When a system is driven across a quantum critical point at a constant rate its evolution must become non-adiabatic as the relaxation time τ diverges at the critical point. According to the Kibble-Zurek mechanism (KZM), the emerging post-transition excited state is characterized by a finite correlation lengthξ set at the timet =τ when the critical slowing down makes it impossible for the system to relax to the equilibrium defined by changing parameters. This observation naturally suggests a dynamical scaling similar to renormalization familiar from the equilibrium critical phenomena. We provide evidence for such KZM-inspired spatiotemporal scaling by investigating an exact solution of the transverse field quantum Ising chain in the thermodynamic limit.
We present a method of extracting information about the topological order from the ground state of a strongly correlated two-dimensional system computed with the infinite projected entangled pair state (iPEPS). For topologically ordered systems, the iPEPS wrapped on a torus becomes a superposition of degenerate, locally indistinguishable ground states. Projectors in the form of infinite matrix product operators (iMPO) onto states with well-defined anyon flux are used to compute topological S and T matrices (encoding mutual-and self-statistics of emergent anyons). The algorithm is shown to be robust against a perturbation driving string-net toric code across a phase transition to a ferromagnetic phase. Our approach provides accurate results near quantum phase transition, where the correlation length is prohibitively large for other numerical methods. Moreover, we used numerically optimized iPEPS describing the ground state of the Kitaev honeycomb model in the toric code phase and obtained topological data in excellent agreement with theoretical prediction.Topologically ordered phases [1] have in recent years attracted significant attention, mostly due to the fact that they support anyonic excitations -exotic quasiparticles that obey fractional statistics. They are of interest not only from a fundamental perspective but also because of the possibility of realizing fault-tolerant quantum computation [2] based on the braiding of non-Abelian anyons. An important challenge is to identify microscopic lattice Hamiltonians that can realize such exotic phases of matter. Apart from a number of exactly solvable models [2-4], verifying whether a given microscopic Hamiltonian realizes a topologically ordered phase and accessing its properties has traditionally been regarded as an extremely hard task.A leading computational approach is to use Density Matrix Renormalization Group (DMRG) [5, 6] on a long cylinder [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In the limit of infinitely long cylinders, DMRG naturally produces ground states with welldefined anyonic flux, from which one can obtain full characterization of a topological order, via so-called topological S and T matrices [21]. Since the proposal of Ref.[21], the study of topological order by computing the ground states of an infinite cylinder with DMRG has become a common practice [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].The cost of a DMRG simulation grows exponentially with the width of cylinder, effectively restricting this approach to thin cylinders. Instead, (infinite) Projected Entangled Pair States (iPEPS) allow for much larger systems [40][41][42]. However, (variationally optimized) iPEPS naturally describe ground states with a superposition of anyonic fluxes. Here we show, starting with one such PEPS, how to produce a PEPS-like tensor network for each ground state with well-defined flux. Such tensor networks are suitable for extracting topological S and T matrices by computing overlaps between ground states. * correspondi...
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