanics when London proposed a model whereby a temperature-independent r−6 law was obtained. The essence of London’s 21 H. B. G. Casimir and D...
only if S[φ] = S[φ]. As a second example, let us probe rotational symmetry: x = Rx, where R ∈ O(m) is a rotation of Euclidean space–time. In this case it would, in general, be unphysical to deﬁne φ(x) = φ(x). To illustrate this point, consider the example of a vector ﬁeld in two dimensions n = m = 2 (see the ﬁgure.) A ...
in the ﬁeld is given by H ≡ familiar expression for the EM energy density. (Hint: Use the vacuum form of Maxwell’s equations and the fact that, for an inﬁnite system, the energy is deﬁned only up to surface terms.) Answer: Following the canonical prescription, let us ﬁrst consider the Lagrangian density, L = − 1 4 FμνF...
2\|λ2 + x1\|λ2x2\|λ1) . In the Dirac bracket representation, the two-body states \|λ1, λ2F(B) corresponding to the wave functions ψF(B)(x1, x2) = (x1\| ⊗ x2\|) \|λ1, λ2F(B) above can be presented as \|λ1, λ2F(B) ≡ 1√ 2 (\|λ1 ⊗ \|λ2 + ζ\|λ2 ⊗ \|λ1) , where ζ = −1 for fermions while ζ = 1 for bosons. Note that the explicit symmetriz...
basis state, the vacuum is denoted by \|0. We will state, soon see why it is convenient to add this strange animal to our family of basis states. The space F is called Fock space and it deﬁnes the principal arena of quantum many-body theory. Vladimir Aleksandrovich Fock 1898–1974 One of the main participants in the hist...
enerate the Fock space in general and any N -particle Hilbert space in particular, it must be possible to represent any operator ˆO1 in an a-representation. n V (ˆxn), where V (x) is a scalar potential, the total spin operator n ˆp2 n Now, although the representation of n-body operators is, after all, quite straightfor...
Nobel Foundation.) Nearly free electron systems For certain materials, notably the elemental metals drawn from groups I–IV of the periodic table, the outermost itinerant conduction electrons behave as if they were “nearly free,” i.e. their dynamic is largely oblivious to both the Coulomb potential created by the positi...
ations of second quantization 55 unitary transformation between Bloch and Wannier states Eq. (2.22) induces an operator transformation a † kσ = 1√ N i eik·Ri a † iσ, a † iσ = 1√ N B.Z. k e−ik·Ri a † kσ. (2.24) We can now use the transformation formulae (2.23) and (2.24) to formulate a Wannier representation of the Hami...
aito, G. Dresselhaus, and M. S. Dresselhaus (Imperial College Press, 1998). 2.2 Applications of second quantization 59 Interaction eﬀects in the tight-binding system Although the pseudopotential of the nearly free electron system accommodates the eﬀects of Coulomb interaction between the conduction and valence band ele...
ing phase and an itinerant electron phase can be realized in two ways. In the ﬁrst case, one can reduce the interaction strength U/t while, in the second, one can introduce charge carriers (or vacancies) into the half-ﬁlled system. Experimentally, the characteristic strength of the interaction is usually tuned by chang...
ductors. Cuprates are built of layers of CuO2 separated by heavy rare earth ions such as lanthanum. Here, the copper ions adopt a square lattice conﬁguration separated by oxygen ions. At half-ﬁlling, electrons in the outermost occupied shell of the copper sites in the plane adopt a partially ﬁlled 3 d9 conﬁguration, wh...
only terms with (k, k, q) (±kF, ±kF, 0), but of Vee. Show that to the summation also terms with (k, k, q) (±kF, ∓kF, 2kF) contribute. When adequately ordered (do it!), these contributions can be arranged into the form of the right-hand side of Eq. (2.36). (For a detailed discussion see, e.g., T. Giamarchi, Quantum Phy...
Firstly, let us recall that the total number operator of a theory described by operators † μHμνbν, λ, bλ b the total number operator commutes with ˆH, i.e. [ ˆN , ˆH] = 0 (exercise). This means that ˆH and ˆN can be simultaneously diagonalized, or, in more physical terms, that the Hamiltonian enjoys the feature of par...
etween neighboring can describe these correlations through models of localized quantum spins – either in chains or, more generally, in higher-dimensional quantum spin lattices. We begin our discussion with the ferromagnetic spin chain. Werner Heisenberg 1901–76 in Nobel Laureate in Physics 1932 “for the creation of qua...
ly bipartite. As before, our strategy will be to expand the Hamiltonian in terms of bosonic operators. However, before doing so, it is convenient to apply a canonical transformation to the Hamiltonian in which the spins on one sublattice, say B, are rotated through 180◦ about the 24 It is straightforward to verify that...
a number, i.e. the bases are equivalent. 2.4 Problems 85 (b) For a given state \|n, (concentrating on a ﬁxed element of the single-particle basis, we suppress the subscript λi throughout), let us choose an integer q such that ˆnaq−1\|n = (n − q + 1)aq−1\|n with n − q + 1 > 0 while n − q ≤ 0. We then obtain 0 ≥ (n − q)n\|a†...
up spin as a particle and a down spin as the vacuum, i.e. \|↑ ≡ \|1 = f †\|0, \| ↓ ≡ \|0 = f \|1. In this representation the spin raising and lowering operators are expressed in the forms ˆS+ = f † and ˆS− = f , while ˆSz = f †f − 1/2. (a) With this deﬁnition, conﬁrm that the spins obey the algebra [ ˆS+, ˆS−] = 2 ˆSz. 2.4 ...
the subspace with m electrons on the impurity (i.e. ˆP0 = σ(1 − ndσ), etc.). 2 n=0 (a) Construct the operators ˆHmn explicitly and explain why ˆH20 = ˆH02 = 0. (b) Since we are interested in the eﬀect of virtual excitations from the \|ψ1 subspace, we may proceed by formally eliminating \|ψ0 and \|ψ2 from the Schr¨odinger ...
ir classical counterparts. In particular, by the Uncertainty Principle, energy conservation can be violated by an amount ΔE over a time ∼ /ΔE (here, and throughout this chapter, we will use for clarity). The connection 2 For this reason, path integration has turned out to be an indispensable tool in ﬁelds such as quant...
, q] = 0 dt L(q, ˙q) = 0 dt [p ˙q − H(p, q)]. t t Before we turn to the discussion of the path integral (3.6), it is useful to recast the integral in an alternative form which will be both convenient in applications and physically instructive. The search for an alternative formulation is motivated by the resemblance of...
is the mathematical identity underlying Wick’s theorem (for real bosonic ﬁelds), to be discussed in more physical terms below. 7 Note that the notation A−1 mn refers to the mn-element of the matrix A−1. 104 Feynman path integral (b) Complex case: The results above are straightforwardly extended to multi-dimensional co...
0, 1, the vector xμ = ∂μφ, and the diagonal matrix g = diag(−1, 1) is the two-dimensional version of a Minkowski metric. (In three spatial dimensions, g would take the form of the standard Minkowski metric of special relativity.) On Wick rotation of the time variable, the factor −1 in the metric changes sign to +1, and...
We thus conclude that P (qi, qf ) = \| det(∂pi/∂qf )\|(2π)−d. Finally, noticing that pi = −∂qi S we arrive at the result of the semiclassical analysis above. In deriving Eq. (3.28) we have restricted ourselves to the consideration of quadratic ﬂuctuations around the classical paths. Under what conditions is this semiclas...
ion of a particle in a double well potential (see ﬁgure). Our aim will be to estimate the quantum probability amplitude for a particle either to stay at the bottom of one of the local minima or to go from one minimum to the other. In doing so, it is understood that the energy range accessible to the particle (i.e. ΔE ∼...
dimensionful) constant absorbing the temporal dimension [time]n introduced by the time integrations, and An(τ1, . . . , τn) is the transition amplitude, evaluated within the semiclassical approximation around a conﬁguration of n instanton bounces at times 0 ≤ τn ≤ τn−1 ≤ · · · ≤ τ1 ≤ τ (see Fig. 3.5). In the following,...
to compute the exponential prefactor K. Although such a computation follows the general principles outlined above and implemented explicitly for the single well, there are some idiosyncrasies in the tunneling system that warrant discussion. According to the general principles outlined in Section 3.2, integrating over G...
For bubbles of too small a diameter, the gain in volume energy is outweighed by the surface energy cost – the bubble will collapse. However, for bubbles beyond a certain critical size, the energy balance is positive. The bubble will grow and, eventually, swallow the entire mass density of the system; the liquid has vap...
quantum particle can lose its phase coherence and with it, its quantum mechanical character. Beginning with the seminal work of Caldeira and Leggett,24 there have been numerous theoretical investigations of the eﬀect of an environment on the quantum mechanical properties of a system. Such eﬀects are particularly acute ...
ger, Theory of the condensation point, Ann. Phys. (NY) 41 (1967), 108–57. 3.3 Applications of the Feynman path integral 133 remain at q = 0 poised precariously on the maximum of the inverted harmonic potential. Contributions from this solution and the associated harmonic ﬂuctuations reproduce terms in the quantum parti...
te of highest weight \| ↑, which is deﬁned as the (normalized) eigenstate of ˆSz with maximum eigenvalue, S (physically: a spin state polarized in the 3-direction). Owing to the irreducibility of the representation, each (normalized) state of the Hilbert space HS can be obtained by applying the Euler-angle-parameterized...
physical picture behind this singularity is as follows: imagine an inﬁnitely thin solenoid running from r = ∞ through the south pole of the sphere to its center. Assuming that the solenoid contains a magnetic ﬂux 4π, the center of the sphere becomes a source of magnetic ﬂux, the so-called Dirac monopole. This picture i...
elevance, impinging on areas such as quantum electron transport in condensed matter systems. The inevitable presence of impurities and imperfections in any macroscopic solid renders the long-time dynamics of electronic charge carriers chaotic. Relying on a loose interpretation of the Heisenberg principle, Δt ∼ /ΔE, i.e...
generalization of this principle, i.e. an equivalence principle relating d-dimensional quantum ﬁeld theory to (d + 1)-dimensional statistical mechanics. However, before exploring this bridge further, we ﬁrst need to generalize the concept of path integration to problems involving quantum ﬁelds. This will be the subjec...
explore a model in which the particle is coupled to the ﬂuctuations of a quantum mechanical “string.” Later, in Section 8.2, we will see that this model provides a description of tunneling through a single impurity in a Luttinger liquid. (a) A quantum particle of mass m is conﬁned by a sinusoidal potential U (q) = 2g s...
ry that takes as its starting point an integration over all conﬁgurations of a given ﬁeld, weighted by an appropriate action. To emphasize the importance of the formulation that, methodologically, represents the backbone of the remainder of the text, we have pruned the discussion to focus only on the essential elements...
nﬁrmed: ai d( ¯φ, φ) e− i ¯φiφi \|φφ\| = = by parts = = − d( ¯φ, φ) e− & i d( ¯φ, φ) d( ¯φ, φ) e− ∂ ¯φi ¯φiφi φi\|φφ\| ' e− ¯φiφi i \|φφ\| i ¯φiφi \|φ ! ∂ ¯φi φ\| d( ¯φ, φ) e− i ¯φiφi \|φφ\|ai, (4.8) i d ¯φi dφi/π. Taking the adjoint of Eq. (4.8), one may where, for brevity, we have set d( ¯φ, φ) ≡ further check that the left-ha...
ation relations of a super-algebra. It therefore seems to be better to abandon the concept of Grassmann complex conjugation altogether. (Unlike with the bosonic case where complex conjugation is inevitable in order to deﬁne convergent Gaussian integrals, no such need arises in the fermionic case.) 164 Functional ﬁeld i...
(4.27) ψ\| ˆH(a†,a)\|ψ ψ\|ψ ≡ H( ¯ψ, ψ) (simiwhere δ = β/N and larly N ( ¯ψ, ψ)) and we have adopted the shorthand ψn = {ψn }, etc. Finally, sending N → ∞ i and taking limits analogous to those leading from Eq. (3.5) to (3.6) we obtain the continuum version of the path integral,8 ijkl Vijkl ¯ψjψ ¯ψiψ ij hij kψ j + ¯ψi = l...
n g(z) that has simple poles at z = iωn. The sumS then emerges as the sum of residues obtained by integrating the product gh along a suitably chosen path in the complex plane. Typical choices of g include g(z) = * β exp(βz)−1 , β exp(βz)+1 , + bosons, fermions, or g(z) = $ β 2 coth(βz/2), β 2 tanh(βz/2), % bosons, ferm...
ns of the global momentum-dependent fermion operator to the vicin† −kF+q, where \|q\| kF. +(q) =ψ ity of the left/right Fermi point, i.e. ψ Fourier transforming this expression we obtain the approximate decomposition † −(q) =ψ † kF+q, ψ † ψ(x) =e ikFxψ+(x) +e −ikFxψ−(x). (4.42) Before proceeding, let us rewrite the actio...
then generated in a second step through multiplication by a Jordan–Wigner string. In fact, there is a certain ambiguity in the deﬁnition of the string variable: deﬁning ˆθ(x) ≡ π −∞ dx ˆρ(x), it is straightforward to show that for m, m odd integers x # eimˆθ(x)ei ˆφ(x), eim ˆθ(x)ei ˆφ(x) $ + ∝ eiπmΘ(x−x) + eiπmΘ(x−x) =...
above will represent the only viable route towards the solution of the problem. 4.4 Summary and outlook This concludes our preliminary introduction to the ﬁeld integral. We have learned how to represent the partition function of a quantum many-body system in terms of a generalized path integral. The ﬁeld integral repre...
n = 2πnT is a bosonic Matsubara frequency.) (b) Another correlation function central to the theory of the interacting Fermi gas (see Section 5.2), the so-called density–density response function, is given by G0(p, iωm)G0(p + q, iωm + iωn) = − 1 Ld ≡ − T Ld nF(ξp) − nF(ξp+q) iωn + ξp − ξp+q q,ωn χd . p p,ωm Again verify...
ger liquid acts as a “catalyst” for the recursive accumulation of a strong potential. In this problem, we will derive the eﬀective low-energy action describing the interplay of interaction and impurity scattering. The actual catalytic ampliﬁcation mechanism outlined above is then explored in Chapter 8 by renormalizatio...
treat the interaction perturbatively, i.e. to develop the expansion I(g) ≈ n gnIn, where, applying Stirling’s approximation, n! n1∼ nne−n, gnIn = (−g)n n! ∞ −∞ dx√ 2π e− 1 2 x2 x4n = (−g)n (4n − 1)!! n! n1∼ − gn e n . This estimate should alarm us: strictly speaking, it states that a series expansion in the “small par...
el in terms of a φ4-model. While the structure of the action could have been guessed on symmetry grounds, the “microscopic” derivation has the advantage that it yields explicit expressions for the coupling constants. There is actually one interesting aspect of the dependence of these constants on the 2 The only diﬀeren...
s been discussed before, so the ﬁrst non-trivial term we have to explore is G(1): @ @ A A A @ G(1)(x, x) = −g φ(x) ddy φ(y)4φ(x) − φ(x)φ(x) ddy φ(y)4 . (5.15) 0 0 0 Since the functional average is now over a Gaussian action, this expression can be evaluated by Wick’s theorem, Eq. (3.21). For example, the functional ave...
t order). One (correctly) expects that these structures, which are diﬃcult to discern from the equivalent analytical representation, will reﬂect themselves in the mathematics of the perturbative expansion. We return to the discussion of this point below. Then there is the issue of combinatorics. The diagrammatic repres...
however, does not mean that the tools developed above are useless: given a system subject to unfamiliar interactions, low-order perturbation theory will usually be applied as a ﬁrst step to qualitatively explore the situation. For example, a malign divergence of the expansion in the interaction operator may signal the ...
s with the Green function discussed in the previous section, the free energy can be expanded in terms of an interaction parameter. To ﬁx a reference scale against which to compare the correlation energies, let us begin by computing the free energy Eq. (4.41) of the non-interacting electron gas: F (0) = −T ln pσ 1 + e−β...
g, closer inspection reveals some structure. Reﬂecting the fact that electronic transport in solids is carried by excitations at the Fermi energy, the electron Green function (5.22) assumes large values for momenta p pF. This implies that only conﬁgurations where all momentum arguments carried by the Green function are...
rection (with the non-interacting result) N (1) = −∂μF to the RPA, NRPA = −∂μFRPA. Noting that ∂μGp = −(Gp)2 we readily ﬁnd (cf. Eq. (5.23)) N (1) = − 2T 2 L3 p,q (Gp)2Gp+qV (q). After a second diﬀerentiation, ρ(1) = ∂μN (1), this expression would lead to the pathological density of states, a consequence of the unscree...
uce a number of general concepts of inﬁnite-order perturbative summations. As should be clear from the discussion above, a meaningful summation over an inﬁnite set of diagrams necessitates the existence of a class of perturbative corrections that is “more important” than others. In practice, what we need is a small par...
um mechanics, especially for his statistical interpretation of the wavefunction.” (Image c The Nobel Foundation.) 5.3 Inﬁnite-order expansions 227 Inspection of the series shows that only diagrams void of crossing interaction lines (cf. the ﬁgure on the right) survive the limit of large N . The approximation – indeed i...
-body physics. In most cases only approximate solutions can be obtained. With our present example, “approximate” means that one sends N to large values and seeks a solution to leading order in N −1. In that limit, the only surviving contribution to the irreducible vertex is the ﬁrst, i.e. a plain interaction line (see ...
− y). Collecting all the terms, we obtain N2 − 18G0(x − x)G4 0(0)G0(y − x) 0(0) −72G0(x − y)G3 ! −(4+y↔y) +18G0(x − x)G4 0(0) −72G0(x − x)G2 ! −2 0(0)G2 0(y − y) 0(0) −9G0(x − x)G4 ! −1 −24G0(x − x)G4 0(y − y) ! −3 = 144G0(x − y)G0(x − y)G2 ! 5 + 144G0(x − y)G0(x − y)G0(y− y)G2 0(y − y)G0(0) 0(0) ! 6 + 96G0(x − y)G0(x...
lowing we focus on the perturbative scheme developed in the original work of Kondo. Later, in Problem 8.8.5, we will introduce a more advanced approach based on the renormalization group. The starting point of the analysis is the eﬀective sd–Hamiltonian (2.51) introduced in Problem 2.4. Setting ˆHsd = ˆH0 + ˆHimp where...
d theoretical formulation is, in fact, much deeper. With our previous examples, the perturbative expansion was benign. However, we already saw some glimpses indicating that more drastic things may happen. For example, for frequencies approaching the plasma frequency, the polarization operator of the weakly interacting ...
ensity” channel; (b) decoupling in the “pairing” or “Cooper” channel; and (c) decoupling in the “exchange” channel. The version of the transformation discussed above corresponds to Fig. 6.1(a). That type of pairing is sometimes referred to as decoupling in the direct channel. The designation becomes more transparent if...
tion. Indeed, there may be sets of degenerate solutions, etc. Often, when new theories describing an unknown territory have been developed, the search for the “correct” mean-ﬁeld turns out to be a matter of long, and sometimes controversial, research. In the present context, spatial and temporal homogeneity translates ...
given number of particles, this equation determines the temperature dependence of the chemical potential, μ(T ). As the temperature is reduced, the distribution function controlling the population of individual states decreases. Since the number of particles must be kept constant, this scaling must be counter-balanced...
ty of the action to the integrand of the toy problem discussed in Section 5.1.) Crucially, the stability of the action is now guaranteed by the interaction vertex, no matter how small is g > 0 (see the schematic plot of the action in the ﬁgure on page 263). Accordingly, we will treat ψ0 no longer as a ﬁxed parameter bu...
cussed in Section 5.1, the absence of a constant contribution to the action (i.e. a contribution that does not vanish in the limit q → 0) signals the existence of long-range power-law correlations in the system. As we will see shortly, the vanishing of 10 The same argument can be formulated for the quantum magnet. 11 I...
of helium, we mention the capability of thin ﬁlms to ﬂow up the walls of a vessel (if the reward is that on the outer side of the container a low-lying basin can be reached – the fountain experiment) or to eﬀortlessly propagate through porous media that no normal ﬂuid may penetrate. Readers interested in learning more ...
his process is governed by two totally diﬀerent time scales. For an electron, it takes a time ∼ E−1 F to traverse the immediate vicinity of a lattice ion and to trigger a distortion out of its equilibrium position into a conﬁguration that both particles ﬁnd energetically beneﬁcial (top right panel of the ﬁgure). –1 E F...
y 1 d ν()F () to replace Ld the momentum sum by an energy integral, and remembering that the pairing interaction is limited to a thin shell around the Fermi surface, we then obtain p F (p) = T Ld p GpG−p = ωD −ωD d ν() 1 − 2nF() 2 ν ωD T d = ν ln , ωD T (6.16) 270 Broken symmetry and collective phenomena where we have ...
ermi sea as E0 = limΔ→0 E\|Ωs. Use this representation (and the solution of the meanﬁeld equation) to verify that the superconductor ground state energy lies below that of the uncorrelated Fermi sea. It is also instructive to ask for the minimum value E\|Ωs may assume 274 Broken symmetry and collective phenomena upon var...
oupling is arranged using a Hubbard–Stratonovich transformation in the Cooper channel (cf. the discussion on page 244) ) * exp g dτ ddr ¯ψ↑ ¯ψ↓ψ↓ψ↑ ) = D( ¯Δ, Δ) exp − dτ ddr 1 g \|Δ\|2 − ¯Δψ↓ψ↑ + Δ ¯ψ↑ ¯ψ↓ * , where Δ(r, τ ) represents a dynamically ﬂuctuating complex ﬁeld. Reﬂecting the behavior of the bilinear ψ↓ψ↑, i...
n change q=0 of the quadratic action of the constant order parameter mode Δ(q = 0). In the vicinity of this point, the constant contribution to the action must scale as ∼ (T − Tc) from which one may conclude that the action assumes the form S(2)[Δ, ¯Δ] = dτ ddr r(T ) 2 \|Δ\|2 + O(∂Δ, ∂τ Δ), where r(T ) ∼ T − Tc and O(∂Δ,...
a certain fraction of the formerly uncorrelated electronic states participate in the condensate, i.e. one may write j = jn + js, where jn, the current carried by the normal states of the system, will not be of further concern to us, while js is the “supercurrent” carried by the condensate. Let us further assume that th...
stitute this result together with the diamagnetic contribution Eq. (6.36) back into the expansion (6.35), partially transform back to real space p + Δ2 p = ξ2 q fqf−q = dτ ddr f 2(τ, r)2, and arrive at the action S[ ˜A] = dτ ddr ( T Ld p −ω2 p − 2Δ2 0 p)2 n + λ2 n + λ2 (ω2 c1 + n 2m − 1 dm2 T Ld p c2 ˜φ2(τ, r) p2(−ω2 (...
the fermionic problems discussed above, where N was proportional to the density of states at the Fermi surface, the action contained a trace over all momentum states. The summation over these states then led to an overall factor N multiplying the action. 292 Broken symmetry and collective phenomena diamagnetic and the ...
[Ψ] = dd+1x L(Ψ, ∂μΨ), containing the isospinor Ψ and its derivatives, be made invariant under non-abelian SU(2) gauge transformations? Referring for a more systematic discussion to Ryder,42 let us brieﬂy sketch the principal idea of non–abelian gauge theory. We ﬁrst notice that a fermion bilinear ∼ ¯Ψ∂μΨ is generally ...
momenta acquire opposite quantum phases. Thus, the pair state is a slowly ﬂuctuating, and therefore stable, object. However, in the presence of a magnetic ﬁeld, the phase factors have to be generalized to r, r\|k ↑, −k ↓ ∼ e−i dr·(k−eA)e−i dr·(−k−eA) ∼ e2ieA·r, where we assumed that the vector potential varies only slow...
itude of unusual quantum phenomena from localization to strong sample-to-sample ﬂuctuations, some of which will be discussed below. The experimental and theoretical study of these phenomena is the central theme of mesoscopic physics. How might one set about modeling an impurity potential in statistical terms? One might...
ψa, ¯ψb] ≡ − γ2 2 m(r) ¯ψb m(r)ψa n(r)ψb ddr ¯ψa mn n(r), (6.49) represents an eﬀective quartic interaction generated by the disorder average. Notice the superﬁcial similarity between Sdis and an attractive short-range “interaction” term. However, in contrast to a dynamically generated interaction, (a) Sdis does not in...
gas 309 Q5: Why is the replica method exact in perturbation theory? A1: Unlike the Fock diagram, where all replica indices are locked to the index of the incoming Green functions, the Hartree diagram contains one free replica summation. This summation yields an excess factor R that, in the limit R → 0, vanishes. For th...

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