In lecture 9 we continued reading the papers on matrix product states that we began in lectures 7 and 8.
1. Lecture 10: the time-dependent variational principle
In this lecture we will discuss the time-dependent variation principle (TDVP), which is a powerful method to simulate the nonequilibrium dynamics of a general quantum system while remaining within a given variational class. The TDVP is due, as far as I’m aware, to Dirac. Additionally, describing the TDVP isn’t especially difficult, so it is rather surprising that this elegant method it hasn’t made into standard textbooks. The general framework of the time-dependent variational principle can be found in [P. Kramer and M. Saraceno, Geometry of the Time-Dependent Variational Principle in Quantum Mechanics (Springer-Verlag, Berlin) (1981)].
These notes can be found in pdf format here.
For the sake of brevity, we henceforth omit the time dependence of and thus of . Stationarity of the action under independent variations of and in the full Hilbert space yields the TDSE and its complex conjugate. But when we only have access to a subspace or manifold , we can still use the calculus of variations with respect to this action to define a time evolution of states ; this is the essence of the TDVP. To this end, we restrict to variations in the tangent plane of at the point . Assume that the manifold can be parametrized as
where we assume the dependence on the complex parameters to be analytic and explicitly denote the anti-analytic dependence . Furthermore, we introduce the notation for and , where we always use barred indices for the complex conjugate variables . Requiring stationarity of with respect to a variation results in the following Euler-Lagrange equations
We can also interpret this as the metric, indeed it is the pullback of the flat metric to , and — assuming linear independence of the tangent vectors — define the inverse metric as , such that . The TDVP thus results in
One can also obtain the Euler-Lagrange equations [Eq. (3)] geometrically, by looking for the coefficients which minimize
Whereas Hamiltonian evolution in is unitary and thus norm-preserving, this is no longer guaranteed for the projected evolution. In order to ensure norm preservation, we can define a modified Lagrangian , which results in
and the gradient of the normalized expectation value
These expressions can be easily interpreted. Under an infinitesimal variation, the norm or phase of a state changes if we move in the direction of . Norm conservation is thus obtained when we subtract from every tangent vector its component along by replacing it with , where the projector is given by
and thus . We indeed find
If the manifold allows for norm and phase variations of states, i.e. if , then we can define the contravariant vector such that . In this paragraph, we henceforth omit all arguments and for the sake of brevity. By definition we have that and we can conclude that has an eigenvalue zero, since , from which we immediately obtain the corresponding eigenvector. (Note that is a Hermitian matrix.) We now also define the covariant vector so that . With these definitions, we can write . Even though is not invertible, we can still define a pseudo-inverse as , so that and . Since we can rewrite as , we are allowed to apply this pseudo-inverse to the Euler-Lagrange equations in order to obtain
In principle, the component of along the zero eigenspace of can be chosen freely but, with the particular solution in the equation above, we satisfy , which is the required condition for norm (and phase) conservation.
If the manifold does not contain the freedom to change the norm and phase of a state, the modified metric would have the same rank as the original metric , which we assumed to be invertible. In particular, if , then and . The Euler-Lagrange equations following from or from are then identical.
Finally, by defining for every pair of functions and a Poisson bracket
we can write down the Euler-Lagrange equations as
For every operator acting on , we can define the expectation value so that its time evolution is governed by . The manifold is thus a symplectic manifold. From the antisymmetry of the Poisson bracket we find , which implies that the energy of the state is conserved under exact integration of the TDVP equations.
The symplectic properties of the time-dependent variational principle also conserve other symmetries. Assume that the Hamiltonian is invariant under the action of a symmetry operator (which should be a unitary operator), such that . In order to be able to transfer this symmetry to the manifold , we need to assume that for any state , the action of is mapped to a new state . Because of the unitarity of , we have , from which we obtain
The condition also allows to conclude and thus
The metric and the gradient thus transform covariantly under the symmetry transformation and can be used to transform Eq. (11) into
and its complex conjugate. By using the injectivity of the map , we can eliminate the Jacobians and in order to obtain the correct flow equations in terms of the new coordinates . One case that is not covered by this general derivation is when is an anti-linear operator, since will then depend on anti-holomorphically. Anti-linear transformations appear in quantum mechanics exclusively for time-reversal transformations. Let us denote with the operator of an elementary time-reversal transformation. Because of the anti-unitarity of and its commutation relation with the Hamiltonian (i.e. since we assume to be time-reversal invariant), we obtain and , which yields
These relations convert Eq. (11) into
or, by eliminating the Jacobian of the transformation,
Note that the signs of the two equations have been switched, which is necessary to revert the time evolution of the new coordinates . For a time-reversal invariant Hamiltonian and a variational manifold that contains the time-reversed state for each of its elements , the flow equations of the time-dependent variational principle are also time-reversal invariant.