Critical exponents with a multiscale entanglement renormalization Ansatz channel

We show how to compute the critical exponents of one-dimensional quantum critical systems in the thermodynamic limit. The method is based on an iterative scheme applied to the multiscale entanglement renormalization Ansatz for the ground-state wave function. We test this scheme to compute the critical exponents of the Ising and $XXZ$ model for which we can compare the method with the exact values. The agreement is at worst within few percent of the exact results already for moderate dimensions of the tensor indices.

Figure 1

(Color online) Graphical representation of a one-dimensional MERA for $L=16$ sites. The red/gray elements correspond to the disentaglers tensors $\chi$ while the blue/dark gray elements are the isometry tensors $\lambda$. The green/light gray element at the top of the MERA is a tensor which plays no role in the $L\to \infty$ limit. The yellow/very light gray box shows the tensor compound which determines the transfer superoperator $\Phi$ associated with the MERA (see Refs. 5, 12 and the note 16 for details).

Figure 2

(Color online) Modulus of the computed critical exponents ${\eta }_{\alpha }$ for the Ising model for $b=1$, $\gamma =1$ (green circle). The dashed red lines correspond to the theoretical values—see Table .

Figure 3

(Color online) Critical exponents and of the $XXZ$ model for various values of $\Delta$ for $m=4$ (black circles and blue crosses). The red dashed lines reports the theoretical values. The inset shows the correspondent error on the ground energy per site.