Real-space renormalization group for the transverse-field Ising model in two and three dimensions

The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent $\nu$. The resulting values of the critical exponent $\nu$ in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on ($2+1$)- and ($3+1$)-dimensional Bravais lattices yields accurate estimates of the critical exponents.

Figure 1

Construction of block spins in one dimension.

Figure 2

Construction of block spins in two dimensions. This block partition preserves the form of the Hamiltonian under renormalization-group transformations.

Figure 3

The order of the renormalization in the horizontal and vertical directions. With $A$, we renormalize the system in the horizontal direction and then in the vertical direction. $B$ has the reverse order.

Figure 4

The renormalization-group flow near the fixed point in the space of $k_h$ and $k_v$. With the symmetrization procedure, $k_{c,h}$ and $k_{c,v}$ become close to each other, the slope of the dominant flow (the thicker line) approaches $1$, and the subdominant direction becomes perpendicular.

Figure 5

The order of renormalization in three dimensions. With $A$, we renormalize the system in the horizontal, then vertical, and finally in the third direction. $B$ realizes the reverse order.