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(New page: Problem Set 3: Ising antiferromagnet on a ”bipartite” lattice, is defined by the Hamiltonian H = J 2 X<ij> SiSj − hXi Si. Note that now the interaction between spins minimizes the ...)
 
m (beta)
 
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Problem Set 3: Ising antiferromagnet on a ”bipartite” lattice,
'''Ising antiferromagnet on a ”bipartite” lattice'''


is defined by the Hamiltonian
Hamiltonian:
H =
J
2 X<ij>
SiSj − hXi
Si.
Note that now the interaction between spins minimizes the energy when the
spins anti-allign, i.e. for Si = −Sj . A bipartite lattice is one that has two sublattices,
so that each spin on sublattice A interacts only with some spin on the
other sublattice B. In this case, in an antiferromagnetic state, each sublattice
assumes a uniform magnetization. We can introduce the magnetization for each
sublattice
mA = DS(A)
i E; mB = DS(B)
i E.


The average magnetization then can be written as
<math> H = \frac{J}{2} \sum_{<ij>} S_i S_j  -− h\sum_i S_i </math>,
m =
 
1
with the sum is over all nearest neighbor sites i and j. Note that now the interaction between spins minimizes the energy when the spins anti-allign, i.e. for <math>S_i = −- S_j\;</math> . A bipartite lattice is one that has two sublattices, so that each spin on sublattice A interacts only with some spin on the other sublattice B. In this case, in an antiferromagnetic state, each sublattice assumes a uniform magnetization. We can introduce the magnetization for each sublattice
2
 
(mA + mB) ,
<math>m_A = < S^{(A)} >; \; \; m_B = < S^{(B)} >.</math>
and the so-called ”staggered” magnetization is defined by the difference between
 
the two sublattices
 
m† =
The average magnetization then can be written as  
1
 
2
<math>m = \frac{1}{2} (m_A + m_B ),</math>
(mA − mB) .
 
For perfect ferromagnetic order m = 1, while for perfect antiferromagnetic order
and the so-called ”staggered” magnetization is defined by the difference between the two sublattices
m† = 1.
 
(a) Use Weiss mean-field decoupling to replace one of the spins in the Hamiltonian
<math>m^{\dagger} = \frac{1}{2} (m_A - m_B ),</math>
by its thermal average. The Weiss field experienced by a given spin is
 
then proportional to the sublattice magnetization on the other sublattice. Write
For perfect ferromagnetic order <math>m = 1\;</math>, while for perfect antiferromagnetic order
down self-consistent equations for mA and mB, and express them through the
<math>m^{\dagger} = 1</math>.
order parameters m and m†.
 
(b) Assume that h = 0, so that m = 0, and solve the mean-field equations
 
by expanding in m†. Determine the Neel (ordering) temperature, and calculate
(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 <math>m_A\;</math> and <math>m_B\;</math>, and express them through the
the order-parameter exponent �.
order parameters <math>m\;</math> and <math>m^{\dagger}</math>.
(c*) 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 � = @m/@h, as a
(2) Assume that <math>h = 0\;</math>, so that <math>m = 0\;</math>, and solve the mean-field equations by expanding in <math>m^{\dagger}</math>. Determine the Neel (ordering) temperature, and calculate the order-parameter exponent <math>\beta</math> .
function of temperature. Show that � has a cusp around TN.
 
(d*) Imagine adding a ”staggered” external field h†, which would be positive
 
on sublattice A, but would be negative on sublattice B. Concentrate on the system
(3) Now consider a small external field <math>h > 0\;</math>, so that both order parameters can assume a nonzero value (Note: <math>m\;</math> will be small). By keeping only the leading terms in <math>h\;</math> and <math>m\;</math>, calculate the uniform spin susceptibility <math>\chi = \partial m/ \partial h</math>, as a function of temperature. Plot <math>\chi\;</math> as a function of temperature, and show that it has a cusp around <math>T_N\;</math>.
with no uniform field (h = 0), and determine the behavior of the staggered
 
susceptibility
 
�† =
(4) Imagine adding a ”staggered” external field <math>h^{\dagger}</math>, which would be positive on sublattice A, but would be negative on sublattice B. Concentrate on the system with no uniform field <math>(h = 0)\;</math>, and determine the behavior of the staggered susceptibility <math>\chi^{\dagger}= \partial m^{\dagger} / \partial h^{\dagger} </math>
@m†
. Show that <math>\chi^{\dagger}</math> blows up at the Neel temperature.
@h†
.
Show that �† blows up at the Neel temperature.

Latest revision as of 20:40, 3 February 2009

Ising antiferromagnet on a ”bipartite” lattice

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 } ,

with the sum is over all nearest neighbor sites i and j. Note that now the interaction between spins minimizes the energy when the spins anti-allign, i.e. 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 S_i = −- S_j\;} . A bipartite lattice is one that has two sublattices, so that each spin on sublattice A interacts only with some spin on the other sublattice B. In this case, in an antiferromagnetic state, each sublattice assumes a uniform magnetization. We can introduce the magnetization for each sublattice

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)} >; \; \; m_B = < S^{(B)} >.}


The average magnetization then can be written as

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 ),}

and the so-called ”staggered” magnetization is defined by 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^{\dagger} = \frac{1}{2} (m_A - m_B ),}

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 = 1\;} , while for perfect antiferromagnetic 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} .


(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}} .


(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 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} .


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


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

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