Quantum transverse-field Ising model on an infinite tree from matrix product states

We give a generalization to an infinite tree geometry of Vidal’s infinite time-evolving block decimation (iTEBD) algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] for simulating an infinite line of quantum spins. We numerically investigate the quantum Ising model in a transverse field on the Bethe lattice using the matrix product state ansatz. We observe a second order phase transition, with certain key differences from the transverse field Ising model on an infinite spin chain. We also investigate a transverse field Ising model with a specific longitudinal field. When the transverse field is turned off, this model has a highly degenerate ground state as opposed to the pure Ising model whose ground state is only doubly degenerate.

Figure 1

The Bethe lattice (infinite Cayley tree).

Figure 2

Two successive Schmidt decompositions on a line allow us to find the $\Gamma$ tensor for the marked site and the two $\lambda$ vectors for the bonds coming out of it.

Figure 3

The three Schmidt decompositions on a tree required to obtain the $\Gamma$ tensor for the marked site and the three $\lambda$ vectors for the bonds emanating from it.

Figure 4

The parametrization and update rules for the infinite line.

Figure 5

(Color online) The two-layer, directed labeling of the tree.

Figure 6

(Color online) The two-step interactions for the infinite tree.

Figure 7

(Color online) Transverse Ising model on an infinite line. Fractional difference of the ground state energy obtained using MPS and the exact ground state, near the phase transition at $s_L=\frac{2}{3}$. The energy scale is logarithmic.

Figure 8

(Color online) Transverse Ising model on an infinite line. The first and second derivative with respect to $s$ of the ground state energy obtained via MPS compared with the exact result.

Figure 9

(Color online) Transverse Ising model on an infinite line. Magnetization obtained using MPS vs $s$, with $B_z={10}^{-8}$.

Figure 10

(Color online) Transverse Ising model on an infinite line. Log-log plot of magnetization vs $x-x_L$. We also plot a line with slope ${\beta }_L=0.125$.

Figure 11

(Color online) Transverse Ising model on an infinite line. Log-log plot of the correlation length vs $\left|x-x_L\right|$. We also plot a line with slope $-{\nu }_L=-1$. For each $\chi$ in the plot, we choose a numerical value of the critical point $s_L$ as the point at which the correlation length is maximal.

Figure 12

(Color online) Transverse Ising model on an infinite tree. The first and second derivative with respect to $s$ of the ground state energy obtained via MPS.

Figure 13

(Color online) Transverse Ising model on an infinite tree. Magnetization vs $s$, with $B_z={10}^{-8}$.

Figure 14

(Color online) Transverse Ising model on an infinite tree. Log-log plot of magnetization vs $x-x_T$, with $B_z={10}^{-8}$. We also plot a line with slope 0.41.

Figure 15

(Color online) Transverse Ising model on an infinite tree. A linear plot of the correlation length vs $s$.

Figure 16

(Color online) The NOT 00 model on an infinite line. The ground state energy and its first two derivatives with respect to $s$.

Figure 17

(Color online) The NOT 00 model on an infinite line. Magnetization as a function of $s$ and correlation length as a function of $s$.

Figure 18

(Color online) Ground state of the NOT 00 model on a line at $s=1$. Comparison of the exact Schmidt coefficients for a division across the middle of a finite chain and of the MPS values $\left(\chi =32\right)$ for an infinite line.

Figure 19

(Color online) The NOT 00 model on an infinite tree. The ground state energy and its first two derivatives with respect to $s$.

Figure 20

(Color online) The NOT 00 model on an infinite tree. Magnetization as a function of $s$ and correlation length as a function of $s$.

Figure 21

A system of classical spins on a lattice whose layers are Cayley trees. $A$ and $B$ denote the Boltzmann factors.

Figure 22

The $M=2$ Cayley tree.

Figure 23

Computing the probability distribution of ${\vec{s}}_0$ and ${\vec{s}}_d$.