CTMRG

The corner transfer matrix renormalization group (CTMRG) algorithm iteratively computes the boundary tensors \(C^k\) and \(E^k\) that approximate the infinite extension of the double-layer tensor network [Nishino1996], [Baxter1978], [Orus2009].

Directional Moves

The boundary tensors are computed by iteratively absorbing rows or columns of the double-layer tensor network. Each absorption is called a directional move. An up-move at row \(y\) inserts the \(y+1\) row of double-layer tensors and absorbs it into the top boundary. The following diagram shows the tensors involved in a single up-move:

../_images/ctmrg_up_move.png

The top boundary tensors \(C^1, E^1, C^2\) and the inserted row \(a'\) are contracted together, and projectors \(P_1, P_2\) are applied to truncate the growing boundary bonds back to dimension \(\chi\):

../_images/ctmrg_absorption.png

The updated boundary tensors are:

\[\begin{split}C^{\prime 1}_{(x,y)} &= \mathrm{contract}\!\left(C^1_{(x,y+1)},\; E^4_{(x,y+1)},\; P^1_x\right) \\ E^{\prime 1}_{(x,y)} &= \mathrm{contract}\!\left(P^2_x,\; E^1_{(x,y+1)},\; a_{(x,y+1)},\; P^1_{x-1}\right) \\ C^{\prime 2}_{(x,y)} &= \mathrm{contract}\!\left(C^2_{(x,y+1)},\; E^2_{(x,y+1)},\; P^2_{x-1}\right)\end{split}\]

Down, left, and right moves are defined analogously. For a unit cell of size \(N_x \times N_y\), a complete absorption from one direction requires \(N_y\) (or \(N_x\)) consecutive moves.

Projector Calculation

Each absorption step increases the boundary bond dimension by a factor of \(D^2\). To maintain a maximum boundary bond dimension \(\chi\), the updated boundary tensors must be projected onto a lower-dimensional subspace. The projectors are computed from a truncated SVD of the contracted environment [Orus2009], [Corboz2014].

1. Build quarter tensors

Four quarter tensors \(Q_k\) for \(k = 1,2,3,4\) are formed by contracting corner, edge, and site tensors at each quadrant of the environment:

../_images/ctmrg_quarter_tensors.png

2. Form half-system tensors

Contract pairs of quarter tensors vertically to form the left and right halves of the environment:

\[R_1 = Q_1 \cdot Q_4, \quad R_2 = Q_2 \cdot Q_3\]

3. SVD and truncation

The full environment is formed by contracting \(R_1\) and \(R_2\) along the bonds to be projected (indicated by the dashed line):

../_images/ctmrg_projector.png

Decompose the contraction \(R_1 \cdot R_2\) via SVD along the bonds to be projected:

\[R_1 \cdot R_2 = U \, \Sigma \, V^\dagger\]

Truncate to the largest \(\chi\) singular values.

4. Form projectors

The projectors are constructed from the truncated singular vectors and values:

\[P_1 = R_1 \cdot U^\dagger \cdot \Sigma^{-1/2}, \quad P_2 = R_2 \cdot V \cdot \Sigma^{-1/2}\]

These projectors approximate a decomposition of the identity and are used in the boundary tensor absorption contractions above. Alternatively, half-system projectors may be formed from only one of \(R_1\) or \(R_2\), discarding correlations in the other half of the network [Corboz2014].

Convergence

CTMRG is iterated — performing directional moves in all four directions — until the boundary tensors reach a fixed point with respect to absorbing a full unit cell. The converged boundary tensors are then used for measurement of observables and, in the case of the full update, for constructing the tensor environment used in the ALS optimisation.

References

[Baxter1978]

    1. Baxter, Variational approximations for square lattice models in statistical mechanics, J. Stat. Phys. 19, 461 (1978).

[Corboz2014] (1,2)

  1. Corboz, T. M. Rice, and M. Troyer, Competing states in the t-J model: Uniform d-wave state versus stripe state, Phys. Rev. Lett. 113, 046402 (2014).