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arXiv:2210.15661 · Physical review. B./Physical review. B

Dynamic melting and condensation of topological dislocation modes

Sanjib Kumar Das, Bitan Roy

Published Oct 19, 2023• 4 citations
arXivHTMLPDF

Abstract

Bulk dislocation lattice defects are instrumental in identifying translationally active topological insulators (TATIs), featuring band inversion at a finite momentum ${\mathbf{K}}_{\mathrm{inv}}$. As such, TATIs host robust gapless modes around the dislocation core, when the associated Burgers vector $\mathbf{b}$ satisfies ${\mathbf{K}}_{\mathrm{inv}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{b}=\ensuremath{\pi}$ (modulo $2\ensuremath{\pi}$). From the time evolution of appropriate density matrices, we show that when a TATI via a real time ramp enters into a trivial or translationally inert topological insulating phase, devoid of gapless dislocation modes, the signatures of the preramp defect modes survive for a long time. More intriguingly, as the system ramps into a TATI phase from any translationally inert insulator, the signature of the dislocation mode dynamically builds up near its core, which is prominent for slow ramps. We exemplify these generic outcomes for two-dimensional time-reversal symmetry breaking insulators. Proposed dynamic responses at the dislocation core can be experimentally observed in quantum crystals, optical lattices, and metamaterials with a time tunable band gap.

DOI
10.1103/physrevb.108.144304

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Bulk dislocation lattice defects are instrumental in identifying translationally active topological insulators (TATIs), featuring band inversion at a finite momentum Kinv{\mathbf{K}}_{\mathrm{inv}}Kinv​. As such, TATIs host robust gapless modes around the dislocation core, when the associated Burgers vector b\mathbf{b}b satisfies Kinv\ifmmode⋅\else\textperiodcentered\fib=\ensuremathπ{\mathbf{K}}_{\mathrm{inv}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{b}=\ensuremath{\pi}Kinv​\ifmmode⋅\else\textperiodcentered\fib=\ensuremathπ (modulo 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ). From the time evolution of appropriate density matrices, we show that when a TATI via a real time ramp enters into a trivial or translationally inert topological insulating phase, devoid of gapless dislocation modes, the signatures of the preramp defect modes survive for a long time. More intriguingly, as the system ramps into a TATI phase from any translationally inert insulator, the signature of the dislocation mode dynamically builds up near its core, which is prominent for slow ramps. We exemplify these generic outcomes for two-dimensional time-reversal symmetry breaking insulators. Proposed dynamic responses at the dislocation core can be experimentally observed in quantum crystals, optical lattices, and metamaterials with a time tunable band gap.
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Nontrivial topological invariant of bulk electronic wavefunctions in two-dimensional quantum crystals leaves its footprints on the edge, dislocation, and corner modes. Here we investigate non-unitary time dynamics of these topological modes in square lattice-based open quantum systems in which the time-dependent Hamiltonian smoothly interpolates between topologically distinct insulators across band gap closing quantum critical points. The temporal dynamics of these modes is described by a Lindblad equation in which the instantaneous Hamiltonian plays the role of the Lindblad operator, thereby allowing the environment to couple with the system through the energy channels (weak measurement protocol). We show that in the presence of such a real time ramp, the survival probability of these modes decreases (increases) in short (long) time scale where the dephasing (quantum Zeno) effect dominates with the increasing amplitude of the system-to-environment coupling, for both slow and fast ramps from a topological to a normal insulating state. For a reverse course of the time evolution, the revival or condensation probability of nucleating such topological modes, otherwise absent in the initial system, increases for stronger system-to-environment coupling. This phenomenon can be attributed to the strong decoherence of the initial mixed state among all the energy eigenstates of the final Hamiltonian which also includes the topological modes, causing their enhanced condensation probability. Our findings can be germane to real open topological materials with time-tunable band gap, and should be applicable to open topological crystals of arbitrary dimension and belonging to any symmetry class.
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It is well known that a bosonic Mott insulator can be realized by condensing vortices of a boson condensate. Usually, a vortex becomes an antivortex (and vice versa) under time reversal symmetry, and the condensation of vortices results in a trivial Mott insulator. However, if each vortex or antivortex interacts with a spin trapped at its core, the time reversal transformation of the composite vortex operator will contain an extra minus sign. It turns out that such a composite vortex condensed state is a bosonic topological insulator (BTI) with gapless boundary excitations protected by U(1)⋊Z2(T) symmetry. We point out that in BTI, an external π-flux monodromy defect carries a Kramers doublet. We propose lattice model Hamiltonians to realize the BTI phase, which might be implemented in cold atom systems or spin-1 solid state systems.
We introduce projected amorphous topological insulators (PATIs) that despite containing only a fraction (xxx) of otherwise randomly selected {\it disconnected} sites of a parent crystal, feature a quantized global topological invariant ({\it strong} PATIs) when its effective Hamiltonian is constructed by integrating out the residual sites of the original lattice. Within the topological regime of the parent model, additionally, there exists a critical xxx below which such systems foster a {\it fragile} PATI that only supports quantized local topological marker on a small fraction of sites therein. Both strong and fragile PATIs accommodate in-gap modes near the boundary. The system also hosts a normal insulator in the entire trivial parameter regime of the lattice-based model for any xxx, devoid of gapless boundary modes, for which both global topological invariant and local topological marker vanish. We demonstrate these {\it possibly} generic outcomes by starting with a parent square lattice-based model for time-reversal symmetry breaking insulators. Then the strong-to-fragile PATI quantum phase transition, tunable via xxx, is characterized by the {\it mean} correlation length exponent ν∈(1.00,1.46)ν\in (1.00,1.46)ν∈(1.00,1.46).