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.

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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...

A system gradually driven through a symmetry-breaking phase transition is subject to the Kibble-Zurek mechanism (KZM). As a consequence of the critical slowing down, its state cannot follow local equilibrium, and its evolution becomes non-adiabatic near the critical point. In the simplest approximation, that stage can be regarded as "impulse" where the state of the system remains unchanged. It leads to the correct KZM scaling laws. However, such "freeze-out" might suggest that the coherence length of the nascent order parameter remains unchanged as the critical region is traversed. By contrast, the original causality-based discussion emphasized the role of the sonic horizon: domains of the broken symmetry phase can expand with a velocity limited by the speed of the relevant sound. This effect was demonstrated in the quantum Ising chain where the dynamical exponent z = 1 and quasiparticles excited by the transition have a fixed speed of sound. To elucidate the role of the sonic horizon, in this paper we study two systems with z > 1 where the speed of sound is no longer fixed, and the fastest excited quasiparticles set the size of the sonic horizon. Their effective speed decays with the increasing transition time. In the extreme case, the dynamical exponent z can diverge such as in the Griffiths region of the random Ising chain where localization of excited quasiparticles freezes the growth of the correlation range when the critical region is traversed. Of particular interest is an example with z < 1 -the long-range extended Ising chain, where there is no upper limit to the velocity of excited quasiparticles with small momenta. Initially, the power-law tail of the correlation function grows adiabatically, but in the non-adiabatic stage it lags behind the adiabatic evolution-in accord with a generalized Lieb-Robinson bound.I.

We generalize the method introduced in Phys. Rev. B 101, 041108 (2020) of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS) to non-Abelian topological order. When wrapped on a torus the unique iPEPS becomes a superposition of degenerate and locally indistinguishable ground states. We find numerically symmetries of the iPEPS, represented by infinite matrix product operators (MPO), and their fusion rules. The rules tell us how to combine the symmetries into projectors onto states with well defined anyon flux. A linear structure of the MPO projectors allows for efficient determination for each state its second Renyi topological entanglement entropy on an infinitely long cylinder directly in the limit of infinite cylinder's width. The same projectors are used to compute topological S and T matrices encoding mutual-and self-statistics of emergent anyons. The algorithm is illustrated by examples of Fibonacci and Ising non-Abelian string net models.

We investigate the Kitaev-Heisenberg (KH) model at finite temperature using the exact environment full update (eeFU), introduced in Phys. Rev. B 99, 035115 (2019), which represents purification of a thermal density matrix on an infinite hexagonal lattice by an infinite projected entangled pair state (iPEPS). We estimate critical temperatures for coupling constants in the stripy and the antiferromagnetic phase. They are an order of magnitude less than the couplings. arXiv:1906.02220v1 [cond-mat.str-el]

We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in previous works [A. Francuz et al., Phys. Rev. B 101, 041108(R) (2020) and A. Francuz and J. Dziarmaga ibid. 102, 235112 (2020)] we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of matrix product states to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological F symbols of the underlying fusion category, which unequivocally identifies the topological order of the ground state. We bring the F symbols to the canonical gauge, and also compute the Drinfeld center of this unitary fusion category to extract the topological S and T matrices encoding mutual statistics and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, Kitaev model, double semion, and twisted quantum double of Z 3 , as well as to non-Abelian double Fibonacci, double Ising, and quantum double of S 3 and Rep(S 3 ) string-net models.