1. Lecture 11: the time-dependent variational principle and mean-field theory
In this and the following lectures we are going to apply the time-dependent variation principle (TDVP) to the variational classes we have met previously, namely, product states for quantum spin systems, mean-field states for bosons and fermions, and matrix product states.
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2. Example 1: a quantum spin system
In this section we apply the TDVP equations of motion to the description of the dynamics of the quantum spin systems studied in lecture 4. As before, we assume that these are an array of spin- degrees of freedom arranged in a regular one-dimensional lattice. Recall that a convenient basis for a single spin- degree of freedom is provided by the eigenstates of the spin operator, written and . The hilbert space of a single spin- is isomorphic to .
The hilbert space for our collection of such spin- degrees of freedom is thus given by
A general hamiltonian for a quantum spin system has the form
where the operator acts nontrivially only on spins and . We’ve identified spins and here, so this systems has periodic boundary conditions.
In this example we apply the TDVP to using as our variational class the set of all states with the form
i.e., the set of all symmetric product states. We don’t directly specify the dependence of , except to require that it is analytic. (Here we are using only one complex parameter to specify ).
In order to apply the TDVP we’re going to need to understand the tangent space to . Thus we differentiate a member with respect to to obtain
This allows us to calculate the Gram matrix which, for us, is simply a complex number (we only have one variational parameter):
which reduces to (using the symmetry of the product)
The other thing we need to calculate is the inner product . This is given, via a similar calculation, by
Suppose, for the sake of argument, that is designed so that it is always normalised (actually this is probably incompatible with the requirement that is analytic), i.e., for all , then
so that we obtain the simplified equation of motion
Otherwise, more generally, we obtain a more complicated expression.
3. Example 2: Bose-Einstein condensates
In this next example it is much more convenient to derive the TDVP directly from a lagrangian representation.
We use the set of field coherent states
Recall that a field coherent state obeys
we can write out the lagrangian:
To apply the TDVP we need to extremise , which we do using variational calculus: the Euler-Lagrange equations give
which reduce to
This is the (time-dependent) Gross-Pitaevskii equation (GPE) which, when , is also known as the nonlinear Schrödinger equation.