• View
• Edit
• History
• Print

PhaseTransitions

Indice

---

Sparse notes on phase transions

Free energy {$F$} is non analytic for some values of {$(x_i)=(p,V,T,\cdots)$} (for infinite systems)

At the critical point {$(x_i)$} two states have equal {$F$}.

Top

---

Ehrenfest:

Order of phase transition {$n$}, lowest integer such that {$d^nF/dx_i^n$} is discontinuous at the critical point, {$x_i$}

Mean-field Landau theory provides a model description for the appearance of these discontinuities and mean-field-corrected Ginzburg-Landau theory complements this description.

The Ehrenfest definition is missing the main distinction between phase transition, which is first-order (as for Ehrenfest) vs. continuous (all the others) and it is properly understood only considering fluctuations.

Top

---

Thermodynamic potentials and Maxwell relations

Since {$dS=dQ_{rev}/T$} the first law for a reversible transformation is

{$dU=TdS-pdV$}

Hence {$U=U(S,V)$}, {$T=\left(\frac{\part U}{\part S}\right)_V$} and {$p=\left(\frac{\part U}{\part V}\right)_S$}. Now define the entalpy

{$H=U+pV$}

whose differential is {$dH=TdS-pdV+pdV+Vdp=TdS+Vdp$}.

Hence {$H=H(S,p)$}, {$T=\left(\frac {\part H}{\part S} \right)_p$} and {$V=\left(\frac {\part H}{\part p}\right)_S$}. Now define the Helmholtz free energy

{$F=U-TS$}

whose differential is {$dF=TdS-pdV-TdS-SdT=-pdV-SdT$}.

Hence {$F=F(V,T)$}, {$p=\left(\frac{\part F}{\part V}\right)_T$} and {$S=\left(\frac{\part F}{\part T}\right)_V$}. Now define the Gibbs free energy

{$G=H-TS$}

whose differential is {$dG=TdS+Vdp-TdS-SdT=Vdp-SdT$}.

Hence {$G=G(p,T)$}, {$V=\left(\frac{\part G}{\part p}\right)_T$} and {$S=\left(\frac{\part G}{\part T}\right)_V$}.

Top

---

Fluctuations

A general thermodynamic relation is that

{$\langle \Delta U^2\rangle = C_V k_B T^2 $}

As long as the specific heat is finite, {$C_V \propto N$} and also {$U\propto N$}, hence the fluctuation ratio

{$\frac {\sqrt{\langle \Delta U^2\rangle}} {U} \propto \frac 1 {\sqrt{N}} $}

goes to zero (fluctuations are negligible) in the thermodynamic limit. However close to a continuous phase transition the specific heat diverges and so does the fluctuation ratio: fluctuations dominate.

Top

---

Critical exponents

The reference is A. Pelissetto, E. Vicari, Physics Reports 368 (2002 ) 549-727

For {$t\equiv\frac{T-Tc} T \rightarrow 0$} various thermodynamic quantities diverge according to power laws. Consider static properties, such as the magnetic susceptibility {$\chi$}, the coherence length {$\xi$}, the order parameter (the uniform or staggered magnetization {$M$}), the specific heat at fixed magnetic field {$C_H$}, the Green function at the critical point {$G$}. The exponent of their power laws are

	Critical exponents
	{$\gamma$}	{$\nu$}	{$\eta$}	{$\alpha$}	{$\beta$}
Models	{$\chi\propto |t|^{-\gamma}$}	{$\xi\propto (-t)^{-\nu}$}	{$G(x)=1/|x|^{d-2+\eta}$}	{$T_cC_H\propto |t|^{-\alpha}$}	{$|M|\propto (-t)^\beta$}
Mean-field *	1	1/2	0	0	1/2
Ising 3D	1.2372(5)	0.6301(4)	0.0364(5)	0.110(1)	0.3265(3)
Ising 2D	7/4	1	-	1/4	1/8
Heisenberg 3D	1.392(5)	0.71(1)	0.037(2)		0.367(3)
Heisenberg 2D (Quantum)	Orders only a T=0 (renormalized classical regime, see below)
XY 3D	1.317(1)	0.671(1)	0.037(1)	-0.015(1)
XY 2D	Kosterlitz-Thouless (not power-laws)

Dynamic properties, such as the characteristic time {$\tau$} that enters spin relaxation as detected from magnetic resonance and neutron diffraction, scale according to {$\tau\propto \xi^z$}.

The critical slowing down of fluctuations is experimentally detected for example in the critical divergence of the NMR or muon relaxation rate. Relaxations are governed by {$\chi^{\prime\prime}({\mathbf q},\omega)$}, hence by static and dynamic aspects. Scaling theory assigns a critical exponent {$n=\nu(z-2+d-\eta)$} to the rate, {$T_{1,2}^{-1}\propto t^{-n}$}. The dynamical exponent is {$z=2+\alpha/\nu$} for the Ising case, and {$z=d/2$}, for the Heisenberg case.

For the 3D case this turns out to be experimentally distinguishable, respectively 0.717 and 0.329 for Isisng and Heisenberg, better than for the static exponent {$\beta$}, respectively 0.3265 and 0.367.

Top

---

The 2D Heisenberg antiferromagnet

Mermin and Wagner's theorem dictates that isotropic magnetic order cannot survive at finite temperatures in 1 and 2 dimensions. The reason is that even at {$T\rightarrow 0$} the dispersion of spin waves ({$\omega=\rho_s q^2$} for ferromagnets (F), but {$\omega=\rho_s q$} for antiferromagnets (AF), {$\rho_s=1.13 J/2\pi k_B$} is called spin stiffness) is populated by enough low energy excitations to destroy order. It is easy to see this with F, according to

{$n_{\mbox{\rm magnons}}\propto \int_0^\infty \frac{g(\omega)d\omega}{\exp(\hbar\omega/k_B T) -1}$}

This expression is finite for {$d=3$} (where {$g(\omega)\propto \sqrt{\omega}$},F, {$\propto \omega^2$} AF ), but the number of magnons diverges for {$d=1,2$}. It is finite again only for {$T=0,D=2$}.

Since the low energy excitations dominate, the critical behaviour of the 2D Heisenberg antiferromagnet (2DHAF) is discussed in the long wavelength approximation, i.e. continuum with a short distance cut-off {$\Lambda^{-1}$}. This is given by the so-called non-linear sigma model, Chakravarty, Halperin, Nelson, with dimensions {$D=d+1$} (taking into account also time, that scales as {$1/k_BT$}), where the exchange interaction is parametrized by the inverse of the spin stiffness {$\rho_s$}, assuming an approximate linear spin wave dispersion with velocity defined at a scale {$\Lambda^{-1}$} (the true spin wave velocity is {$c=\sqrt2JaZ_c/\hbar$}, with {$Z_c=1.18$}).

Calculations with the renormalization group take care of what happens when the scale changes (typically, it diverges for reduced temperature {$t\rightarrow 0$}, and that is why a continuum, hydrodynamic limit will do for the critical regime).

In this approximation the {$d=2$} case, having no magnetic order at finite temperatures, is further shown to display an ordered Neel state at T=0 depending on the strength of the coupling, i.e. the inverse spin stiffness. The figure shows the coupling vs. T phase diagram of the 2DHAF (mind that it refers to an inverse coupling {$g\propto \rho_s^{-1}$}, with the zero-temperature-order line for values less than a critical value {$1/\rho_{sc}$} and the three regimes . The difference may be described when cooling a sample thorugh the hashed regions: for Renormalized Classical systems, such as the cuprates, along path (1), the correlation length {$\xi$} diverges exponentially for T going to zero, according to {$\xi=\frac {0.27\hbar c}{2\pi \rho_s} \, \frac {\exp{2\pi\rho_s/T}}{1+T/2\pi\rho_s} $} whereas for Quantum Disordered system, along path (2), it reaches a temperature independent value.

Real life 2D Heisenberg AF systems will have some residual interaction causing them to deviate from the ideal case. Both small anisotropy terms ora residual spin coupling between 2D layers - say, {$J^\prime$} - will produce a three dimensional ordering at a temperature that scales as {$J/\log(J/J^\prime)$}. Thus measuring {$\xi(T)$} close enough to {$T_N$} one will observe a cross-over to the classical 3D critical exponent {$\nu$}, but above that and still within the hashed region one does observe divergence towards {$T=0$}.

Top

---

The 2DHAF by NMR

(From Imai et al.). The spin-lattice relaxation of ⁶³Cu in La₂CuO₄ is given by the usual Moriya formula with the imaginary susceptibility of the S=1/2 of the Cu square lattice

{$\frac 1 {T_1} = \frac {2 k_B T}{g^2 \mu_B^2 \hbar^2} \sum_{\mathbf{q}} A_{\mathbf{q}} \frac {\chi^{\prime\prime}(\omega_L,\mathbf{q})}{\omega_L}$}

where the spatial FT of the main on-site hyperfine coupling {$A$} acts as a structure factor. Hence one gets

{$ \frac 1 {T_1} = \frac {0.35}{Zc}\, \frac {A_{\mathbf{q}}}{J\hbar} \, \frac {\xi}{a}\, \frac {(T/2\pi\rho_s)^{3/2}}{(1+T/2\pi\rho_s)^2}$}

and the diverging exponential dependence wins over the vanishing power law, as {$T\rightarrow0$}

Top

---

Dynamic critical exponents

(Follows the introduction of Hohenberg, Halperin RPM 49 453 (1977)) The physics that we have described above regards static properties, i.e. single time correlation functions and linear response to time-independent perturbations. Thermodynamics (single-time, equilibrium distributions) suffices.

Dynamic properties such as those measured e.g. by inelastic neutron and light (Raman) scattering, relaxation rates in magnetic resonance and transport coefficients. The typical one is the q-dependent width of the susceptibility {$\chi({\mathbf q},\omega)$}, {$\Gamma({\mathbf q})$}. Its critical exponent, z, is defined as {$\Gamma({\mathbf q})\propto\xi^{-z}$}. For instance, in neutron scattering the width of the energy scan at the mode of wave-vector q is proportional to {$\Gamma({\mathbf q})$}. In NMR, close to an AF phase transition, one can expect that the largest contribution to relaxation rates comes from the lifetime {$\tau_{AF}=\Gamma({\mathbf q}_{AF})^{-1}$} of the corresponding mode.

Van Hove's original idea was that these quantities are determined by interactions on an atomic length-scale, so that they should scale with the static short length correlation functions, i.e. {$\Gamma({\mathbf q})\propto\chi({\mathbf q})$}. This justifies qualitatively the observed critical slowing-down of fluctuations. However the proper application of renormalization group methods to this issue (dynamic scaling) shows that there is considerable mode-coupling, and that the critical behaviour of {$\Gamma({\mathbf q})$} as well as of the other two-times correlation functions may deviate completely from this simplified prediction.

Semi-microscopic models are conceptually required: starting from full microscopic models one can integrate out variations with wave-lengths less than a cut-off {$\Lambda^{-1}$}, intermediate between the lattice parameter and the sample dimensions ({$\Lambda$} is a wave-vector). Therefore these models are conceptually equivalent to hydrodynamics.

---

Top

---

Indice