Density-Matrix Renormalization Group

The DMRG algorithm was an iterative procedure to approximately diagonalize Hamiltonians, which was shown to produce matrix-product states. These days, a slightly more modern version of the algorithm is used, which directly operates in the MPS formalism.

The goal of this algorithm, as we are going to use it, is to compute the minimum of an energy functional:

\[\mathrm{argmin}_\psi E[\psi] := \mathrm{argmin}_\psi \frac{\langle{\psi|H|\psi}}{\langle{\psi|\psi}},\]

where the state \(\psi\) is a matrix product state. The algorithm works by reinterpreting this functional as a function over the individual tensors.

In a simple version of the algorithm, we could define \(E(X^{(k)})\) as the energy that results from substituting the k-th tensor by \(X^{(k)}\). As the picture below shows, the numerator and denominator are quadratic forms of the only tensor involved:

We can therefore write something like

\[E_{2,3}[\vec{X}] = \frac{\vec{X}^T H_\text{eff} \vec{X}}{\vec{X}^T\vec{X}},\]

where \(H_\text{eff}\) is an effective quadratic form that results from contracting all MPS and MPO tensors except \(X\) (which we have reshaped as a vector \(\vec{X}\)). This functional can be minimized by solving the eigenvalue equation

\[H_\text{eff} \vec{X} = E_\text{min} \vec{X}.\]

In that algorithm, we would find the best tensor for the site k=1, substitute that tensor into the state; compute the best tensor for the site k=2, and so on and so forth.

A more clever version works by operating on pairs of tensors. The idea is that we re-create the same MPS by joining two tensors from neighboring sites. For instance, the figure below sketches how we would look at the expectation value of an MPO Hamiltonian \(H\) over a state \(\psi\) which is in canonical form with respect to the second site, and where we have replaced the second and third tensor by a combined one \(X\).

We would apply the same principles as before, solving an eigenvalue problem for the two-site tensor \(X\), but now we add a second step in which this tensor is optimally split into two smaller tensors, as explained in this manual. Note that, while in the single-site algorithm the size of the tensor \(X\) is fixed, in this algorithm the splitting of the two-site tensor will produce objects whose size will dynamically adapt the size of the virtual dimension—i.e. the amount of entanglement and correlations—to the needs of the problem-

The DMRG in SeeMPS implements this variant of the algorithm, sweeping over pairs of sites—e.g., (1,2), then (2,3), then (3,4), and so on—back and forth until the energy and the state converge.

dmrg(H[, guess, maxiter, tol, tol_up, ...])

Compute the ground state of a Hamiltonian represented as MPO using the two-site DMRG algorithm.