Abstract
We propose the following definition of topological quantum phases valid for mixed states: two states are in the same phase if there exists a time independent, fast and local Lindbladian evolution driving one state into the other. The underlying idea, motivated by \cite{Konig2014}, is that it takes time to create new topological correlations, even with the use of dissipation.We show that it is a good definition in the following sense: (1) It divides the set of states into equivalent classes and it establishes a partial order between those according to their level of ``topological complexity''. (2) It provides a path between any two states belonging to the same phase where observables behave smoothly.We then focus on pure states to relate the new definition in this particular case with the usual definition for quantum phases of closed systems in terms of the existence of a gapped path of Hamiltonians connecting both states in the corresponding ground state path. We show first that if two pure states are in the same phase in the Hamiltonian sense, they are also in the same phase in the Lindbladian sense considered here.We then turn to analyse the reverse implication, where we point out a very different behaviour in the case of symmetry protected topological (SPT) phases in 1D. Whereas at the Hamiltonian level, phases are known to be classified with the second cohomology group of the symmetry group, we show that symmetry cannot give any protection in 1D in the Lindbladian sense: there is only one SPT phase in 1D independently of the symmetry group.We finish analysing the case of 2D topological quantum systems. There we expect that different topological phases in the Hamiltonian sense remain different in the Lindbladian sense. We show this formally only for theZnquantum double modelsD(Zn). Concretely, we prove that, ifmis a divisor ofn, there cannot exist any fast local Lindbladian connecting a ground state ofD(Zm)with one ofD(Zn), making rigorous the initial intuition that it takes long time to create those correlations present in theZncase that do not exist in theZmcase and that, hence, theZnphase is strictly more complex in the Lindbladian case than theZmphase. We conjecture that such Lindbladian does exist in the opposite direction since Lindbladians can destroy correlations.
Highlights
We show that it is a good definition in the following sense: (1) It divides the set of states into equivalent classes and it establishes a partial order between those according to their level of “topological complexity”. (2) It provides a path between any two states belonging to the same phase where observables behave smoothly
Whereas at the Hamiltonian level, phases are known to be classified with the second cohomology group of the symmetry group, we show that symmetry cannot give any protection in 1D in the Lindbladian sense: there is only one symmetry protected topological (SPT) phase in 1D independently of the symmetry group
Let us comment on another topic that resonates with what we study in this work, but differs in several fundamental aspects, namely dynamical quantum phase transitions (DQPT)
Summary
We say that a state ρ0 can be driven fast to another state ρ1, both supported on HS, and we write ρ0 →− ρ1, if it exists a dissipative evolution generated by a local and time-independent Lindbladian L such that for t poly log N , etL(ρ0) − ρ1 1 poly(N ) e−μt. Notice that after a time t poly log N , the upper bound will vanish in the thermodynamic limit. For this reason, the time scale μ−1 can be chosen constant or of order log N. We say that two states belong to the same phase if there exist two local Lindbladian evolutions such that ρ0 →− ρ1 and ρ0 ←− ρ1, and in this case we write ρ0 ↔ ρ1
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.