Topological order of mixed states in correlated quantum many-body systems

Topological order has become a new paradigm to distinguish ground states of interacting many-body systems without conventional long-range order. Here, we discuss possible extensions of this concept to density matrices describing statistical ensembles. For a large class of quasithermal states, which can be realized as thermal states of some quasilocal Hamiltonian, we generalize earlier definitions of density-matrix topology to generic many-body systems with strong correlations. We point out that the robustness of topological order, defined as a pattern of long-range entanglement, depends crucially on the perturbations under consideration. While it is intrinsically protected against local perturbations of arbitrary strength in an infinite closed quantum system, purely local perturbations can destroy topological order in open systems coupled to baths if the coupling is sufficiently strong. We discuss our classification scheme using the finite-temperature quantum Hall states and point out that the classical Hall effect can be understood as a finite-temperature topological phase.

Figure 1

In a pure quantum state, topological order corresponds to a pattern of long-range entanglement. For an ensemble of quantum states, all from a certain subspace of the full Hilbert space, the notion of topological order can be generalized by allowing arbitrary basis changes $\widehat{U}$ within this restricted subspace. If this is sufficient to map the entire ensemble on an ensemble of topologically trivial states, the ensemble can be considered topologically trivial.

Figure 2

Illustration of topological order representing nonlocal (long-range) entanglement in a wave function, shared by two spatially separated parties ($A$ and $B$) in this case. The long-range entanglement (i.e., the topological order) is robust to arbitrary local basis changes, corresponding to LU transformations generated by local Hamiltonians ${\widehat{\mathcal{H}}}_{A,B}$.

Figure 3

The spectrum of $-log\widehat{\rho }$ corresponding to a generic density matrix $\widehat{\rho }$ can be divided into manifolds $\mu$ of states separated by gaps ${\mathrm{\Delta }}_{\rho }^{\left(\mu \right)}$. Adiabatic deformations of the density matrix $\widehat{\rho }$ leave this spectral structure invariant. Two density matrices with the same spectral structure are defined to be topologically equivalent if all manifolds are equivalent; two manifolds of states from different density matrices are topologically equivalent if they can be transformed into one another by a combination of arbitrary unitary transformations $\widehat{U}$ within the manifold, and a $\mathrm{LU}$ transformation acting on the entire Hilbert space $H$. We sketch an example where the first manifold is long-range entangled and the second manifold is equivalent to an ensemble of topologically trivial states.

Figure 4

(a) As a toy model that possesses density-matrix topological order, we discuss a two-band integer Chern insulator ($N$ particles) at finite temperatures. (b) The spectral structure of the thermal density matrix $-log\widehat{\rho }=\beta \widehat{\mathcal{H}}$ has gaps ${\mathrm{\Delta }}_{\rho }^{\left(\mu \right)}$ at low energies if the combined bandwidth of the two bands is smaller than the band gap $2J<\mathrm{\Delta }$. If on the other hand $2J>\mathrm{\Delta }$, only the first gap ${\mathrm{\Delta }}_{\rho }^1$ remains open (c). The many-body Chern numbers $\mathcal{C}$ of the resulting manifolds of states are indicated. As a result, we obtain the phase diagram shown in (d), where regions with different topological order are indicated by different colors.

Figure 5

(a) We study the Hofstadter-Hubbard model in a superlattice potential. Fermions with nearest-neighbor interactions (strength $U$) tunnel between sites of a square lattice (hopping $t$) in a magnetic field (flux per plaquette $\alpha$). On the sites indicated by solid black dots, an attractive superlattice potential (strength $g$) is switched on, which drives a topological phase transition in the ground state for $g_c\approx 1.5t$. We used $N=3$ particles, $\alpha =\frac{1}{8}$, and $U=t$ and set $t=1$. (b) The winding of the Wilson loops $W\left({\theta }_y\right)$ for the first (red) and second (blue) manifold of states at $g=0$ is shown as a function of the twist angle ${\theta }_y$ introduced in the periodic boundary conditions. (c) The full spectrum of the thermal density matrix, $\widehat{\mathcal{H}}=-\beta log\widehat{\rho }$, is shown. The spectral structure changes around $g_c\approx 1.5t$ (green dashed line).

Figure 6

Topological order is not robust when couplings to local baths are considered. They can destroy the long-range entanglement in the system, which is the hallmark of topological order, in a finite time. This destruction of nonlocal entanglement between two parties $A$ and $B$ by the action of local Lindblad generators ${\widehat{L}}_{A,B}$ is illustrated in the figure.