Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture XXVII
Last time we discussed a formal relationship between the long-wavelength limit of the classical
Ising chain in one dimension and the quantum dynamics of a single quantum spin, with Hamiltonian
HQ=E0−∆
2ˆσx−˜
hˆσz.(1)
Our goals in the first part of the lecture will be to make the nature of this “quantum-classical
mapping” clearer and to understand how it can be used to understand the phase diagram of these
magnetic systems.
The steps in what we did last time were to write the partition function of the classical chain in
the form
Z=T r(T1T2)M(2)
where T1T2is the transfer matrix. For large correlation length (low temperature) the transfer
matrix could be rewritten as T1T2≈e−aHQ, so that the partition function is effectively that of the
single quantum spin:
Z=T r exp(−HQ/T ) (3)
with 1/T =Lτ=Ma. Note that as M→ ∞, the effective temperature in the quantum model goes
to 0.
Recall that in deriving the above results we had to assume that the correlation length of the
classical model was much larger than the correlation length. When this is true, the physics is
largely independent of short-distance details: for instance, adding a next-nearest-neighbor interac-
tion would change the problem quantitatively by changing the effective splitting ∆ in the quantum
model, but not change the qualitative picture. This means that the quantum Hamiltonian HQdoes
not correspond to just one classical model in one higher dimension; it corresponds to the scaling
limit of many different classical Hamiltonians, since HQis only sensitive to the long-length-scale
physics of the classical problem. One usually just tries to choose the simplest classical model from
the set of classical models that relate to a given HQ.
An intuitive picture of the above quantum-classical (QC) mapping is that a quantum model in
ddimensions at temperature Tis equivalent to the scaling limit of a classical model in a strange
“slab” geometry: the classical model has dinfinite dimensions and a d+ 1th dimension of length
proportional to β= (kT )−1. Note also that the scaling limit depended on taking certain combi-
nations in the classical model to be finite and correspond to the parameters E0,˜
h, and ∆ of the
quantum model.
We argued in class, and can see explicitly using the methods developed in 212, that it is the long-
length-scale physics of the scaling limit that controls the universal properties near a continuous (a.
k. a. second-order) phase transition. This means that quantum phase transitions in the quantum
model can be related using the QC mapping to classical transitions in the classical analogue.
The case we studied so far, the classical Ising chain in one dimension and single Ising spin, has no
interesting phase transition. In one more dimension, we start to see interesting results: the quantum
(transverse-field) Ising chain in one dimension at zero temperature has a transition as a function of
g(the ratio of magnetic field to Ising coupling) that is equivalent to the 2D classical Ising model,
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