Real-space renormalization in statistical mechanics

This review compares the conceptualization and practice of early real-space renormalization group methods with the conceptualization of more recent real-space transformations based on tensor networks. For specificity, it focuses upon two basic methods: the “potential-moving” approach most used in the period 1975–1980 and the “rewiring method” as it has been developed in the last five years. The newer method, part of a development called the tensor renormalization group, was originally based on principles of quantum entanglement. It is specialized for computing approximations for tensor products constituting, for example, the free energy or the ground state energy of a large system. It can attack a wide variety of problems, including quantum problems, which would otherwise be intractable. The older method is formulated in terms of spin variables and permits a straightforward construction and analysis of fixed points in rather transparent terms. However, in the form described here it is unsystematic, offers no path for improvement, and of unknown reliability. The new method is formulated in terms of index variables which may be considered as linear combinations of the statistical variables. Free energies emerge naturally, but fixed points are more subtle. Further, physical interpretations of the index variables are often elusive due to a gauge symmetry which allows only selected combinations of tensor entries to have physical significance. In applications, both methods employ analyses with varying degrees of complexity. The complexity is parametrized by an integer called $\chi$ (or $D$ in the recent literature). Both methods are examined in action by using them to compute fixed points related to Ising models for small values of the complexity parameter. They behave quite differently. The old method gives a reasonably good picture of the fixed point, as measured, for example, by the accuracy of the measured critical indices. This happens at low values of $\chi$, but there is no known systematic way of getting more accurate results within the old method. In contrast, the rewiring method seems to work poorly in fixed point calculations at low $\chi$. This stands in contrast to the known excellent performance of these newer methods in calculations of free energy, but not fixed points, at large values of $\chi$. Speculations are offered with a particular eye to seeing the reasons why the results of these two approaches are so different.

Figure 1

The setup for a potential-moving scheme on a square lattice. The old spins ($\sigma$) are marked by solid circles located at the vertices of a square lattice. Note that each such spin belongs to four different squares. These squares form the “blocks” for our calculation. The new spins $\mu$ appear in one-quarter of the blocks and are marked as empty circles. The thick dashed lines emanating from these new spins denote coupling terms that link these to the old spins. Each such coupling connects a single old spin to a new one. The potential moving places all the interactions between old spins in one-quarter of the squares (solid colored). The old spins around every one of these squares are connected only to themselves and to the neighboring new-spin variables. They can be summed over, giving a new effective coupling between adjacent new spins. The basic blocks in the starting situation are shown as squares of different textures in the upper left panel. The blocks in the final situation are shown outlined by the dashed lines in the bottom left panel. In the course of the calculation, the lengths of the sides of the blocks have been increased by a factor of 2.

Figure 2

Identification of the spin variables located at the vertices of a square unit cell.

Figure 3

The basic tensor network used here for the SVD renormalization calculations. Tensors are represented by solid shaded squares. Solid circles denote the position of the tensor’s indices. Every tensor has four indices. Every index assumes integer values between 1 and $\chi$ and is shared between exactly two tensors. Four indices determine the configuration of the statistical variable, and the corresponding tensor entry gives the statistical weight of the configuration. Note that the interactions represented by the tensors occupy half the available space. The left inset shows the labeling of the indices; the right inset shows the same tensor in the stick figure usually used in the literature.

Figure 4

The tensor network after rewiring. The old four-legged tensors are shown lightly shaded. They have disappeared and have been replaced by the three-index tensors $S^A$ and $S^B$ shown in darker shades. Each three-index tensor appears as a triangle with two of the old tensor indices and one new index at its vertices. These are, respectively, shown as solid circles and empty circles. Note the white squares are all empty of interactions. These squares are of two kinds: the ones flanked by colored triangles (three-legged tensors) and the one flanked by shaded triangles (the ghosts of disappeared four tensors). Each first-kind square permits the summation over the four old index variable at its corners and thereby the generation of interactions among the new indices. These white squares together with their four bounding triangles become the new tensors on the rescaled system.

Figure 5

The thermal exponent $x_T$ vs $\chi$. Empty circles denote the results that [25] obtained for an hexagonal lattice tensor product. Squares denote the exponents for a square lattice obtained here, and diamonds denote the results for a hexagonal lattice obtained here. All the preceding arise from a fixed-point–response-matrix analysis. In contrast, the solid circles denote exponents obtained through the fitting of the free energy in the vicinity of the critical point.