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We define the Uhlmann number as an extension of the Chern number, and we use this quantity to describe the topology of 2D translational invariant Fermionic systems at finite temperature. We consider two paradigmatic systems and we study the changes in their topology through the Uhlmann number. Through the linear response theory we linked two geometrical quantities of the system, the mean Uhlmann curvature and the Uhlmann number, to directly measurable physical quantities, i.e. the dynamical susceptibility and to the dynamical conductivity, respectively.
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We enquire into the quasi-many-body localization in topologically ordered states of matter, revolving around the case of Kitaev toric code on ladder geometry, where different types of anyonic defects carry different masses induced by environmental errors. Our study verifies that random arrangement of anyons generates a complex energy landscape solely through braiding statistics, which suffices to suppress the diffusion of defects in such multi-component anyonic liquid. This non-ergodic dynamic suggests a promising scenario for investigation of quasi-many-body localization. Computing standard diagnostics evidences that, in such disorder-free many-body system, a typical initial inhomogeneity of anyons gives birth to a glassy dynamics with an exponentially diverging time scale of the full relaxation. A by-product of this dynamical effect is manifested by the slow growth of entanglement entropy, with characteristic time scales bearing resemblance to those of inhomogeneity relaxation. This setting provides a new platform which paves the way toward impeding logical errors by self-localization of anyons in a generic, high energy state, originated in their exotic statistics.
Prethermalization, by introducing emergent quasiconserved observables, plays a crucial role in protecting Floquet many-body phases over exponentially long time, while the ultimate fate of such quasiconserved operators can signal thermalization to infinite temperature. To elucidate the properties of prethermal quasiconservation in many-body Floquet systems, here we systematically analyze infinite temperature correlations between observables. We numerically show that the late-time behavior of the autocorrelations unambiguously distinguishes quasiconserved observables from non-conserved ones, allowing to single out a set of linearly-independent quasiconserved observables. By investigating two Floquet spin models, we identify two different mechanism underlying the quasi-conservation law. First, we numerically verify energy quasiconservation when the driving frequency is large, so that the system dynamics is approximately described by a static prethermal Hamiltonian. More interestingly, under moderate driving frequency, another quasiconserved observable can still persist if the Floquet driving contains a large global rotation. We show theoretically how to calculate this conserved observable and provide numerical verification. Having systematically identified all quasiconserved observables, we can finally investigate their behavior in the infinite-time limit and thermodynamic limit, using autocorrelations obtained from both numerical simulation and experiments in solid state nuclear magnetic resonance systems.
We study the geometric Uhlmann phase of entangled mixed states in a composite system made of two coupled spin-$frac 1 2$ particles with a magnetic field acting on one of them. Within a depolarizing channel setup, an exact analytical expression for such a phase in each subsystem is derived. We find an explicit connection to the concurrence of the depolarizing channel density matrix, which allows to characterize the features of the Uhlmann phase in terms of the degree of entanglement in the system. In the space of field direction and coupling parameter, it exhibits a phase singularity revealing a topological transition between orders with different winding numbers. The transition occurs for fields lying in the equator of the sphere of directions and at critical values of the coupling which can be controlled by tuning the depolarization strength. Notably, under these conditions the concurrence of the composite system is bounded to the range $[0,1/2]$. We also compare the calculated Uhlmann phase to an interferometric phase, which has been formulated as an alternative for density matrices. The latter does not present a phase vortex, although they coincide in the weak entanglement regime, for vanishing depolarization (pure states). Otherwise they behave clearly different in the strong entanglement regime.
The presence of flat bands is a source of localization in lattice systems. While flat bands are often unstable with respect to interactions between the particles, they can persist in certain cases. We consider a diamond ladder with transverse hopping that possesses such stable flat bands and show that many-body localization appears in the presence of interactions. We demonstrate that the eigenstate thermalization hypothesis is violated and verify localization by time evolution of local observables, revival probabilities, and participation ratios. Thus, this system appears to be an example for many-body localization without disorder.
My previous work [arXiv:1902.00977] studied the dynamics of Renyi entanglement entropy $R_alpha$ in local quantum circuits with charge conservation. Initializing the system in a random product state, it was proved that $R_alpha$ with Renyi index $alpha>1$ grows no faster than diffusively (up to a sublogarithmic correction) if charge transport is not faster than diffusive. The proof was given only for qubit or spin-$1/2$ systems. In this note, I extend the proof to qudit systems, i.e., spin systems with local dimension $dge2$.
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