Slow Propagation in Some Disordered Quantum Spin Chains

Abstract

We introduce the notion of transmission time to study the dynamics of disordered quantum spin chains and prove results relating its behavior to many-body localization properties. We also study two versions of the so-called Local Integrals of Motion (LIOM) representation of spin chain Hamiltonians and their relation to dynamical many-body localization. We prove that uniform-in-time dynamical localization expressed by a zero-velocity Lieb–Robinson bound implies the existence of a LIOM representation of the dynamics as well as a weak converse of this statement. We also prove that for a class of spin chains satisfying a form of exponential dynamical localization, sparse perturbations result in a dynamics in which transmission times diverge at least as a power law of distance, with a power for which we provide lower bound that diverges with increasing sparseness of the perturbation.

Appendix A: Lieb–Robinson Bounds

In this appendix we develop a bound on the velocity of propagation under the Heisenberg dynamics which ignores interaction terms supported in a given subset of the lattice. We use the results of [36], in which Lieb–Robinson bounds which do not depend on on-site interactions are developed for Hamiltonians expressed in terms of time-dependent interactions.

Let $(\Gamma ,d)$ denote a countable metric space, and let $\mathcal {P}_0(\Gamma )$ denote the collection of finite subsets of $\Gamma$. Assign a spin Hilbert space $\mathcal {H}_x$ to each $x\in \Gamma$. The algebra of local observables is given by $\mathcal {A}^\text {loc}=\cup _{X\in \mathcal {P}_0(\Gamma )} \mathcal {A}_X$, where $\mathcal {A}_X=\bigotimes _{x\in X}\mathcal {B}(\mathcal {H})$. A time-dependent interaction $\Phi :\mathbb {R}\times \mathcal {P}_0(\Gamma )$ is called continuous if $t\mapsto \Phi (t,X)$ is norm continuous for every $X\in \mathcal {P}_0(\Gamma )$.

To measure the spatial decay of the interaction we introduce the notion of an F-function. Let $(\Gamma ,d)$ denote a countable metric space. Then an F-function on $(\Gamma ,d)$ is a function $F:[0,\infty )\rightarrow (0,\infty )$ such that

• (1) F is non-increasing.

• (2) F is integrable, i.e.,

  $$\begin{aligned} \Vert F\Vert =\sup _{x\in \Gamma }\sum _{y\in \Gamma } F(d(x,y))<\infty . \end{aligned}$$

  (A.1)

• (3) F satisfies the convolution identity,

  $$\begin{aligned} C_F=\sup _{x,y\in \Gamma }\frac{1}{F(d(x,y))}\sum _{z\in \Gamma }F(d(x,z))F(d(z,y))<\infty . \end{aligned}$$

  (A.2)

If $\mu >0$, it is easy to show that $F_\mu (x)=e^{-\mu x}F(x)$ also defines an F function on $(\Gamma ,d)$ with $\Vert F_\mu \Vert \le \Vert F\Vert$ and $C_{F_\mu }\le C_F$.

Given an F-function F, we denote by $\mathcal {B}_F$ the set of continuous interactions $\Phi :\mathbb {R}\times \mathcal {P}_0(\Gamma )\rightarrow \mathcal {A}^\text {loc}$ such that the function on $\mathbb {R}$

$$\begin{aligned} t\mapsto \sup _{x,y\in \Gamma } \frac{1}{F(d(x,y))} \sum _{\begin{array}{c} x,y\in X\\ |X|>1 \end{array}}\Vert \Phi (t,X)\Vert \end{aligned}$$

(A.3)

is locally bounded.

Theorem A.1

(Theorem 3.1 in [36]) Let $\Phi \in \mathcal {B}_{F_\mu }$ for some F-function F and $\mu >0$, and let $X,Y\in \mathcal {P}_0(\Gamma )$ with $X\cap Y=\emptyset$. Then for any $\Lambda \in \mathcal {P}_0(\Gamma )$ with $X\cup Y\subseteq \Lambda$, we have

$$\begin{aligned} \sup _{\begin{array}{c} A\in \mathcal {A}_X^1\\ B\in \mathcal {A}_Y^1 \end{array}}\Vert [\tau _t^{H_\Lambda }(A),B]\Vert \le \frac{2 \Vert F\Vert }{C_{F_\mu }}\min \{|X|,|Y|\} (e^{2 C_{F_\mu } I(t)}-1)e^{-\mu d(X,Y)} \end{aligned}$$

(A.4)

for every $t\in \mathbb {R}$, where

$$\begin{aligned} I(t)=\int _{\min \{0,t\}}^{\max \{0,t\}} \sup _{x,y\in \Gamma } \frac{e^{\mu d(x,y)}}{F(d(x,y))} \sum _{\begin{array}{c} x,y\in X\\ |X|>1 \end{array}}\Vert \Phi (s,X)\Vert ds. \end{aligned}$$

(A.5)

We will now apply the previous theorem to obtain a Lieb–Robinson bound which ignores interaction terms in certain parts of the lattice. For simplicity we restrict ourselves to one-dimensional finite volume systems. Neither of these restrictions is essential.

Suppose that we have a quantum spin chain $\mathcal {H}=\bigotimes _{x=0}^n\mathcal {H}_x$ on the interval $\Lambda _n=[0,n]\subset \mathbb {Z}_+$ together with a time-dependent Hamiltonian H(t) generated by an interaction $\Phi (t):\mathcal {P}(\Lambda _n)\rightarrow \mathcal {B}(\mathcal {H})$. Let $\mathcal {I}=\{I_j\}_{j=1}^m$ be a collection of disjoint subintervals $I_j=[a_j,b_j]\subset \Lambda _n$, satisfying $b_{j}<a_{j+1}$. For purposes of notation let $b_0=0$ and $a_{m+1}=n$. We seek to define an equivalent spin chain in which the spins located on the sites $[b_j,a_{j+1}]$ are identified. Define the contracted lattice $\Gamma _{\mathcal {I}}$ by,

$$\begin{aligned} \Gamma _{\mathcal {I}}=\cup _{j=1}^m [a_j,b_j)\cup \{n\} \end{aligned}$$

Define a map $\mathcal {C}:\Lambda _n\rightarrow \Gamma _{\mathcal {I}}$ by,

$$\begin{aligned} \mathcal {C}(x)={\left\{ \begin{array}{ll} a_{j} &{} \text { if }x\in [b_{j-1},a_{j}] \text { for some }j=1,2,\ldots ,m+1\\ x &{} \text { Otherwise} \end{array}\right. } \end{aligned}$$

(A.6)

Note that $\mathcal {C}$ maps a site in $\Lambda _n$ to its corresponding site in $\Gamma _\mathcal {I}$. For each $x\in \Gamma _{\mathcal {I}}$, define

$$\begin{aligned} \mathcal {H}_x'= \bigotimes _{z\in \mathcal {C}^{-1}(\{x\})}\mathcal {H}_z \end{aligned}$$

(A.7)

Then $\bigotimes _{x=0}^n\mathcal {H}_x=\bigotimes _{x\in \Gamma _{\mathcal {I}}}\mathcal {H}_x'$, and an observable which has support X in $\mathcal {A}_{\Lambda _n}$ has support $\mathcal {C}(X)$ in $\mathcal {A}_{\Gamma _\mathcal {I}}$. Define an interaction $\tilde{\Phi }(t)$ on $\Gamma _{\mathcal {I}}$ by,

$$\begin{aligned} \tilde{\Phi }(t)(X)=\sum _{\begin{array}{c} Z\subseteq \Lambda _n\\ \mathcal {C}(Z)=X \end{array}}\Phi (t)(Z) \end{aligned}$$

(A.8)

Then $\tilde{\Phi }$ and $\Phi$ generate the same Hamiltonian. With this setup we have the following proposition.

Theorem A.2

Suppose d is a metric on $\Gamma _{\mathcal {I}}$. Let $\mu >0$ and let F denote any F-function on $(\Gamma _{\mathcal {I}},d)$. Then for any $X,Y\subseteq \Lambda _n$ with $\mathcal {C}(X)\cap \mathcal {C}(Y)=\emptyset$ we have,

$$\begin{aligned} \sup _{\begin{array}{c} A\in \mathcal {A}_{X}^1\\ B\in \mathcal {A}_Y^1 \end{array}}\Vert [\tau _t^{H}(A),B]\Vert \le \frac{2\Vert F\Vert }{C_{F_\mu }}\min \{|\mathcal {C}(X)|,|\mathcal {C}(Y)|\}(e^{2C_{F_\mu } I(t)}-1)e^{-\mu d(\mathcal {C}(X),\mathcal {C}(Y))} \end{aligned}$$

(A.9)

holds for all $t\in \mathbb {R}$, where

$$\begin{aligned} I(t)=\int _{\min \{0,t\}}^{\max \{0,t\}}\sup _{x,y\in \Gamma _\mathcal {I}}\frac{e^{\mu d(x,y)}}{F(d(x,y))}\sum _{\begin{array}{c} X\subseteq \Gamma _{\mathcal {I}}:\\ x,y\in X, \\ |X|>1 \end{array}}\Vert \tilde{\Phi }(s)(X)\Vert ds. \end{aligned}$$

(A.10)

Proof

Apply Theorem A.1 to the spin model $\tilde{\Phi }$. $\square$

A few remarks about this theorem need to be made. Note that

$$\begin{aligned} \sum _{\begin{array}{c} X\subseteq \Gamma _\mathcal {I}:\\ x,y\in X,\\ |X|>1 \end{array}}\Vert \tilde{\Phi }(t)(X)\Vert =\sum _{\begin{array}{c} X\subseteq \Gamma _\mathcal {I}:\\ x,y\in X,\\ |X|>1 \end{array}} \Vert \sum _{\begin{array}{c} Z\subseteq \Lambda _n\\ \mathcal {C}(Z)=X \end{array}}\Phi (t)(Z) \Vert \le \sum _{\begin{array}{c} Z\subseteq \Lambda _n:\\ x,y\in Z,\\ |\mathcal {C}(Z)|>1 \end{array}}\Vert \Phi (t)(Z)\Vert \end{aligned}$$

(A.11)

for any pair $x,y\in \Gamma _\mathcal {I}$. If $Z\subset [b_{j-1},a_j]$ for some j, then $\mathcal {C}(Z)$ will contain at most one point of $\Gamma _\mathcal {I}$. Therefore Theorem A.2 provides an upper bound on the speed of propagation which excludes elements from the original interaction with support Z.

While Theorem A.2 was stated for an arbitrary metric d on $\Gamma _\mathcal {I}$, there are two natural metrics which both allow $(\Gamma _\mathcal {I},d)$ to be isometrically embedded into $\mathbb {Z}_+$. One choice to simply restrict the usual metric on $\mathbb {Z}_+$ to $\Gamma _\mathcal {I}$. Another choice is to define d so that $(\Gamma _{\mathcal {I}},d)$ isometrically embeds into [0, L], where $L=\sum _{j=1}^m(b_j-a_j)$. With either of these metrics, given an F-function F on $\mathbb {Z}_+$ with the usual metric, the constants in Theorem A.2 can be chosen to be $c_0=2\Vert F\Vert /C_{F_\mu }$ and $c_1=2 C_{F_\mu }$. In particular, these constants do not depend on n or the collection of intervals $\mathcal {I}$. This follows from the fact that $\Gamma _\mathcal {I}$ isometrically embeds into $(\mathbb {Z}_+,|\cdot |)$ when equipped with either of these metrics.