Can Translation Invariant Systems Exhibit a Many-Body Localized Phase?

Abstract

We review some recent works related to the exploration of Many-Body Localization in the absence of quenched disorder. We stress that, for systems where not all eigenstates of the Hamiltonian are expected to be localized, as it is generically the case for translation invariant systems with short range interactions, some rare large ergodic spots constitute a possible mechanism for thermalization, even though such spots occur just as well in systems with strong quenched disorder, where all eigenstates are localized. Nevertheless, we show that there is a regime of asymptotic localization for some translation invariant Hamiltonians.

Appendices

Appendix 1

We show a sequence of possible steps to connect $| \eta \rangle$ to $| \eta ' \rangle$ via resonant transitions on Fig. 1. We start from $| \eta \rangle$ in the upper left corner, go from left to right and from top to bottom, and end up with $| \eta ' \rangle$ in the lower right corner. Occupation numbers marked in red are the ones that will get swapped.

Appendix 2

We prove (17 and 18). Consider an initial classical state $| \eta \rangle$ and another classical state $| \eta ' \rangle$ such that $\langle \eta ' | H^{(0)} | \eta \rangle \ne 0$. There are thus points $x,y$ satisfying $|x-y|_1 = 1$ and such that $\eta '_x = \eta _x + 1$ and $\eta '_y = \eta _y - 1$, while $\eta '_z = \eta _z$ for all $z\ne x,y$. For concreteness, let us assume that $x=(x_1,x_2)$ and $y=(x_1,x_2+1)$ (other cases analogous), let us write $\eta _x=a$, $\eta _y=b$ and let us denote by $r,s, \dots , w$ the occupation numbers on the neighboring sites. The transition from $|\eta \rangle$ to $| \eta ' \rangle$ is represented as

Because transitions must preserve the interaction energy according to (14), and because of the genericity condition (16), the states $| \eta \rangle$ and $| \eta ' \rangle$ must contain “the same oriented nearest neighbors pairs with the same multiplicity”, meaning that all the patterns $(w,b)$, $(b,s)$, $(v,a)$, $(a,t)$, $\left( {\begin{array}{c}r\\ b\end{array}}\right)$, $\left( {\begin{array}{c}b\\ a\end{array}}\right)$, $\left( {\begin{array}{c}a\\ u\end{array}}\right)$ must be found in $\eta '$. This imposes strong constraints on $| \eta \rangle$ for the transition to be possible. First we find that we need to have $b=a+1$, and then that $| \eta \rangle$ must actually be a configuration of one of the following two types

for some $s$ and $v$. Now, for $| \eta \rangle$ satisfying (17), we find a configuration depicted as

In order for $| \eta \rangle$ to be connected by $H^{(0)}$ to a state $|\eta ' \rangle$ where at least one of the values $e,f,g$ or $h$ has changed, a particle must hop along one of the twelve bonds appearing on the last figure. By the above, this is not possible. This shows (18).