Landau theory

{{main|Mean field theory}}

Landau was motivated to suggest that the free energy of any system should obey two conditions:

• Be analytic in the order parameter and its gradients.
• Obey the symmetry of the Hamiltonian.

Given these two conditions, one can write down (in the vicinity of the critical temperature, T_c) a phenomenological expression for the free energy as a Taylor expansion in the order parameter.

File:LandauFreeEnergy.svg

Consider a system that breaks some symmetry below a phase transition, which is characterized by an order parameter $\backslash eta$. This order parameter is a measure of the order before and after a phase transition; the order parameter is often zero above some critical temperature and non-zero below the critical temperature. In a simple ferromagnetic system like the Ising model, the order parameter is characterized by the net magnetization $m$, which becomes spontaneously non-zero below a critical temperature $T\_c$. In Landau theory, one considers a free energy functional that is an analytic function of the order parameter. In many systems with certain symmetries, the free energy will only be a function of even powers of the order parameter, for which it can be expressed as the series expansion{{cite book|last1=Landau|first1=L.D.|last2=Lifshitz|first2=E.M.|title=Statistical Physics|volume=5|publisher=Elsevier|year=2013|isbn=978-0080570464}}

:$F(T,\backslash eta)\; -\; F\_0\; =\; a(T)\; \backslash eta^2\; +\; \backslash frac\{b(T)\}\{2\}\; \backslash eta^4\; +\; \backslash cdots$

In general, there are higher order terms present in the free energy, but it is a reasonable approximation to consider the series to fourth order in the order parameter, as long as the order parameter is small. For the system to be thermodynamically stable (that is, the system does not seek an infinite order parameter to minimize the energy), the coefficient of the highest even power of the order parameter must be positive, so $b(T)>0$. For simplicity, one can assume that $b(T)\; =\; b\_0$, a constant, near the critical temperature. Furthermore, since $a(T)$ changes sign above and below the critical temperature, one can likewise expand $a(T)\; \backslash approx\; a\_0\; (T-T\_c)$, where it is assumed that $a>0$ for the high-temperature phase while for the low-temperature phase, for a transition to occur. With these assumptions, minimizing the free energy with respect to the order parameter requires

:$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; \backslash eta\}\; =\; 2a(T)\; \backslash eta\; +\; 2b(T)\; \backslash eta^3\; =\; 0$

The solution to the order parameter that satisfies this condition is either $\backslash eta=\; 0$, or

:$\backslash eta\_0^2\; =\; -\backslash frac\{a\}\{b\}\; =\; -\; \backslash frac\{a\_0\}\{b\_0\}(T-T\_c)$

File:LandauTheoryTransitions.svg

It is clear that this solution only exists for

:$\backslash eta(T)\; \backslash propto\; \backslash left|\; T\; -\; T\_c\; \backslash right|^\{1/2\}$

below the critical temperature, indicating a critical exponent $\backslash beta\; =\; 1/2$ for this Landau mean-theory model.

The free-energy will vary as a function of temperature given by

:$F\; -\; F\_0\; =$

\begin{cases}

- \dfrac{a_0^2}{2b_0} (T-T_c)^2, & T

0, & T>T_c

\end{cases}

From the free energy, one can compute the specific heat,

:$c\_p\; =\; -T\backslash frac\{\backslash partial^2\; F\}\{\backslash partial\; T^2\}\; =$

\begin{cases}

\dfrac{a_0^2}{b_0} T, & T

0, & T>T_c

\end{cases}

which has a finite jump at the critical temperature of size $\backslash Delta\; c\; =\; a\_0^2\; T\_c/b\_0$. This finite jump is therefore not associated with a discontinuity that would occur if the system absorbed latent heat, since $T\_c\; \backslash Delta\; S\; =\; 0$. It is also noteworthy that the discontinuity in the specific heat is related to the discontinuity in the second derivative of the free energy, which is characteristic of a second-order phase transition. Furthermore, the fact that the specific heat has no divergence or cusp at the critical point indicates its critical exponent for $c\backslash sim\; |T-T\_c|^\{-\backslash alpha\}$ is $\backslash alpha=0$.

Landau expanded his theory to consider the restraints that it imposes on the symmetries before and after a transition of second order. They need to comply with a number of requirements:

• The distorted (or ordered) symmetry needs to be a subgroup of the higher one.
• The order parameter that embodies the distortion needs to transform as a single irreducible representation (irrep) of the parent symmetry
• The irrep should not contain a third order invariant
• If the irrep allows for more than one fourth order invariant, the resulting symmetry minimizes a linear combination of these invariants

In the latter case more than one daughter structure should be reacheable through a continuous transition. A good example of this are the structure of MnP (space group Cmca) and the low temperature structure of NbS (space group P6₃mc). They are both daughters of the NiAs-structure and their distortions transform according to the same irrep of that spacegroup.{{Cite journal |last=Franzen |first=H.F. |last2=Haas |first2=C. |last3=Jellinek |first3=F. |date=1974 |title=Phase transitions between NiAs- and MnP-type phases |journal=Phys. Rev. B |volume=10 |issue=4 |pages=1248–1251 |bibcode=1974PhRvB..10.1248F |doi=10.1103/PhysRevB.10.1248}}

In many systems, one can consider a perturbing field $h$ that couples linearly to the order parameter. For example, in the case of a classical dipole moment $\backslash mu$, the energy of the dipole-field system is $-\backslash mu\; B$. In the general case, one can assume an energy shift of $-\backslash eta\; h$ due to the coupling of the order parameter to the applied field $h$, and the Landau free energy will change as a result:

:$F(T,\backslash eta)\; -\; F\_0\; =\; a\_0\; (T-T\_c)\; \backslash eta^2\; +\; \backslash frac\{b\_0\}\{2\}\; \backslash eta^4\; -\; \backslash eta\; h$

In this case, the minimization condition is

:$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; \backslash eta\}\; =\; 2\; a(T)\; \backslash eta\; +\; 2b\_0\; \backslash eta^3\; -\; h\; =\; 0$

One immediate consequence of this equation and its solution is that, if the applied field is non-zero, then the magnetization is non-zero at any temperature. This implies there is no longer a spontaneous symmetry breaking that occurs at any temperature. Furthermore, some interesting thermodynamic and universal quantities can be obtained from this above condition. For example, at the critical temperature where $a(T\_c)=0$, one can find the dependence of the order parameter on the external field:

:$\backslash eta(T\_c)\; =\; \backslash left(\; \backslash frac\{h\}\{2b\_0\}\; \backslash right)^\{1/3\}\; \backslash propto\; h^\{1/\backslash delta\}$

indicating a critical exponent $\backslash delta\; =\; 3$.

File:LandauTheorySusceptibility.svg

Furthermore, from the above condition, it is possible to find the zero-field susceptibility $\backslash chi\backslash equiv\; \backslash partial\; \backslash eta/\backslash partial\; h|\_\{h=0\}$, which must satisfy

:$0\; =\; 2\; a\; \backslash frac\{\backslash partial\; \backslash eta\}\{\backslash partial\; h\}\; +\; 6b\; \backslash eta^2\; \backslash frac\{\backslash partial\; \backslash eta\}\{\backslash partial\; h\}\; -\; 1$

:$[2\; a\; +\; 6b\; \backslash eta^2]\; \backslash frac\{\backslash partial\; \backslash eta\}\{\backslash partial\; h\}\; =\; 1$

In this case, recalling in the zero-field case that $\backslash eta^2\; =\; -a/b$ at low temperatures, while $\backslash eta^2=0$ for temperatures above the critical temperature, the zero-field susceptibility therefore has the following temperature dependence:

:

which is reminiscent of the Curie-Weiss law for the temperature dependence of magnetic susceptibility in magnetic materials, and yields the mean-field critical exponent $\backslash gamma=1$.

It is noteworthy that although the critical exponents so obtained are incorrect for many models and systems, they correctly satisfy various exponent equalities such as the Rushbrooke equality: $\backslash alpha\; +\; 2\backslash beta\; +\; \backslash gamma\; =\; 2$.

Landau theory can also be used to study first-order transitions. There are two different formulations, depending on whether or not the system is symmetric under a change in sign of the order parameter.

Here we consider the case where the system has a symmetry and the energy is invariant when the order parameter changes sign.

A first-order transition will arise if the quartic term in $F$ is negative. To ensure that the free energy remains positive at large $\backslash eta$, one must carry the free-energy expansion to sixth-order,{{cite book|title=The Landau Theory of Phase Transitions|chapter=Chapter 5: First-Order Transitions|last1=Tolédano|first1=J.C.|last2=Tolédano|first2=P.|publisher=World Scientific Publishing Company|date=1987|url=https://books.google.com/books?id=6gU8DQAAQBAJ&q=landau+first+order|isbn=9813103949}}{{cite book|last1=Stoof|first1=H.T.C.|last2=Gubbels|first2=K.B.|last3=Dickerscheid|first3=D.B.M.|title=Ultracold Quantum Fields|publisher=Springer|year=2009|isbn=978-1-4020-8763-9}}

:$F(T,\backslash eta)\; =\; A(T)\; \backslash eta^2\; -\; B\_0\; \backslash eta^4\; +\; C\_0\; \backslash eta^6,$

where $A(T)=A\_0(T-T\_0)$, and $T\_0$ is some temperature at which $A(T)$ changes sign. We denote this temperature by $T\_0$ and not $T\_c$, since it will emerge below that it is not the temperature of the first-order transition, and since there is no critical point, the notion of a "critical temperature" is misleading to begin with. $A\_0,\; B\_0,$ and $C\_0$ are positive coefficients.

We analyze this free energy functional as follows: (i) For $T\; >\; T\_0$, the $\backslash eta^2$ and $\backslash eta^6$ terms are concave upward for all $\backslash eta$, while the $\backslash eta^4$ term is concave downward. Thus for sufficiently high temperatures $F$ is concave upward for all $\backslash eta$, and the equilibrium solution is $\backslash eta\; =\; 0$. (ii) For , both the $\backslash eta^2$ and $\backslash eta^4$ terms are negative, so $\backslash eta\; =\; 0$ is a local maximum, and the minimum of $F$ is at some non-zero value $\backslash pm\backslash eta\_0(T)$, with

. (iii) For $T$ just above $T\_0$, $\backslash eta\; =\; 0$ turns into a local minimum, but the minimum at $\backslash eta\_0(T)$ continues to be the global minimum since it has a lower free energy. It follows that as the temperature is raised above $T\_0$, the global minimum cannot continuously evolve from

$\backslash eta\_0(T)$ to 0. Rather, at some intermediate temperature $T\_*$, the minima at $\backslash eta\_0(T\_*)$ and $\backslash eta\; =\; 0$ must become degenerate. For $T\; >\; T\_*$, the global minimum will jump discontinuously from $\backslash eta\_0(T\_*)$ to 0.

To find $T\_*$, we demand that free energy be zero at $\backslash eta\; =\; \backslash eta\_0(T\_*)$ (just like the $\backslash eta=0$ solution), and furthermore that this point should be a local minimum. These two conditions yield two equations,

:$0=A(T)\; \backslash eta^2\; -\; B\_0\; \backslash eta^4\; +\; C\_0\; \backslash eta^6,$

:$0=2A(T)\; \backslash eta\; -\; 4\; B\_0\; \backslash eta^3\; +\; 6\; C\_0\; \backslash eta^5,$

File:LandauFirstOrderTransition.svg

which are satisfied when $\backslash eta^2(T\_*)\; =\; \{B\_0\}/\{2C\_0\}$. The same equations also imply that $A(T\_*)\; =\; A\_0(T\_*-T\_0)\; =\; B\_0^2/4C\_0$. That is,

:$T\_*\; =\; T\_0\; +\; \backslash frac\{B\_0^2\}\{4\; A\_0\; C\_0\}.$

From this analysis both points made above can be seen explicitly. First, the order parameter suffers a discontinuous jump from $(B\_0/2C\_0)^\{1/2\}$ to 0. Second, the transition temperature $T\_*$ is not the same as the temperature $T\_0$ where $A(T)$ vanishes.

At temperatures below the transition temperature,

:$\backslash eta\_0^2\; =\; \backslash frac\{B\_0\}\{3C\_0\}\; \backslash left[\; 1\; +\; \backslash sqrt\{1\; -\; \backslash frac\{3A(T)\; C\_0\}\{B\_0^2\}\}\; \backslash right]$

which is plotted to the right. This shows the clear discontinuity associated with the order parameter as a function of the temperature. To further demonstrate that the transition is first-order, one can show that the free energy for this order parameter is continuous at the transition temperature $T\_*$, but its first derivative (the entropy) suffers from a discontinuity, reflecting the existence of a non-zero latent heat.

Next we consider the case where the system does not have a symmetry. In this case there is no reason to keep only even powers of $\backslash eta$ in the expansion of $F$, and a cubic term must be allowed (The linear term can always be eliminated by a shift $\backslash eta\; \backslash to\; \backslash eta$ + constant.) We thus consider a free energy functional

:$F(T,\backslash eta)\; =\; A(T)\; \backslash eta^2\; -\; C\_0\; \backslash eta^3\; +\; B\_0\; \backslash eta^4\; +\; \backslash cdots.$

Once again $A(T)=A\_0(T-T\_0)$, and $A\_0,\; B\_0,\; C\_0$ are all positive. The sign of the cubic term can always be chosen to be negative as we have done by reversing the sign of $\backslash eta$ if necessary.

We analyze this free energy functional as follows: (i) For , we have a local maximum at $\backslash eta\; =\; 0$, and since the free energy is bounded below, there must be two local minima at nonzero values and $\backslash eta\_+(T)\; >\; 0$. The cubic term ensures that $\backslash eta\_+$ is the global minimum since it is deeper. (ii) For $T$ just above $T\_0$, the minimum at $\backslash eta\_-$ disappears, the maximum at $\backslash eta\; =\; 0$ turns into a local minimum, but the minimum at $\backslash eta\_+$ persists and continues to be the global minimum. As the temperature is further raised, $F(T,\backslash eta\_+(T))$ rises until it equals zero at some temperature $T\_*$. At $T\_*$ we get a discontinuous jump in the global minimum from $\backslash eta\_+(T\_*)$ to 0. (The minima cannot coalesce for that would require the first three derivatives of $F$ to vanish at $\backslash eta\; =\; 0$.)

To find $T\_*$, we demand that free energy be zero at $\backslash eta\; =\; \backslash eta\_+(T\_*)$ (just like the $\backslash eta=0$ solution), and furthermore that this point should be a local minimum. These two conditions yield two equations,

:$0=A(T)\; \backslash eta^2\; -\; C\_0\; \backslash eta^3\; +\; B\_0\; \backslash eta^4,$

:$0=2A(T)\; \backslash eta\; -\; 3\; C\_0\; \backslash eta^2\; +\; 4\; B\_0\; \backslash eta^3,$

which are satisfied when $\backslash eta(T\_*)\; =\; \{C\_0\}/\{2B\_0\}$. The same equations also imply that $A(T\_*)\; =\; A\_0(T\_*-T\_0)\; =\; C\_0^2/4B\_0$. That is,

:$T\_*\; =\; T\_0\; +\; \backslash frac\{C\_0^2\}\{4\; A\_0\; B\_0\}.$

As in the symmetric case the order parameter suffers a discontinuous jump from $(C\_0/2B\_0)$ to 0. Second, the transition temperature $T\_*$ is not the same as the temperature $T\_0$ where $A(T)$ vanishes.

It was known experimentally that the liquid–gas coexistence curve and the ferromagnet magnetization curve both exhibited a scaling relation of the form $|T\; -\; T\_c|^\{\backslash beta\}$, where $\backslash beta$ was mysteriously the same for both systems. This is the phenomenon of universality. It was also known that simple liquid–gas models are exactly mappable to simple magnetic models, which implied that the two systems possess the same symmetries. It then followed from Landau theory why these two apparently disparate systems should have the same critical exponents, despite having different microscopic parameters. It is now known that the phenomenon of universality arises for other reasons (see Renormalization group). In fact, Landau theory predicts the incorrect critical exponents for the Ising and liquid–gas systems.

The great virtue of Landau theory is that it makes specific predictions for what kind of non-analytic behavior one should see when the underlying free energy is analytic. Then, all the non-analyticity at the critical point, the critical exponents, are because the equilibrium value of the order parameter changes non-analytically, as a square root, whenever the free energy loses its unique minimum.

The extension of Landau theory to include fluctuations in the order parameter shows that Landau theory is only strictly valid near the critical points of ordinary systems with spatial dimensions higher than 4. This is the upper critical dimension, and it can be much higher than four in more finely tuned phase transitions. In Mukhamel's analysis of the isotropic Lifschitz point, the critical dimension is 8. This is because Landau theory is a mean field theory, and does not include long-range correlations.

This theory does not explain non-analyticity at the critical point, but when applied to superfluid and superconductor phase transition, Landau's theory provided inspiration for another theory, the Ginzburg–Landau theory of superconductivity.

Consider the Ising model free energy above. Assume that the order parameter $\backslash Psi$ and external magnetic field, $h$, may have spatial variations. Now, the free energy of the system can be assumed to take the following modified form:

:$F\; :=\; \backslash int\; d^D\; x\; \backslash \; \backslash left(\; a(T)\; +\; r(T)\; \backslash psi^2(x)\; +\; s(T)\; \backslash psi^4(x)\; \backslash \; +\; f(T)\; (\backslash nabla\; \backslash psi(x))^2\; \backslash \; +h(x)\; \backslash psi(x)\backslash \; \backslash \; +\; \backslash mathcal\{O\}(\backslash psi^6\; ;\; (\backslash nabla\; \backslash psi)^4)\; \backslash right)$

where $D$ is the total spatial dimensionality. So,

:$\backslash langle\; \backslash psi(x)\; \backslash rangle\; :=\; \backslash frac\{\backslash text\{Tr\}\backslash \; \backslash psi(x)\; \{\backslash rm\; e\}^\{-\backslash beta\; H\}\}\{Z\}$

Assume that, for a localized external magnetic perturbation $h(x)\; \backslash rightarrow\; 0\; +\; h\_0\; \backslash delta(x)$, the order parameter takes the form $\backslash psi(x)\; \backslash rightarrow\; \backslash psi\_0\; +\; \backslash phi(x)$. Then,

:$\backslash frac\{\backslash delta\; \backslash langle\; \backslash psi(x)\; \backslash rangle\}\{\backslash delta\; h(0)\}\; =\; \backslash frac\{\backslash phi(x)\}\{h\_0\}\; =\; \backslash beta\; \backslash left\; (\; \backslash langle\; \backslash psi(x)\; \backslash psi(0)\; \backslash rangle\; -\; \backslash langle\; \backslash psi(x)\; \backslash rangle\; \backslash langle\; \backslash psi(0)\; \backslash rangle\; \backslash right\; )$

That is, the fluctuation $\backslash phi(x)$ in the order parameter corresponds to the order-order correlation. Hence, neglecting this fluctuation (like in the earlier mean-field approach) corresponds to neglecting the order-order correlation, which diverges near the critical point.

One can also solve "Equilibrium Statistical Physics" by Michael Plischke, Birger Bergersen, Section 3.10, 3rd ed for $\backslash phi(x)$, from which the scaling exponent, $\backslash nu$, for correlation length $\backslash xi\; \backslash sim\; (T-T\_c)^\{-\backslash nu\}$ can deduced. From these, the Ginzburg criterion for the upper critical dimension for the validity of the Ising mean-field Landau theory (the one without long-range correlation) can be calculated as:

:$D\; \backslash ge\; 2\; +\; 2\; \backslash frac\{\backslash beta\}\{\backslash nu\}$

In our current Ising model, mean-field Landau theory gives $\backslash beta\; =\; 1/2\; =\; \backslash nu$ and so, it (the Ising mean-field Landau theory) is valid only for spatial dimensionality greater than or equal to 4 (at the marginal values of $D=4$, there are small corrections to the exponents). This modified version of mean-field Landau theory is sometimes also referred to as the Landau–Ginzburg theory of Ising phase transitions. As a clarification, there is also a Ginzburg–Landau theory specific to superconductivity phase transition, which also includes fluctuations.

• Ginzburg–Landau theory
• Landau–de Gennes theory
• Ginzburg criterion
• Stuart–Landau equation

• Landau L.D. Collected Papers (Nauka, Moscow, 1969)
• Michael C. Cross, Landau theory of second order phase transitions, [http://www.pmaweb.caltech.edu/~mcc/Ph127/b/Lecture6.pdf] (Caltech statistical mechanics lecture notes).
• Yukhnovskii, I R, Phase Transitions of the Second Order – Collective Variables Method, World Scientific, 1987, {{ISBN|9971-5-0087-6}}

Category:Statistical mechanics

Category:Phase transitions

Category:Lev Landau