How does one bound the growth of the support of local operators in the Transverse-Field Ising Chain?

Consider the transverse-field Ising chain (TFIC) in a transverse-field $B$:

$$H_{TFIC}(B)\equiv -\sum_{j=1}^{N-1} \sigma^x_j\sigma^x_{j+1}+B\sum_{j=1}^N \sigma^z$$

At $B=0$, we have the classical Ising chain, and so there is a set of groundstates for this system with zero entanglement, namely the all-up and all-down states, respectively.

Let's try to understand the ferromagnetic phase of this model: we start by adiabatically turning on $B$, represented by the time-dependent unitary evolution operator $U(B)$ defined below:

$$U(B)\equiv \lim_{T\to \infty}\mathcal T\exp\left(-i\int_0^Tdt\, H(tB/T) \right)$$

From the perspective of the groundstate subspace, because the ferromagnetic phase $B<1$ of the TFIC is gapped, then $U(B)$ must be a local unitary (LU) operator as long as $B<1$:

$$|\psi_0(B)\rangle \,=U(B)|\uparrow\rangle^{\otimes N},\,\,\,\,\,\,\,~~~~~|\psi_1(B)\rangle \,=U(B)|\downarrow\rangle^{\otimes N}$$

Since $U(B)$ is a LU operator, the symmetry-broken groundstates $|\psi_0(B)\rangle, |\psi_1(B)\rangle$ of the transverse-field Ising chain are short-range entangled (SRE) states.

My question is about the Heisenberg picture for the ferromagnetic phase of the TFIC, which is a much more general viewpoint from the perspective of calculating observables. An observable (e.g. magnetization) at $B\neq 0$ looks like

$$\langle\uparrow|^{\otimes N}U^\dagger (B)\, \sigma^x_j\, U(B)|\uparrow\rangle^{\otimes N}$$

Let's now just examine the operator in the middle:

$$\sigma^x_j(B) \equiv U^\dagger (B)\, \sigma^x_j\, U(B)$$

Clearly, since the TFIC is short-range entangled, $\text{supp}(\sigma^x_j(B))$ decays exponentially with distance from $j$. At $B=0$, the support vanishes except at $j$. How quickly does this support grow as $B<1$ increases from zero?

• The constants you get in the LR bound are easy to compute: The proof is constructive. Maybe you want to know the real velocity with which information can spread in the TFIM? – Norbert Schuch Sep 14 '17 at 5:44
• I guess a simple example is the unequal time commutator of $\sigma^z_j$ with $\sigma^z_k$. This should uniformly vanish at B=0, but then become some exponentially decaying function for nonzero B. Anything that captures that B-dependence is helpful. Perhaps I need to rewrite the question. – David Roberts Sep 14 '17 at 6:04
• Just rewrote the question. – David Roberts Sep 14 '17 at 6:26
• This is quite a different question now! It is not even clear how LR-bounds would help here. One could instead do quasi-adiabatic evolution (to which a LR-bound would apply), but this does sth. different to operators. What are you actually interested in? – Norbert Schuch Sep 18 '17 at 9:30
• On a different note, it is not clear to me whether the $U$ is well-defined the way you define it. (Not even when asking for its action on some quasi-local algebra of operators.) – Norbert Schuch Sep 18 '17 at 9:31