Out-of-equilibrium dynamics and thermalization of string order: from spin-1 chains to optical lattices

Leonardo Mazza, Davide Rossini, Manuel Endres, Rosario Fazio

Quantum phases of matter are characterized by the correlations of the underlying many-body
wavefunction of the system. Although they are typically captured by a local order parameter, it has
been shown that a broad class of systems, e.g. topologically ordered ones, possesses a non-local
form order. I will begin by showing that such non-local order can be experimentally observed, as
reported in some recent work on string order in 1D optical lattices [1, 2].

Afterwards, I will discuss the equilibration dynamics of string order in one-dimensional quantum
systems. The problem is rather intriguing, as string order is a non-local quantity, and cannot be
framed into the well-established picture of equilibration of local order parameters.

After initializing a spin-1 chain in the Haldane phase, the time evolution of non-local correlations
following a sudden quench is studied by means of matrix-product-state-based algorithms. It is
observed that thermalization occurs only for scales up to a horizon growing at a well defined speed,
due to the finite maximal velocity at which string correlations can propagate, related to a
Lieb-Robinson bound (which can also be formulated for string order).

The persistence of string ordering at finite times is non-trivially related to symmetries of the
quenched Hamiltonian. A complete classification of when string order persists to a Hamiltonian
global quench is given; the result is interestingly related to the concept of “symmetry protected
topological order” [3].

A qualitatively similar behavior is found for the string order of the Mott insulating phase in the
Bose-Hubbard chain. This paves the way towards an experimental testing of our results in present
cold-atom setups.

Finally, I discuss how to extend this work to the study of the out-of-equilibrium dynamics of
topological order, addressing the dynamics of the entanglement spectrum in one-dimensional
symmetry protected systems.

References:
[1] M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauss, C. Gross,
L. Mazza, M.C. Banuls, L. Pollet, I. Bloch, S. Kuhr, Science 334, 200 (2011).
[2] M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauss, C. Gross,
L. Mazza, M.C. Banuls, L. Pollet, I. Bloch, S. Kuhr, Appl. Phys. B 113, 27 (2013).
[3] L. Mazza, D. Rossini, M. Endres, R. Fazio, arXiv:1311.2615 (2013),
accepted for publication in Phys. Rev. B Rapid Comm.