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7.8: Ginzburg-Landau Theory

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Ginzburg-Landau free energy

Including gradient terms in the free energy, we write F[m(x),h(x)]=ddx{f0+12am2+14bm4+16cm6 hm+12κ(m)2+} . In principle, any term which does not violate the appropriate global symmetry will turn up in such an expansion of the free energy, with some coefficient. Examples include hm3 (both m and h are odd under time reversal), m2(m)2, We now ask: what function m(x) extremizes the free energy functional F[m(x),h(x)]? The answer is that m(x) must satisfy the corresponding Euler-Lagrange equation, which for the above functional is am+bm3+cm5hκ2m=0 . If a>0 and h is small (we assume b>0 and c>0), we may neglect the m3 and m5 terms and write (aκ2)m=h , whose solution is obtained by Fourier transform as ˆm(q)=ˆh(q)a+κq2 , which, with h(x) appropriately defined, recapitulates the result in Equation [mhqeqn]. Thus, we conclude that ˆχ(q)=1a+κq2 , which should be compared with Equation [xhiheqn]. For continuous functions, we have ˆm(q)=ddxm(x)eiqxm(x)=ddq(2π)dˆm(q)eiqx . We can then derive the result m(x)=ddxχ(xx)h(x) , where χ(xx)=1κddq(2π)deiq(xx)q2+ξ2 , where the correlation length is ξ=κ/a(TTc)1/2, as before.

If a<0 then there is a spontaneous magnetization and we write m(x)=m0+δm(x). Assuming h is weak, we then have two equations a+bm20+cm40=0(a+3bm20+5cm40κ2)δm=h . If a>0 is small, we have m20=a/3b and δˆm(q)=ˆh(q)2a+κq2 ,

Domain wall profile

A particularly interesting application of Ginzburg-Landau theory is its application toward modeling the spatial profile of defects such as vortices and domain walls. Consider, for example, the case of Ising (Z2) symmetry with h=0. We expand the free energy density to order m4: F[m(x)]=ddx{f0+12am2+14bm4+12κ(m)2} . We assume a<0, corresponding to T<Tc. Consider now a domain wall, where m(x)=m0 and m(x+)=+m0, where m0 is the equilibrium magnetization, which we obtain from the Euler-Lagrange equation, am+bm3κ2m=0 , assuming a uniform solution where m=0. This gives m0=|a|/b. It is useful to scale m(x) by m0, writing m(x)=m0ϕ(x). The scaled order parameter function ϕ(x) interpolates between ϕ()=1 and ϕ(+)=1.

It also proves useful to rescale position, writing x=(2κ/|a|)1/2ζ. Then we obtain 122ϕ=ϕ+ϕ3 . We assume ϕ(ζ)=ϕ(ζ) is only a function of one coordinate, ζζ1. Then the Euler-Lagrange equation becomes d2ϕdζ2=2ϕ+2ϕ3Uϕ , where U(ϕ)=12(ϕ21)2 . The ‘potential’ U(ϕ) is an inverted double well, with maxima at ϕ=±1. The equation ¨ϕ=U(ϕ), where dot denotes differentiation with respect to ζ, is simply Newton’s second law with time replaced by space. In order to have a stationary solution at ζ± where ϕ=±1, the total energy must be E=U(ϕ=±1)=0, where E=12˙ϕ2+U(ϕ). This leads to the first order differential equation dϕdζ=1ϕ2 , with solution ϕ(ζ)=tanh(ζ) . Restoring the dimensionful constants, m(x)=m0tanh(x2ξ) , where the coherence length ξ(κ/|a|)1/2 diverges at the Ising transition a=0.

Derivation of Ginzburg-Landau free energy

We can make some progress in systematically deriving the Ginzburg-Landau free energy. Consider the Ising model, ˆHkBT=12i,jKijσiσjihiσi+12iKii , where now Kij=Jij/kBT and hi=Hi/kBT are the interaction energies and local magnetic fields in units of kBT. The last term on the RHS above cancels out any contribution from diagonal elements of Kij. Our derivation makes use of a generalization of the Gaussian integral, dxe12ax2bx=(2πa)1/2eb2/2a . The generalization is \[\int\limits_{-\infty}^\infty\!\!\!dx\ns_1\cdots\!\!\!\int\limits_{-\infty}^\infty\!\!\!dx\ns_N\> e^{-{1\over 2} A\ns_{ij} x\ns_ix\ns_j - b\ns_i x\ns_i}={(2\pi)^{N/2}\over \sqrt

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    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07:_Mean_Field_Theory_of_Phase_Transitions/7.08:_Ginzburg-Landau_Theory), /content/body/div[3]/p[1]/span[3], line 1, column 2
\!\!\int\limits_{-\infty}^\infty\!\!\!d\phi\ns_1 \cdots\!\!\!\int\limits_{-\infty}^\infty\!\!\!d\phi\ns_N\> e^{-{1\over 2} K^{-1}_{ij}\phi\ns_i\phi\ns_j}\,\Tra e^{(\phi\ns_i+h\ns_i)\sigma\ns_i}\\ &={det}^{-1/2}(2\pi K)\> e^{-{1\over 2} K\ns_{ii}}\!\!\int\limits_{-\infty}^\infty\!\!\!d\phi\ns_1 \cdots\!\!\!\int\limits_{-\infty}^\infty\!\!\!d\phi\ns_N\> e^{-{1\over 2} K^{-1}_{ij}\phi\ns_i\phi\ns_j}\,e^{\sum_i \ln\left[2\cosh(\phi\ns_i+h\ns_i)\right]}\\ &\equiv \int\limits_{-\infty}^\infty\!\!\!d\phi\ns_1\cdots\!\!\!\int\limits_{-\infty}^\infty\!\!\!d\phi\ns_N\> e^{-\RPhi(\phi\ns_1,\ldots,\phi\ns_N)}\ , \end{split}\] where Φ=12i,jK1ijϕiϕjilncosh(ϕi+hi)+12lndet(2πK)+12TrKNln2 . We assume the model is defined on a Bravais lattice, in which case we can write ϕi=ϕRi. We can then define the Fourier transforms, ϕR=1NqˆϕqeiqRˆϕq=1NRϕReiqR and ˆK(q)=RK(R)eiqR .

A few remarks about the lattice structure and periodic boundary conditions are in order. For a Bravais lattice, we can write each direct lattice vector R as a sum over d basis vectors with integer coefficients, viz. R=dμ=1nμaμ , where d is the dimension of space. The reciprocal lattice vectors bμ satisfy aμbν=2πδμν , and any wavevector q may be expressed as q=12πdμ=1θμbμ . We can impose periodic boundary conditions on a system of size M1×M2××Md by requiring ϕR+dμ=1lμMμaμ=ϕR . This leads to the quantization of the wavevectors, which must then satisfy eiMμqaμ=eiMμθμ=1 , and therefore θμ=2πmμ/Mμ , where mμ is an integer. There are then M1M2Md=N independent values of q, which can be taken to be those corresponding to mμ{1,,Mμ}.

Let’s now expand the function Φ(\Vphi) in powers of the ϕi, and to first order in the external fields hi. We obtain Φ=12q(ˆK1(q)1)|ˆϕq|2+112Rϕ4RRhRϕR+O(ϕ6,h2)+12TrK+12Trln(2πK)Nln2 On a d-dimensional lattice, for a model with nearest neighbor interactions K1 only, we have ˆK(q)=K1δeiqδ, where δ is a nearest neighbor separation vector. These are the eigenvalues of the matrix Kij. We note that Kij is then not positive definite, since there are negative eigenvalues19. To fix this, we can add a term K0 everywhere along the diagonal. We then have ˆK(q)=K0+K1δcos(qδ) . Here we have used the inversion symmetry of the Bravais lattice to eliminate the imaginary term. The eigenvalues are all positive so long as K0>zK1, where z is the lattice coordination number. We can therefore write ˆK(q)=ˆK(0)αq2 for small q, with α>0. Thus, we can write ˆK1(q)1=a+κq2+ . To lowest order in q the RHS is isotropic if the lattice has cubic symmetry, but anisotropy will enter in higher order terms. We’ll assume isotropy at this level. This is not necessary but it makes the discussion somewhat less involved. We can now write down our Ginzburg-Landau free energy density: F=aϕ2+12κ|ϕ|2+112ϕ4hϕ , valid to lowest nontrivial order in derivatives, and to sixth order in ϕ.

One might wonder what we have gained over the inhomogeneous variational density matrix treatment, where we found F=12qˆJ(q)|ˆm(q)|2qˆH(q)ˆm(q)+kBTi{(1+mi2)ln(1+mi2)+(1mi2)ln(1mi2)} . Surely we could expand ˆJ(q)=ˆJ(0)12aq2+ and obtain a similar expression for F. However, such a derivation using the variational density matrix is only approximate. The method outlined in this section is exact.

Let’s return to our complete expression for Φ: Φ(\Vphi)=Φ0(\Vphi)+Rv(ϕR) , where Φ0(\Vphi)=12qG1(q)|ˆϕ(q)|2+12Tr(11+G1)+12Trln(2π1+G1)Nln2 . Here we have defined v(ϕ)=12ϕ2lncoshϕ=112ϕ4145ϕ6+172520ϕ8+ and G(q)=ˆK(q)1ˆK(q) . We now want to compute Z=D\Vphi eΦ0(\Vphi)eRv(ϕR) where D\Vphidϕ1dϕ2dϕN . We expand the second exponential factor in a Taylor series, allowing us to write Z=Z0(1Rv(ϕR)+12RRv(ϕR)v(ϕR)+) , where Z0=D\Vphi eΦ0(\Vphi)lnZ0=12Tr[ln(1+G)G1+G]+Nln2 and F(\Vphi)=D\VphiFeΦ0D\VphieΦ0 .

To evaluate the various terms in the expansion of Equation [ZZZ], we invoke Wick’s theorem, which says xi1xi2xi2L=dx1dxNe12G1ijxixjxi1xi2xi2L/dx1dxNe12G1ijxixj=all distinctpairingsGj1j2Gj3j4Gj2L1j2L , where the sets {j1,,j2L} are all permutations of the set {i1,,i2L}. In particular, we have x4i=3(Gii)2 . In our case, we have ϕ4R=3(1NqG(q))2 . Thus, if we write v(ϕ)112ϕ4 and retain only the quartic term in v(ϕ), we obtain FkBT=lnZ0=12Tr[G1+Gln(1+G)]+14N(TrG)2Nln2=Nln2+14N(TrG)214Tr(G2)+O(G3) . Note that if we set Kij to be diagonal, then ˆK(q) and hence G(q) are constant functions of q. The O(G2) term then vanishes, which is required since the free energy cannot depend on the diagonal elements of Kij.

Ginzburg criterion

Let us define A(T,H,V,N) to be the usual ( thermodynamic) Helmholtz free energy. Then eβA=Dm eβF[m(x)] , where the functional F[m(x)] is of the Ginzburg-Landau form, given in Equation [DWFE]. The integral above is a functional integral. We can give it a more precise meaning by defining its measure in the case of periodic functions m(x) confined to a rectangular box. Then we can expand m(x)=1Vqˆmqeiqx , and we define the measure Dmdm0qqx>0dReˆmqdImˆmq . Note that the fact that m(x)R means that ˆmq=ˆmq. We’ll assume T>Tc and H=0 and we’ll explore limit TT+c from above to analyze the properties of the critical region close to Tc. In this limit we can ignore all but the quadratic terms in m, and we have eβA=Dmexp(12βq(a+κq2)|ˆmq|2)=q(πkBTa+κq2)1/2 . Thus, A=12kBTqln(a+κq2πkBT) . We now assume that a(T)=αt, where t is the dimensionless quantity t=TTcTc , known as the reduced temperature.

We now compute the heat capacity CV=T2AT2. We are really only interested in the singular contributions to CV, which means that we’re only interested in differentiating with respect to T as it appears in a(T). We divide by NSkB where NS is the number of unit cells of our system, which we presume is a lattice-based model. Note NSV/ad where V is the volume and a the lattice constant. The dimensionless heat capacity per lattice site is then cCVNS=α2ad2κ2Λddq(2π)d1(ξ2+q2)2 , where ξ=(κ/αt)1/2|t|1/2 is the correlation length, and where Λa1 is an ultraviolet cutoff. We define R(κ/α)1/2, in which case c=R4adξ4d12Λξddˉq(2π)d1(1+ˉq2)2 , where ˉqqξ. Thus, c(t){const.if d>4lntif d=4td22if d<4 .

For d>4, mean field theory is qualitatively accurate, with finite corrections. In dimensions d4, the mean field result is overwhelmed by fluctuation contributions as t0+ ( as TT+c). We see that MFT is sensible provided the fluctuation contributions are small, provided R4adξ4d1 , which entails \boldsymbol{t\gg t\ns_\ssr{G}}, where \boldsymbol{t\ns_\ssr{G}=\bigg({\Sa\over R\ns_*}\bigg)^{\!{2d\over 4-d}}} is the Ginzburg reduced temperature. The criterion for the sufficiency of mean field theory, namely \boldsymbol{t\gg t\ns_\ssr{G}}, is known as the Ginzburg criterion. The region \boldsymbol{|t|<t\ns_\ssr{G}} is known as the critical region.

In a lattice ferromagnet, as we have seen, Ra is on the scale of the lattice spacing itself, hence \boldsymbol{t\ns_\ssr{G}\sim 1} and the critical regime is very large. Mean field theory then fails quickly as TTc. In a (conventional) three-dimensional superconductor, R is on the order of the Cooper pair size, and R/a102103, hence \boldsymbol{t\ns_\ssr{G}=(a/R\ns_*)^6\sim 10^{-18} - 10^{-12}} is negligibly narrow. The mean field theory of the superconducting transition – BCS theory – is then valid essentially all the way to T=Tc.


This page titled 7.8: Ginzburg-Landau Theory is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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