Problem Set 3: Difference between revisions
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Hamiltonian: | Hamiltonian: | ||
<math> H = \frac{J}{2} \sum_{<ij>} S_i S_j − h\sum_i S_i </math> | <math> H = \frac{J}{2} \sum_{<ij>} S_i S_j -− h\sum_i S_i </math> | ||
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 | 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 | ||
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<math>m^{\dagger} = 1</math>. | <math>m^{\dagger} = 1</math>. | ||
( | |||
by its thermal average. The Weiss field experienced by a given spin is | (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 | ||
then proportional to the sublattice magnetization on the other sublattice. Write | order parameters <math>m</math> and <math>m^{\dagger}</math>. | ||
down self-consistent equations for | |||
order parameters m and | |||
( | (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. | ||
by expanding in | |||
the order-parameter exponent | |||
( | (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. Show that <math>\chi</math> has a cusp around <math>T_N</math>. | ||
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 | |||
function of temperature. Show that | (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 | ||
( | |||
on sublattice A, but would be negative on sublattice B. Concentrate on the system | <math>\chi^{\dagger}= \partial m^{\dagger} / \partial h^{\dagger} </math> | ||
with no uniform field (h = 0), and determine the behavior of the staggered | |||
. | . | ||
Show that | Show that <math>\chi^{\dagger}</math> blows up at the Neel temperature. | ||
Revision as of 15:49, 27 January 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 }
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_A = < S^{(A)} >.}
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.
(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. 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}
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.