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Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture XVIII
The last lecture used the Holstein-Primakoff boson representation of spin operators
S+^ = (
√ 2 S − nb)b, S−^ = b†(
√ 2 S − nb), Sz^ = S − nb. (1)
in order to understand spin waves around both ferromagnetic and antiferromagnetic ground states. We first complete the antiferromagnetic case from last time, then use the coherent state represen- tation to define a path integral for calculating spin correlation functions.
Our result for the spin-wave spectrum of the antiferromagnet was
H 1 =
∑
k
ωk(α† kαk +
JSzN 2
ωk = |J|Sz
√ 1 − γ k^2. (2)
These operators αk were the result of a Bogoliubov transformation to eliminate anomalous products like bb, b†b†:
αk = cosh θkbk − sinh θkb†−k bk = cosh θkαk + sinh θkα†−k. (3)
Here tanh 2θk = −γk, (4)
and γk, introduced in the last lecture, was essentially the Fourier transform of the nearest-neighbor points (e.g., cos(ka) in one dimension).
Clearly the lowest-energy state corresponds to zero occupancy of all the α bosonic states. Hence it may seem that at zero temperature our spin-wave theory is justified since one of our requirements was that the number of bosons satisfy 〈n〉 2 S. But remember that this statement was about the number of the original b bosons, not the rotated α bosons! So we need to find out what 〈nb〉 is in the state with nα = 0. The reduction of the staggered moment per site is just, from the same calculation as in the ferromagnetic case,
∆ ˜m 0 = −
N
∑
i
〈b† i bi〉 = −
∑
k
〈b† kbk〉
= −
N
∑
k
〈(cosh θkα† k + sinh θkα−k)(cosh θkαk + sinh θkα†−k)〉
N
〈cosh^2 θkα† kαk + sinh^2 θkα−kα†−k〉. (5)
At zero temperature the only contribution from this is from the second term αα†. So at T = 0,
∆ ˜m 0 = −
N
∑
k
sinh^2 θbk = −
N
∑
k
cosh 2θk − 1 2
N
∑
k
√ 1 − tanh^2 2 θk
N
∑
k
√ 1 − γ k^2
Now consider the above sum in one dimension, where γk = cos(ka). Then the sum is loga- rithmically divergent at small |k| (can anyone explain to me why Auerbach has it diverging as 1 /k?), and there is no long-range order in the ground state. In two dimensions, as before, we have order at T = 0 but no order for finite temperature. However, the moment is reduced by quantum fluctuations even in the ground state relative to the maximum moment in the N´eel state, unlike in the ferromagnet.
Finally we can make a table to summarize what spin-wave theory has taught us about the ferromagnet and antiferromagnet in various dimensions on a bipartite lattice. We write ??? for the 1D antiferromagnet at zero temperature because, as we will see in the next few lectures, its behavior is quite complicated: for integer spin it is gapped and truly disordered (correlations fall off exponentially), while for half-integer spin it is critical (correlations fall off algebraically).
Ferromagnet Antiferromagnet d = 1, T = 0 Ordered ??? d = 1, T > 0 Disordered Disordered d = 2, T = 0 Ordered Ordered d = 2, T > 0 Disordered Disordered d = 3, T = 0 Ordered Ordered d = 3, T > 0 Ordered (low T ) Ordered (low T )
Our goal in the next two lectures will be, first, to develop a way of calculating measurable quantities at finite temperature, which will involve using some deep principles of statistical me- chanics to relate the response of a system to a small perturbation, to its correlation functions in imaginary time. We will see along the way that there is an interesting connection between quantum problems at zero temperature in d dimensions and classical problems at nonzero temperature in d + 1 dimensions.
The motivation for the path-integral method is essentially as follows: previously we didn’t bother to introduce path integrals for ordinary fermionic correlation functions (Green’s functions), and instead used Wick’s theorem to evaluate products of fields with respect to the noninteracting Hamiltonian. Unfortunately for spins Wick’s theorem is not as useful because the geometry of spin space is ignored if we just use a quadratic Lagrangian so that Wick’s theorem is valid. In general path integrals, like the one we develop in the next lecture, are good for gaining intuition about possible nonperturbative behavior; operator methods, like what we did before for the Green’s function, are more reliable for perturbation theory.
