Shielding property for thermal equilibrium states in the quantum Ising model

We show that Gibbs states of nonhomogeneous transverse Ising chains satisfy a shielding property. Namely, whatever the fields on each spin and exchange couplings between neighboring spins are, if the field in one particular site is null, then the reduced states of the subchains to the right and to the left of this site are exactly the Gibbs states of each subchain alone. Therefore, even if there is a strong exchange coupling between the extremal sites of each subchain, the Gibbs states of the each subchain behave as if there is no interaction between them. In general, if a lattice can be divided into two disconnected regions separated by an interface of sites with zero applied field, then we can guarantee a similar result only if the surface contains a single site. Already for an interface with two sites we show an example where the property does not hold. When it holds, however, we show that if a perturbation of the Hamiltonian parameters is done in one side of the lattice, then the other side is completely unchanged, with regard to both its equilibrium state and dynamics.

Figure 1

Representations of the original and of the dual chains. The sites of the dual chain correspond to the links of the original chain and vice-versa.

Figure 2

(a) An example of sets $A$, $X$, $S$, $B$, and $Y$. (b) Example of a system with two sites on the interface, which does not satisfy the shielding property. The reduced state on site 4 has dependence on the external magnetic field $h_1$ applied on site 1.

Figure 3

Two different arrangements are considered here for the same lattice. The subset between the red lines is the set $S$, where the external magnetic field is null. Outside the red lines there is the subset $B$ and inside the set $A$.

Figure 4

Temporal evolution of the magnetization of each site of a transverse quantum Ising chain. The external magnetic field of site 15 was fixed null and the evolution was calculated via tDMRG.

Figure 5

Board game associated to a chain when $h_3=0$.

Figure 6

Scheme to understand the product ${\kappa }_{\left[1\right]}{\kappa }_{\left[2\right]}{\kappa }_{\left[3\right]}$ as a “board game.” The rows represent different steps ($n=3$) of a board game with $L=3$ white pieces. The pieces can move from one vertex to another only when they are connected by continuous lines.

Figure 7

An example of a more general lattice.

Figure 8

Scheme for the proof analogous to Fig. 6.

Figure 9

Graphics of the value of the magnetization $\langle {\sigma }_4^x\rangle$ of site 4 in function of the external magnetic field $h_1$ applied in site 1. Each graphic was done for one different value of $\beta$, which are $\beta =1$, 4, and 7. Note that the scale of each graphic is different.