Solution to Set 3

Ferromagnetism

Group 3 is the best

Ferromagnetism Ferromagnetic ordering of microscopic magnets (the magnetic moments of individual particles) Books Topics Magnetic Order Ferromagnetism Weiss field Solid State Physics Section 8.3 Section 8.4 Soft Condensed Matter Section 3.2

Given Information

• Classical Ising antiferromagnet on a ”bipartite” lattice given by Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \frac{J}{2} \sum_{<ij>} S_i S_j -− h\sum_i S_i } ,

• Magnetization for Lattice A: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_A = < S^{(A)} > \;}

• Magnetization for Lattice B: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_B = < S^{(B)} > \;}

• Average magnetization: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \frac{1}{2} (m_A + m_B ),}

• "Staggered” magnetization: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger} = \frac{1}{2} (m_A - m_B ),}

(Note: It's the difference between the two sublattices)

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 1\;} for perfect ferromagnetic order

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger} = 1} for perfect antiferromagnetic order

Part 1

Use Weiss mean-field decoupling to replace one of the spins in the Hamiltonian by its thermal average. The Weiss field experienced by a given spin is then proportional to the sublattice magnetization on the other sublattice. Write down self-consistent equations for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_A\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_B\;} , and express them through the order parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger}} .

Given the Hamiltonian for a classical Ising antiferromagnet on a ”bipartite” lattice:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \frac{J}{2} \sum_{<ij>} S_i S_j -− h\sum_i S_i }

where

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\;} = Hamiltonian, the total energy of the system
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J\;} = Interaction Energy where J < 0 because it’s anti-ferromagnetic
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\;} = Spin with value of 1 or -1
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h\;} = Magnetic field energy from external sources that breaks the symmetry

Mean field approximation

Each spin on lattice A feels an average magnetization due to the sins on lattice B (and vice versa).

The magnetization on each sublattice:

• Sublattice A: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_A = < S_i^{A} > \;}
• Sublattice B: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_B = < S_j^{B} > \;}

Weiss Mean Field Approximation

Use this to decouple the spins to find the energy of each spin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\epsilon A \;}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{i}^{A} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{J}{2}S_i \sum_{j \epsilon B}^{z}\left \langle S_{j}^{B} \right \rangle - hS_i} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\left ( \frac{J}{2}zm_B - h \right ) S_i} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = B_{eff}^{B}S_i}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{i}^{B} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{J}{2}S_j \sum_{j \epsilon A}^{z}\left \langle S_{i}^{A} \right \rangle - hS_j} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\left ( \frac{J}{2}zm_A - h \right ) S_j} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = B_{eff}^{A}S_j}

Conditions of the magnetization

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{A} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left \langle S_{i}^{A} \right \rangle \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = Z^{-1} \sum_{\left \{S \right \}} S_{i}e^{-\beta \left ( \frac{J}{2}zm_{B}-h \right )S_{i}} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \tanh \left ( -\frac{J\beta}{2}zm_{B}+h\beta \right ) \;}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{B} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left \langle S_{i}^{B} \right \rangle \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = Z^{-1} \sum_{\left \{S \right \}} S_{i}e^{-\beta \left ( \frac{J}{2}zm_{A}-h \right )S_{i}} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \tanh \left ( -\frac{J\beta}{2}zm_{A}+h\beta \right ) \;}

Average Magnetization

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\equiv \frac{1}{2}\left ( m_{A} + m_{B} \right ) \;}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \frac{1}{2}\left [ \tanh \left ( -\frac{J\beta }{2}z\left ( m-m^{\dagger} \right ) + h\beta \right ) + \tanh \left ( -\frac{J\beta }{2}z\left ( m+m^{\dagger} \right ) + h\beta \right ) \right ] \;}

Staggered Magnetization

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger} \equiv \frac{1}{2}\left ( m_{A} - m_{B} \right ) \;}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger} = \frac{1}{2}\left [ \tanh \left ( -\frac{J\beta }{2}z\left ( m-m^{\dagger} \right ) - h\beta \right ) + \tanh \left ( -\frac{J\beta }{2}z\left ( m+m^{\dagger} \right ) + h\beta \right ) \right ] \;}

Part 2

Assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = 0} , so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 0} , and solve the mean-field equations by expanding in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger}} .

Determine the Neel (ordering) temperature, and calculate the order-parameter exponent.

The Néel Temperature

Chapter 8.4 of Solid State Physics

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_N = \frac{C \lambda}{2}}

where

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_N\;} is Néel Temperature, the onset temperature for antiferromagnetism

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = \frac {N \mu_0 g^2 \mu_b^2}{k_B}} is Curie constant

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = \frac{4 \sum_{j \neq i} \jmath_{i j}} {N \mu_0 g^2 \mu_b^2} = - \frac{4 z \jmath} {N \mu_0 g^2 \mu_b^2}}

Order Parameter Exponent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \;}

Expand Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger}} to the 3rd order

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{A} = -m_{B}}

$2m^{\dagger }=\tanh \left({\frac {J\beta }{2}}zm^{\dagger }\right)-\tanh \left(-{\frac {J\beta }{2}}zm^{\dagger }\right)\;$

$2m^{\dagger }=2\left({\frac {J\beta }{2}}zm^{\dagger }-{\frac {\left({\frac {J\beta }{2}}zm^{\dagger }\right)^{3}}{3}}\right)\;$

$1={\frac {J\beta }{2}}z-{\frac {\left({\frac {J\beta }{2}}z\right)^{3}m^{\dagger 2}}{3}}\;$

$m^{\dagger }={\frac {1}{{\frac {J\beta }{2}}z}}z{\sqrt {3\left(1-{\frac {2T}{Jz}}\right)}}\;$

$m^{\dagger }={\frac {2T}{Jz}}{\sqrt {3\left({\frac {2Jz}{2Jz}}-{\frac {2T}{Jz}}\right)}}\;$

$m^{\dagger }={\frac {2T}{Jz}}{\sqrt {\frac {2}{Jz}}}{\sqrt {3\left({\frac {Jz}{2}}-T\right)}}\;$

$m^{\dagger }=T\left({\frac {2}{Jz}}\right)^{\frac {3}{2}}{\sqrt {3\left(T_{N}-T\right)}}\;$

$\beta ={\frac {1}{2}}$

Part 3

Now consider a small external field $h>0\;$, so that both order parameters can assume a nonzero value (Note: $m\;$ will be small). By keeping only the leading terms in $h\;$ and $m\;$, calculate the uniform spin susceptibility $\chi =\partial m/\partial h$, as a function of temperature. Plot $\chi \;$ as a function of temperature, and show that it has a cusp around $T_{N}\;$.

Average magnetization as a function of small external field m(h)

$m\equiv {\frac {1}{2}}\left(m_{A}+m_{B}\right)\;$

$m={\frac {1}{2}}\left[\tanh \left(-{\frac {J\beta }{2}}z\left(m-m^{\dagger }\right)+h\beta \right)+\tanh \left(-{\frac {J\beta }{2}}z\left(m+m^{\dagger }\right)+h\beta \right)\right]\;$

$2m=\left[\beta -\beta \tanh ^{2}\left({\tfrac {1}{2}}J\beta zm^{\dagger }\right)\right]h+J\beta zm\left[-1+\tanh ^{2}\left({\tfrac {1}{2}}J\beta zm^{\dagger }\right)\right]$

$m(h)={\frac {\left[\beta -\beta \tanh ^{2}\left({\tfrac {1}{2}}J\beta zm^{\dagger }\right)\right]h}{2+J\beta zm\left[1-\tanh ^{2}\left({\tfrac {1}{2}}J\beta zm^{\dagger }\right)\right]}}\;$

Uniform Spin Susceptibility

$\chi (h)={\frac {\partial m}{\partial h}}\;$ $={\frac {\left[\beta -\beta \tanh ^{2}\left({\tfrac {1}{2}}J\beta zm^{\dagger }\right)\right]}{2+J\beta z\left[1-\tanh ^{2}\left({\tfrac {1}{2}}J\beta zm^{\dagger }\right)\right]}}\;$

Susceptibility vs. Temperature Graph

There is a cusp around $T_{N}$

Part 4

Imagine adding a ”staggered” external field $h^{\dagger }$, which would be positive on sublattice A, but would be negative on sublattice B. Concentrate on the system with no uniform field $(h=0)\;$, and determine the behavior of the staggered susceptibility $\chi ^{\dagger }=\partial m^{\dagger }/\partial h^{\dagger }$. Show that $\chi ^{\dagger }$ blows up at the Neel temperature.

Staggered Spin Susceptibility

$\chi ^{\dagger }(h)={\frac {\partial m^{\dagger }}{\partial h^{\dagger }}}\;$

Case $T<T_{N}$:

$\lim _{h\to 0}{\frac {\partial m^{\dagger }}{\partial h^{\dagger }}}\;$ $={\frac {1-\tanh ^{2}\left({\tfrac {1}{2}}Jz\left({\tfrac {2}{Jz}}\right)^{\tfrac {3}{2}}{\sqrt {3\left|T_{N}-T\right|}}\right)}{2T+Jz\left[1-\tanh ^{2}\left({\tfrac {1}{2}}Jz\left({\tfrac {2}{Jz}}\right)^{\tfrac {3}{2}}{\sqrt {3\left|T_{N}-T\right|}}\right)\right]}}\;$ $={\frac {1}{2T+Jz}}\;$

Case $T>T_{N}$:

$\lim _{h\to 0}{\frac {\partial m^{\dagger }}{\partial h^{\dagger }}}\;$ $={\frac {1}{T+Jz}}\;$

At Neel Temperature

$\lim _{T\to T_{N}}\chi ^{\dagger }(h)={\frac {d\chi }{dT}}|_{T\to T_{N}^{-}}-{\frac {d\chi }{dT}}|_{T\to T_{N}^{+}}\;$ $={\frac {1}{2T+Jz}}-{\frac {1}{T+Jz}}\;$ $={\frac {-T}{2T^{2}+3TJz+J^{2}z^{2}}}\;$ $=\mathbf {constant} \;$