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phys212ln26 - Physics 212 Statistical mechanics II Fall...

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Physics 212: Statistical mechanics II, Fall 2006 Lecture XXV-XXVI For the past few lectures, we have been discussing some of the complex dynamical phenomena that can appear in classical statistical mechanics. In this final lecture, we turn to quantum systems where the dynamics and statics are both derived from the quantum-mechanical Hamiltonian. Our goal will be to show that in some cases there is a connection between quantum phase transitions at zero temperature, and classical phase transitions at finite temperature. We previously gave an argument that, because of loss of phase coherence at sufficiently long times and distances, finite-temperature critical points were essentially unmodified (at least as far as universal, long-distance properties) by quantum mechanics. At zero temperature, this no longer holds: if a system has a transition between two ground states as a function of some parameter, then at this transition quantum effects will generally be very important. We can divide such quantum phase transitions into two classes according to whether there is or is not a phase transition in the classical system at finite temperature. Experimentally, it is now possible in a number of systems to see signs of quantum critical behavior at low but nonzero temperature. An example we discuss that is of relevance to some magnetic materials is the quantum or transverse-field Ising model. First consider the ordinary Ising model in one dimension, H = - K i σ z i σ z i +1 - z i . (1) We can either think of the spins as classical Ising spins or as quantum spin-half variables, since even if the spins in the above are quantum spins, the above calculation gives the same result since we can go to the z basis where all operators in the Hamiltonian are diagonal. The quantum or transverse-field Ising model is just H = - K i σ z i σ z i +1 - x i . (2) It turns out that having the magnetic field point in a different direction than the easy-axis of the spins dramatically changes the physics. It turns out that this “quantum Ising model” has a zero- temperature phase transition that is in the universality class of the classical 2D Ising model, which was solved by Onsager. The reason you may see textbooks writing h = gK for the quantum Ising model is that, since the ground state is unchanged if we just multiply the entire Hamiltonian by a constant, we can think of the ground state as just a function of the dimensionless number g . The classical Ising partition function in one dimension can be written as, with M the number of spins, Z = σ z i M i =1 T 1 ( σ z , σ z i +1 ) T 2 ( σ z i ) (3) where T 1 ( σ z 1 , σ z 2 ) = exp( z 1 σ z 2 ), T 2 ( σ z i ) = exp( z i ). (These give the same transfer matrix T = T 1 T 2 that was derived earlier in the course.)But this product is equivalent to a matrix product: for 1
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