Step Density Profiles in Localized Chains

Abstract

We consider two types of strongly disordered one-dimensional Hamiltonian systems coupled to baths (energy or particle reservoirs) at the boundaries: strongly disordered quantum spin chains and disordered classical harmonic oscillators. These systems are believed to exhibit localization, implying in particular that the conductivity decays exponentially in the chain length L. We ask however for the profile of the (very slowly) transported quantity in the steady state. We find that this profile is a step-function, jumping in the middle of the chain from the value set by the left bath to the value set by the right bath. This is confirmed by numerics on a disordered quantum spin chain of 9 spins and on much longer chains of harmonic oscillators. From theoretical arguments, we find that the width of the step grows not faster than $\sqrt{L}$, and we confirm this numerically for harmonic oscillators. In this case, we also observe a drastic breakdown of local equilibrium at the step, resulting in a heavily oscillating temperature profile.

Appendix: Alternative Derivation of the \(\lambda \rightarrow 0\) Limit

We propose a second way to derive Eq. (2.10), i.e. the expression of the NESS profile in the limit $\lambda \rightarrow 0$. This is somewhat less straightforward but possibly more intuitive. We obtain also a little bit more: we derive an effective dynamics in the limit $\lambda \rightarrow 0$.

1.1 Equations of Motion

We massage the equations of motion to arrive at (3.9) below. First, it is convenient to adopt a matrix notation:

$$\begin{aligned} {\mathrm d}|q \rangle&= |p \rangle {\mathrm d}t, \\ {\mathrm d}|p \rangle&= - H |q \rangle {\mathrm d}t - \lambda \sum _{x\in \partial \Lambda } p_x |x \rangle {\mathrm d}t + \sqrt{2\lambda } \sum _{x\in \partial \Lambda } \sqrt{T}_x {\mathrm d}b_x(t) |x \rangle . \end{aligned}$$

Next, we introduce an effective time $\tau = \lambda t$, so that the equations become

$$\begin{aligned} {\mathrm d}|q \rangle&= \lambda ^{-1} |p \rangle {\mathrm d}\tau , \\ {\mathrm d}|p \rangle&= - \lambda ^{-1} H |q \rangle {\mathrm d}\tau - \sum _{x\in \partial \Lambda } p_x |x \rangle {\mathrm d}\tau + \sqrt{2} \sum _{x\in \partial \Lambda } \sqrt{T}_x {\mathrm d}b_x(\tau ) |x \rangle . \end{aligned}$$

Next we move to the normal modes of the isolated system. Recall (2.9). Let us introduce the notations

$$\begin{aligned} q(k) = \sum _x \psi _k(x) q_x, \quad p(k) = \sum _x \psi _k(x) p_x \end{aligned}$$

Then the equations of motion can be recast as

$$\begin{aligned} {\mathrm d}q(k)&= \lambda ^{-1}p(k) {\mathrm d}\tau , \\ {\mathrm d}p(k)&= - \lambda ^{-1} \omega _k^2q(k) {\mathrm d}\tau - \sum _{x\in \partial \Lambda } p_x \psi _k(x) {\mathrm d}\tau + \sqrt{2} \sum _{x\in \partial \Lambda } \sqrt{T}_x \psi _k (x) {\mathrm d}b_x(\tau ). \end{aligned}$$

It is convenient to pass from the (q, p) coordinates to the $(a^*,a)$ coordinates:

$$\begin{aligned}&a(k) = q(k) + \frac{\mathrm {i}}{\omega _k} p(k), \quad a^*(k) = q(k) - \frac{\mathrm {i}}{\omega _k} p(k), \\&q(k) = \frac{1}{2} \big ( a(k) + a^*(k) \big ), \quad p(k) = \frac{\omega _k}{2 \mathrm {i}} \big (a(k) - a^*(k)\big ). \end{aligned}$$

The equations become

$$\begin{aligned} {\mathrm d}a(k) = \frac{- \mathrm {i}\omega _k}{\lambda } a(k) {\mathrm d}\tau + \frac{\mathrm {i}}{\omega _k} \left( - \sum _{x\in \partial \Lambda } p_x \psi _k(x) {\mathrm d}\tau + \sqrt{2} \sum _{x\in \partial \Lambda } \sqrt{T}_x \psi _k (x) {\mathrm d}b_x(\tau ) \right) .\quad \end{aligned}$$

(3.9)

1.2 Limit $\lambda \rightarrow 0$: Resonant Averaging

We now consider the limit $\lambda \rightarrow 0$. We follow the method used by Dymov in [45]. The first term in the rhs of (3.9) is dominant on short time scales, and if the second term was absent, the solution would be given by $a(k,\tau ) = {\mathrm e}^{-\mathrm {i}\omega _k \tau /\lambda } a(k,0)$. This motivates the change of variables

$$\begin{aligned} a(k,\tau ) = {\mathrm e}^{-\mathrm {i}\omega _k \tau /\lambda } A(k,\tau ). \end{aligned}$$

The evolution equation for A(k) is

$$\begin{aligned} {\mathrm d}A(k) = \frac{\mathrm {i}\, {\mathrm e}^{\mathrm {i}\omega _k \tau /\lambda }}{\omega _k} \left( - \sum _{x\in \partial \Lambda } p_x \psi _k(x) {\mathrm d}\tau + \sqrt{2} \sum _{x\in \partial \Lambda } \sqrt{T}_x \psi _k (x) {\mathrm d}b_x(\tau ) \right) . \end{aligned}$$

The rhs is oscillating fast at any frequency $\omega _k$ (the spectrum of H is bounded away from 0). We exploit this to obtain an expression in the limit $\lambda \rightarrow 0$. First we analyze the noise term and then the dissipative term.

1.2.1 Noise Term

A computation of second moments shows that, in general,

$$\begin{aligned} \sqrt{2} B_{k,x}(t) : = B^{(1)}_{k,x} (t) + \mathrm {i}B^{(2)}_{k,x} (t) : = \lim _{\lambda \rightarrow 0} \sqrt{2} \int _0^t {\mathrm e}^{-\mathrm {i}\omega _k s/\lambda } {\mathrm d}b_{x}(s) \end{aligned}$$

where $B^{(i)}_{k,x}, i=1,2$ are standard independent Brownian motions and the limit is in law, jointly for the collection of processes indexed by k, x, i. As noticed in [45], for a single k, x, the noise ‘bifurcates’ in the limit, since two independent copies ($i=1,2$) are generated from a single process. There is more here: each Brownian motion ‘$2|\Lambda |-$furcates’, since for each mode k the corresponding noises are independent. Hence the noise term in our effective equation will be given by

$$\begin{aligned} \frac{\mathrm {i}\sqrt{2}}{\omega _k} \sum _{x\in \partial \Lambda } \sqrt{T}_x \psi _k (x) {\mathrm d}B_{k,x}(\tau ). \end{aligned}$$

(3.10)

1.2.2 Dissipative Term

We work out

$$\begin{aligned} p_x= & {} \sum _k \psi _k^* (x) p(k) = \sum _k \psi _k^* (x) \frac{\omega _k}{2 \mathrm {i}} (a_k - a_k^*) \\= & {} \frac{1}{2 \mathrm {i}} \sum _k \psi _k^*(x) \omega _k \big ( {\mathrm e}^{-\mathrm {i}\omega _k \tau /\lambda } A_k - {\mathrm e}^{\mathrm {i}\omega _k \tau /\lambda } A_k^* \big ). \end{aligned}$$

This term will be multiplied by ${\mathrm e}^{\mathrm {i}\omega _k \tau /\lambda }$. Hence, in the limit $\lambda \rightarrow 0$, only the term with the factor ${\mathrm e}^{-\mathrm {i}\omega _k \tau / \lambda }$ remains. We obtain the dissipation term

$$\begin{aligned} - \frac{1}{2} \sum _{x\in \partial \Lambda } |\psi _k(x)|^2 A_k. \end{aligned}$$

(3.11)

In conclusion, we find that the effective equation in the limit $\lambda \rightarrow 0$ is given by

$$\begin{aligned} dA(k) = \frac{\mathrm {i}\sqrt{2}}{\omega _k} \sum _{x\in \partial \Lambda } \sqrt{T}_x \psi _k (x) {\mathrm d}B_{k,x}(\tau ) - \frac{1}{2} \sum _{x\in \partial \Lambda } |\psi _k(x)|^2 A_k \end{aligned}$$

Since all modes have now been decoupled, it is straightforward to recover the temperature profile (2.10).

1.3 Comments

This derivation requires the limit $\lambda \rightarrow 0$ to be taken before the thermodynamic limit $\Lambda \rightarrow {\mathbb Z}^d$. Indeed, when arguing that the noises corresponding to the different modes k become independent, we relied on $\lambda$ being much smaller than the difference between any two frequencies $\omega _k$. This derivation would hence fail if the $\omega _k$ were to form a continuum. For localized systems, the situation is more subtle: since each site in the chain sees effectively only a few modes, the treatment above becomes correct again. A more careful argument for this was given in Sect. 2.2.