Landau theory of charge order: symmet...

Landau theory of charge order: symmetry breaking

Landau theory is a good way to describe the symmetry breaking of of charge order using group theory.

Suppose the charge pattern can be described as

\rho(\vec{r})=\rho_0(\vec{r})+\delta\rho(\vec{r})

where ρ is the total charge density, ρ₀ is the charge density of the high temperature phase corresponding to symmetry group G₀ and δρ correspond to one or more irreducible representations (except the identity) of G₀.

At high temperature δρ=0, giving the high symmetry phase. While at low temperature, δρ≠0, giving the low symmetry phase.

A good example of cubic to tetragonal structural phase transition can be very revealling.

In this case:

\rho(\vec{r})=\sum_i\rho_0(\vec{r}-\vec{R}_i)+\rho_1(\vec{r}-\vec{R}_i+z_0)

where R_i represents the cubic lattice and z₀ is the distortion along z direction.

One can look at the character table (below) and recognize that the z₀ distortion corresponds T_1u irreducible representation (IR).



A closer look at the character table shows that only C₄, σ_h and σ_d leave z₀ invariant. Therefore, in the low temperature (low symmetry phase), the group has E, C₄, σ_d and σ_h, which gives a new group C_4v, corresponding to the symmetry of the low T phase.

Useful special symbols

Useful special symbols

Greek letters	α	β	γ	δ	ε	ζ	η	θ	ι	κ
	λ	μ	ν	ξ	ο	π	ρ	ς	σ	τ
	υ	φ	χ	ψ	ω
	Α	Β	Γ	Δ	Ε	Ζ	Η	Θ	Ι	Κ
	Λ	Μ	Ν	Ξ	Ο	Π	Ρ		Σ	Τ
	Υ	Φ	Χ	Ψ	Ω
Roman letters	ℂ	℃	ℇ	℉	ℐ	ℑ	ℒ	ℏ	℞	Ω
	ℋ	ℛ	ℱ	Å	ℰ
	𝓖
Additional Greek letters	ħ	ϴ	ϵ
Arrows	↔	↕	￫	￩	￬	￪	→	←	↓	↑
	↙	↘	↗	￨
Operators	±	∓	×	÷	∙	†	≈	≣	≡	≠
	∫	∏	∑	∆	∇	∂	∅
	√	∛	∜
	∝	∞
Geometry	‖	⊥
Fractions	¼	½	¾
Units	Å

Conductivity of conductor

Conductivity of conductor

How to define a conductor? The very natural way to say that a conductor is a material that conducts electric current. But for a physicist, that's not enough, most materials do have a measurable conductivity, just the numbers vary by 20 order of magnitude. It is hard to draw the lines between conductors, semiconductors and insulators. However, some good examples (300 K) should be able to at least give us some idea.

Material	Conductivity Sm-1 or 1/(Ωm)	Ref
Silver	63.0 × 106
Copper	59.6 × 106
Gold	45.2 × 106
Mercury	1.0× 106
Carbon	2.8 × 104
Fe3O4	103
LuFe2O4	~1	[1] After breakdown 60V/cm-1
Germanium	2.2
LuFe2O4	10-2	[1] Before breakdown 10V/cm-1
Silicon	1.5× 10-3
BiFeO3	~1× 10-3	[2] 1kV/cm-1
Deionized water	5.5 × 10-6
Glass	10-10 -10-14
Paraffin	10-17
Teflon	10-22 - 10-24

Reference:
[1] Title: Nonlinear current-voltage behavior and electrically driven phase transition in charge-frustrated LuFe2O4
Author(s): Zeng LJ, Yang HX, Zhang Y, et al.
Source: EPL   Volume: 84   Issue: 5 Article Number: 57011   Published: DEC 2008

[2]Title: Switchable Ferroelectric Diode and Photovoltaic Effect in BiFeO3
Author(s): Choi T, Lee S, Choi YJ, et al.
Source: SCIENCE   Volume: 324   Issue: 5923   Pages: 63-66   Published: APR 3 2009

Frustrated spin triangles

Frustrated Ising spin triangles

When we talk about spin liquid, one often invoke the frustrated spin system, for which the classical example is the Ising spin triangle with antiferromagnetic interaction.

Let's then look at such a model system see how it behaves, which will be very revealing for understanding more complicated system.

1. Hamiltonian and basis

Suppose there is a (localized) spin triangle of sites A,B,C, in the language of second quantization, there are 8 possible states, which we take as the basis:

$\phi_1 =|\uparrow_A\uparrow_B\uparrow_C>$;
$\phi_2 =|\uparrow_A\uparrow_B\downarrow_C>$;
$\phi_3 =|\uparrow_A\downarrow_B\uparrow_C>$;
$\phi_4 =|\uparrow_A\downarrow_B\downarrow_C>$;
$\phi_5 =|\downarrow_A\uparrow_B\uparrow_C>$;
$\phi_6 =|\downarrow_A\uparrow_B\downarrow_C>$;
$\phi_7 =|\downarrow_A\downarrow_B\uparrow_C>$;
$\phi_8 =|\downarrow_A\downarrow_B\downarrow_C>$;

The spin Hamiltonian will be:
$H=\frac{1}{2}[\sum_{i \ne j}^{}J_{ij} S^z_iS^z_j+(S^+_iS^-j+S^-_iS^+_j)/2]$

2. Eigenstates

We can diagonalize the Hamitonian and get the eigenstates and eigenenergies.

$\xi _i=\sum_{j}^{}c_{ij}\phi_{j}$


Table:

	E=-3/4J				E=3/4J
	ξ1	ξ2	ξ3	ξ4	ξ5	ξ6	ξ7	ξ8
ci1	0.000	0.000	0.000	0.000	1.000	0.000	0.000	0.000
ci2	-0.816	0.000	0.000	0.000	0.000	0.577	0.000	0.000
ci3	0.408	0.707	0.000	0.000	0.000	0.577	0.000	0.000
ci4	0.000	0.000	0.000	0.816	0.000	0.000	-0.577	0.000
ci5	0.408	-0.707	0.000	0.000	0.000	0.577	0.000	0.000
ci6	0.000	0.000	0.707	-0.408	0.000	0.000	-0.577	0.000
ci7	0.000	0.000	-0.707	-0.408	0.000	0.000	-0.577	0.000
ci8	0.000	0.000	0.000	0.000	0.000	0.000	0.000	1.000

It turns out that the system become two energy subspaces.

The first subspace corresponds to degenerate ground states with energy -3/4J.
One can see that |S_z|= 0 for those states.

The second subspace corresponds to a spin quartet with S=3/2.

3. Field and temperature dependence:  magnetization plateau

By varying temperature and magnetic field, one can study the magnetization change.
Above is the result we found for different J value. Starting from left are J=0,1,2,3,4,5,6,7,9,10,11

1) J=0 corresponds to isolated spins, which is actually Brolluvin function
2) J>=9: there is a clear plateau at low magnetization which corresponds to the saturation magnetization of the first subspace of the eigenstates.
3) J=1-7: intermediate cases

4. Field and temperature dependence.

In real experiment, one can not vary J unfortunately. Instead, we only have control over B and T. Next, we show the B,T dependence of the magnetization with fixed J.

In this picture, we can see that at very low temperature, the two plateau is obvious.

One can show that a system with 6 Ising spin which form triangular lattice also has similar behavior, in the sense that there is a low magnetization plateau of 1/3 of the magnitude. The 10 Ising spin system is not calculable for me however due to the computational difficulty but one can imaging the similarity. The key is that for the frustrate the spin system, there is a ground state with huge degeneracy which can behave like a paramagnetic system with reduced spin magnitude.

5. specific heat

As shown in the above figure (J=1,2,3 from the left), the specific heat has a peak at a energy scale proportional to the exchange interaction.

This is actually very typical behavior of two-level system.

6. specific field in magnetic fields

From left to right, the curves correspond to different magnetic fields. One can see that at intermediate fields, there are two peaks in the specific heat.