Problem Set 3: Difference between revisions
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<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>, | ||
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 | 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 | ||
<math>m_A = < S^{(A)} >; \; \; | <math>m_A = < S^{(A)} >; \; \; m_B = < S^{(B)} >.</math> | ||
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<math>m^{\dagger} = \frac{1}{2} (m_A - m_B ),</math> | <math>m^{\dagger} = \frac{1}{2} (m_A - m_B ),</math> | ||
For perfect ferromagnetic order <math>m = 1</math>, while for perfect antiferromagnetic order | For perfect ferromagnetic order <math>m = 1\;</math>, while for perfect antiferromagnetic order | ||
<math>m^{\dagger} = 1</math>. | <math>m^{\dagger} = 1</math>. | ||
(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 | (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 | ||
order parameters <math>m\;</math> and <math>m^{\dagger}</math>. | order parameters <math>m\;</math> and <math>m^{\dagger}</math>. | ||
(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. | (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> . | ||
(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>. | (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>. | ||
(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> | (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> | ||
. Show that <math>\chi^{\dagger}</math> blows up at the Neel temperature. | . Show that <math>\chi^{\dagger}</math> 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 (syntax error): {\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 (syntax error): {\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
The average magnetization then can be written as
and the so-called ”staggered” magnetization is defined by the difference between the two sublattices
For perfect ferromagnetic order , while for perfect antiferromagnetic order .
(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 and , and express them through the
order parameters and .
(2) Assume that , so that , and solve the mean-field equations by expanding in . Determine the Neel (ordering) temperature, and calculate the order-parameter exponent .
(3) Now consider a small external field , so that both order parameters can assume a nonzero value (Note: will be small). By keeping only the leading terms in and , calculate the uniform spin susceptibility , as a function of temperature. Plot as a function of temperature, and show that it has a cusp around .
(4) Imagine adding a ”staggered” external field , which would be positive on sublattice A, but would be negative on sublattice B. Concentrate on the system with no uniform field , and determine the behavior of the staggered susceptibility
. Show that blows up at the Neel temperature.
