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Richard Liu
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Note on Lieb-Robinson Bound(1)

物理笔记

This note mainly focus on the basis of Lieb-Robinson Bound, considering the application towards correlation function, topological order, and nonrelativistic Goldstone theorem. The main references are:

[1] An Online Chinese Note

[2] M. B. Hastings, Locality in Quantum Systems.

Part. 1 Lieb-Robinson Bound

Consider H=∑ZHZH=\sum_ZH_Z, where ZZ is denoted as a set of lattice site and ∥HZ∥\| H_Z\| decays exponentially.

Theorem 1

Suppose for site i, we have:

∑X∋i∥HX∥∣X∣exp⁡[μdiam(X)]≤s<∞,   μ,s>0\sum_{X\ni i}\|H_X\||X|\exp[\mu {\rm diam}(X)]\leq s< \infty, ~~~\mu,s>0

Let A,BA,B to be bosonic operators supported on sets X,YX,Y(dist(X,Y)>0{\rm dist}(X,Y)>0), then:

∥[A(t),B]∥≤2∥A∥∥B∥∑i∈Xexp⁡[−μdist(i,Y)](e2s∣t∣−1)≤2∥A∥∥B∥∣X∣exp⁡[−μdist(X,Y)](e2s∣t∣−1)\begin{aligned} \|[A(t),B]\|&\leq 2\|A\|\|B\|\sum_{i\in X}\exp[-\mu{\rm dist}(i,Y)](e^{2s|t|}-1)\\ &\leq 2\|A\|\|B\||X|\exp[-\mu{\rm dist}(X,Y)](e^{2s|t|}-1)\\ \end{aligned}

Proof:

Let tn=tNn=nϵt_n=\frac{t}{N}n=n\epsilon, with N is large, and IX=∑Z:Z∩X≠∅HZI_X = \sum_{Z:Z \cap X \neq\varnothing}H_Z.

∥[A(t),B]∥−∥[A(0),B]∥=∑n=0N−1ϵ∥[A(t<!−−swig0−−>),B]∥−∥[A(tn),B]∥ϵ\|[A(t),B]\|-\|[A(0),B]\|= \sum_{n=0}^{N-1}\epsilon\frac{\|[A(t_NaN),B]\|-\|[A(t_n),B]\|}{\epsilon}
∥[A(t<!−−swig1−−>),B]∥−∥[A(tn),B]∥=∥[A(ϵ),B(−tn)]∥−∥[A,B(−tn)]∥≤∥[A+iϵ[H,A],B(−tn)]∥−∥[A,B(−tn)]∥+O(ϵ2)=∥[A+iϵ[IX,A],B(−tn)]∥−∥[A,B(−tn)]∥+O(ϵ2)\begin{aligned} \|[A(t_NaN),B]\|-\|[A(t_n),B]\| &= \|[A(\epsilon),B(-t_n)]\|-\|[A,B(-t_n)]\|\\ &\leq\|[A+i\epsilon[H,A],B(-t_n)]\|-\|[A,B(-t_n)]\|+\mathcal{O}(\epsilon^2)\\ &=\|[A+i\epsilon[I_X,A],B(-t_n)]\|-\|[A,B(-t_n)]\|+\mathcal{O}(\epsilon^2)\\ \end{aligned}
∥[A+iϵ[IX,A],B(−tn)]∥=∥[eiϵIXAe−iϵIX,B(−tn)]∥+O(ϵ2)=∥[A,e−iϵIXB(−tn)eiϵIX]∥+O(ϵ2)≤∥[A,B(−tn)]∥+ϵ∥[A,[IX,B(−tn)]]∥+O(ϵ2)\begin{aligned} \|[A+i\epsilon[I_X,A],B(-t_n)]\|&=\|[e^{i\epsilon I_X}Ae^{-i\epsilon I_X},B(-t_n)]\|+\mathcal{O}(\epsilon^2)\\ &=\|[A,e^{-i\epsilon I_X}B(-t_n)e^{i\epsilon I_X}]\|+\mathcal{O}(\epsilon^2)\\ &\leq\|[A,B(-t_n)]\|+\epsilon\|[A,[I_X,B(-t_n)]]\|+\mathcal{O}(\epsilon^2)\\ \end{aligned}

Then we obtain:

∥[A(t),B]∥−∥[A(0),B]∥=2∥A∥∑n=0N−1ϵ∑Z:Z∩X≠∅∥[HZ,B(−tn)]∥+O(ϵ)=2∥A∥∑n=0N−1∑Z:Z∩X≠∅ϵ∥[HZ(tn),B]∥+O(ϵ)=2∥A∥∑Z:Z∩X≠∅∫0∣t∣dx∥[HZ(x),B]∥\begin{aligned} \|[A(t),B]\|-\|[A(0),B]\|&=2\|A\| \sum_{n=0}^{N-1}\epsilon\sum_{Z:Z \cap X \neq\varnothing}\|[H_Z,B(-t_n)]\|+\mathcal{O}(\epsilon)\\ &=2\|A\| \sum_{n=0}^{N-1}\sum_{Z:Z \cap X \neq\varnothing}\epsilon\|[H_Z(t_n),B]\|+\mathcal{O}(\epsilon)\\ &=2\|A\|\sum_{Z:Z \cap X \neq\varnothing}\int_0^{|t|}dx\|[H_Z(x),B]\|\\ \end{aligned}

Approximation: using the upper limit to control the bound.

Define CB(X,t):=sup⁡A∈AX∥[A(t),B]∥∥A∥C_B(X,t):=\sup_{A\in\mathcal{A}_X}\frac{\|[A(t),B]\|}{\|A\|}, then we have CB(X,0)=0C_B(X,0)=0 fordist(X,Y)>0{\rm dist}(X,Y)>0 and :CB(Z,0)≤2∥B∥C_B(Z,0)\leq2\|B\|

CB(X,t)≤2∥HZ∥∑Z:Z∩X≠∅∫0∣t∣dxCB(Z,x)≤2∥HZ1∥∑Z1:Z1∩X≠∅∫0∣t∣dxCB(Z,0)+22∥HZ1∥∥HZ2∥∑Z1:Z1∩X≠∅∑Z2:Z2∩Z1≠∅∫0∣t∣∫0∣x∣dxdyCB(Z,y)≤2(2∣t∣)∥B∥∑Z1:Z1∩X≠∅,Z1∩Y≠∅∥HZ1∥+2(2∣t∣)22!∥B∥∑Z1:Z1∩X≠∅∑Z2:Z2∩Z1≠∅,Z2∩Y≠∅∥HZ1∥∥HZ2∥+⋯\begin{aligned} C_B(X,t)&\leq2\|H_Z\|\sum_{Z:Z \cap X \neq\varnothing}\int_0^{|t|}dxC_B(Z,x)\\ &\leq2\|H_{Z_1}\|\sum_{Z_1:Z_1 \cap X \neq\varnothing}\int_0^{|t|}dxC_B(Z,0)\\ &+2^2\|H_{Z_1}\|\|H_{Z_2}\|\sum_{Z_1:Z_1 \cap X \neq\varnothing}\sum_{Z_2:Z_2 \cap Z_1 \neq\varnothing}\int_0^{|t|}\int_0^{|x|}dxdyC_B(Z,y)\\ &\leq2(2|t|)\|B\|\sum_{Z_1:Z_1 \cap X \neq\varnothing,Z_1\cap Y\neq\varnothing}\|H_{Z_1}\|+2\frac{(2|t|)^2}{2!}\|B\|\sum_{Z_1:Z_1 \cap X \neq\varnothing}\sum_{Z_2:Z_2 \cap Z_1 \neq\varnothing,Z_2\cap Y\neq\varnothing}\|H_{Z_1}\|\|H_{Z_2}\|+\cdots \end{aligned}

For the first term, we have:

∑Z1:Z1∩X≠∅,Z1∩Y≠∅∥HZ1∥≤∑i∈X∑Z1∋i,Z1∩Y≠∅∥HZ1∥\sum_{Z_1:Z_1 \cap X \neq\varnothing,Z_1\cap Y\neq\varnothing}\|H_{Z_1}\|\leq\sum_{i\in X}\sum_{Z_1\ni i,Z_1\cap Y\neq\varnothing}\|H_{Z_1}\|

In case that Z1∩Y≠∅Z_1\cap Y\neq\varnothing, then dist(i,Y)≤diam(Z){\rm dist}(i,Y)\leq {\rm diam}(Z), which means:

∑Z1:Z1∩X≠∅,Z1∩Y≠∅∥HZ1∥≤∑i∈Xexp⁡(−μdist(i,Y))\sum_{Z_1:Z_1 \cap X \neq\varnothing,Z_1\cap Y\neq\varnothing}\|H_{Z_1}\|\leq \sum_{i\in X}\exp(-\mu{\rm dist}(i,Y))

For the second term, we have:

∑Z1:Z1∩X≠∅∑Z2:Z2∩Z1≠∅,Z2∩Y≠∅∥HZ1∥∥HZ2∥=∑i∈X∑Z1∋i∑j∈Z1∑Z2∋j,Z2∩Y≠∅∥HZ1∥∥HZ2∥≤∑i∈X∑Z1∋i∑j∈Z1∑Z2∋j,Z2∩Y≠∅∥HZ1∥∥HZ2∥exp⁡(−dist(i,Y))exp⁡(dist(i,j))exp⁡(dist(j,Y))=∑i∈X∑Z1∋i∑j∈Z1exp⁡(−μdist(i,Y))exp⁡(μdist(i,j))∑Z2∋j,Z2∩Y≠∅∥HZ1∥∥HZ2∥exp⁡(μdist(j,Y))≤∑i∈X∑Z1∋i∑j∈Z1exp⁡(−μdist(i,Y))exp⁡(μdist(i,j))∑Z2∋j,Z2∩Y≠∅∥HZ1∥∥HZ2∥exp⁡(μdiam(Z2))≤∑i∈X∑Z1∋i∑j∈Z1exp⁡(−μdist(i,Y))exp⁡(μdist(i,j))∥HZ1∥s≤∑i∈X∑Z1∋iexp⁡(−μdist(i,Y))exp⁡(μdiam(Z1))∥HZ1∥s∣Z1∣≤∑i∈Xexp⁡(−μdist(i,Y))s2\begin{aligned} &\sum_{Z_1:Z_1 \cap X \neq\varnothing}\sum_{Z_2:Z_2 \cap Z_1 \neq\varnothing,Z_2\cap Y\neq\varnothing}\|H_{Z_1}\|\|H_{Z_2}\|\\ &=\sum_{i\in X}\sum_{Z_1 \ni i}\sum_{j\in Z_1}\sum_{Z_2\ni j,Z_2\cap Y\neq \varnothing}\|H_{Z_1}\|\|H_{Z_2}\|\\ &\leq\sum_{i\in X}\sum_{Z_1 \ni i}\sum_{j\in Z_1}\sum_{Z_2\ni j,Z_2\cap Y\neq \varnothing}\|H_{Z_1}\|\|H_{Z_2}\|\exp(-{\rm dist}(i,Y))\exp({\rm dist}(i,j))\exp({\rm dist}(j,Y))\\ &= \sum_{i\in X}\sum_{Z_1 \ni i}\sum_{j\in Z_1}\exp(-{\mu\rm dist}(i,Y))\exp(\mu{\rm dist}(i,j))\sum_{Z_2\ni j,Z_2\cap Y\neq \varnothing}\|H_{Z_1}\|\|H_{Z_2}\|\exp(\mu{\rm dist}(j,Y))\\ &\leq \sum_{i\in X}\sum_{Z_1 \ni i}\sum_{j\in Z_1}\exp(-\mu{\rm dist}(i,Y))\exp(\mu{\rm dist}(i,j))\sum_{Z_2\ni j,Z_2\cap Y\neq \varnothing}\|H_{Z_1}\|\|H_{Z_2}\|\exp(\mu{\rm diam}(Z_2))\\ &\leq \sum_{i\in X}\sum_{Z_1 \ni i}\sum_{j\in Z_1}\exp(-\mu{\rm dist}(i,Y))\exp(\mu{\rm dist}(i,j))\|H_{Z_1}\|s\\ &\leq \sum_{i\in X}\sum_{Z_1 \ni i}\exp(-\mu{\rm dist}(i,Y))\exp(\mu{\rm diam}(Z_1))\|H_{Z_1}\|s|Z_1|\\ &\leq \sum_{i\in X}\exp(-\mu{\rm dist}(i,Y))s^2\\ \end{aligned}

Higher order terms follows the same procedure. Then,

CB(X,t)≤2∥B∥∑i∈Xexp⁡[−μdist(X,Y)](e2s∣t∣−1)C_B(X,t)\leq2\|B\|\sum_{i \in X}\exp[-\mu{\rm dist}(X,Y)](e^{2s|t|}-1)

Q.E.D.

We now have Lieb-Robinson bound with the form:

∥[A(t),B]∥≤2∥A∥∥B∥∣X∣exp⁡[−μdist(X,Y)](e2s∣t∣−1)\|[A(t),B]\|\leq 2\|A\|\|B\||X|\exp[-\mu{\rm dist}(X,Y)](e^{2s|t|}-1)\\

We can then deduce another form setting a constant vLRv_{LR} such that for t≤dist(X,Y)/vLRt\leq {\rm dist}(X,Y)/v_{LR},

∥[A(t),B]∥≤vLRtl∥A∥∥B∥∣X∣g(l),l=dist(X,Y)\|[A(t),B]\|\leq \frac{v_{LR}t}{l}\|A\|\|B\||X|g(l),l={\rm dist}(X,Y)

Apparently, vLR=4s/μv_{LR}=4s/\mu is a plausible choice.

Richard Liu · 生活如歌，至死少年。

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