The ground state fidelity per lattice site is shown to be able to detect quantum phase transitions for the Kitaev model on the honeycomb lattice, a prototypical example of quantum lattice systems with topological order. It is found that, in the thermodynamic limit, the ground state fidelity per lattice site is non-analytic at the phase boundaries: the second-order derivative of its logarithmic function with respect to a control parameter describing the interaction between neighboring spins is logarithmically divergent. A finite size scaling analysis is performed, which allows us to extract the correlation length critical exponent from the scaling behaviors of the ground state fidelity per lattice site.
We study the reduced fidelity and reduced fidelity susceptibility in the Kitaev honeycomb model. It is shown that the reduced fidelity susceptibility of two nearest site manifest itself a peak at the quantum phase transition point, although the one-site reduced fidelity susceptibility vanishes. Our results directly reveal that the reduced fidelity susceptibility can be used to characterize the quantum phase transition in the Kitaev honeycomb model, and thus suggest that the reduced fidelity susceptibility is an accurate marker of the topological phase transition when it is properly chosen, des…
We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped A phase with Abelian anyon excitations to a gapless B phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be $1/ξ=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]$, which diverges ar…
The Kitaev model is a fascinating example of an exactly solvable model displaying a spin-liquid ground state in two dimensions. However, deviations from the original Kitaev model are expected to appear in real materials. In this Letter, we investigate the fate of Kitaev's spin liquid in the presence of disorder -- bond defects or vacancies -- for an extended version of the model. Considering static flux backgrounds, we observe a power-law divergence in the low-energy limit of the density of states with a nonuniversal exponent. We link this power-law distribution of energy scales to weakly coup…
We study the behavior of the defect and heat densities under sudden quenching near the quantum critical points in the two-dimensional Kitaev honeycomb model both in the thermodynamic and non-thermodynamic limits. We consider quenches starting from a quantum critical point into the gapped as well as the gapless phases. We choose points on the lines of anisotropic quantum critical points as well as different points of intersection of these lines as the initial points from where the quenching starts. We find that the defect and heat densities display the expected power-law scalings along with log…
The fidelity per site between two ground states of a quantum lattice system corresponding to different values of the control parameter defines a surface embedded in a Euclidean space. The Gaussian curvature naturally quantifies quantum fluctuations that destroy orders at transition points. It turns out that quantum fluctuations wildly distort the fidelity surface near the transition points, at which the Gaussian curvature is singular in the thermodynamic limit. As a concrete example, the one-dimensional quantum Ising model in a transverse field is analyzed. We also perform a finite size scalin…
Fidelity approach has been widely used to detect various types of quantum phase transitions, including some that are beyond the Landau symmetry breaking theory, in condensed matter models. However, challenges remain in locating the transition points with precision in several models with unconventional phases such as the quantum spin liquid phase in spin-1 Kitaev-Heisenberg model. In this work, we propose a novel approach, which we named the fidelity map, to detect quantum phase transitions with higher accuracy and sensitivity as compared to the conventional fidelity measures. Our scheme extend…
The Kitaev model on the honeycomb lattice is a paradigmatic system known to host a wealth of nontrivial topological phases and Majorana edge modes. In the static case, the Majorana edge modes are nondispersive. When the system is periodically driven in time, such edge modes can disperse and become chiral. We obtain the full phase diagram of the driven model as a function of the coupling and the driving period. We characterize the quantum criticality of the different topological phase transitions in both the static and driven model via the notions of Majorana-Wannier state correlation functions…
We propose a quantum protocol that allows preparing a ground state (GS) of the honeycomb Kitaev model. Our approach efficiently uses underlying symmetries and techniques from topological error correction. It is based on the stabilization procedure, the developed centralizer ansatz, and utilizes the vortex basis description as the most advantageous for qubit-based simulations. We demonstrate the high fidelity preparation of spin liquid ground states for the original Kitaev model, getting the exact GS for N=24 spins using 230 two-qubit operations. We then extend the variational procedure to non-…
The Kitaev model is a beautiful example of frustrated interactions giving rise to deep and unexpected phenomena. In particular, its classical version has remarkable properties stemming from exponentially large ground state degeneracy. Here, we present a study of magnetic clusters with spin-$S$ moments coupled by Kitaev interactions. We focus on two cluster geometries -- the Kitaev square and the Kitaev tetrahedron -- that allow us to explicitly enumerate all classical ground states. In both cases, the classical ground state space (CGSS) is large and self-intersecting, with non-manifold charact…
We present a reentrant phase transition in a bridging model between two different topological models: Kitaev and Haldane chains. This model is activated by introducing a bond alternation into the Kitaev chain [A. Yu Kitaev, Phys.-Usp. {\bf 44} 131 (2001)]. Without the bond alternation, finite pairing potential induces a topological state defined by zero-energy Majorana edge mode, while finite bond alternation without the pairing potential makes a different topological state similar to the Haldane state, which is defined by local Berry phase in the bulk. The topologically-ordered state correspo…
To provide a physical example of quantum scars, we study the many-body scars in the spin-1 Kitaev chain where the so-called PXP Hamiltonian is exactly embedded in the spectra. Regarding the conserved quantities, the Hilbert space is fragmented into disconnected subspaces and we explore the associated constrained dynamics. The continuous revivals of the fidelity and the entanglement entropy when the initial state is prepared in $\vert\mathbb{Z}_k\rangle$ ($k=2,3$) state illustrate the essential physics of the PXP model. We study the quantum phase transitions in the one-dimensional spin-1 Kitaev…
We study the interplay between the Kitaev and Ising interactions on both ladder and two dimensional lattices. We show that the ground state of the Kitaev ladder is a symmetry-protected topological (SPT) phase, which is protected by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. It is confirmed by the degeneracy of the entanglement spectrum and non-trivial phase factors (inequivalent projective representations of the symmetries), which are obtained within infinite matrix-product representation of numerical density matrix renormalization group. We derive the effective theory to describe the topo…
We study the quantum phase transitions in the two-dimensional spin-orbit models in terms of fidelity susceptibility and reduced fidelity susceptibility. An order-to-order phase transition is identified by fidelity susceptibility in the two-dimensional Heisenberg XXZ model with Dzyaloshinsky-Moriya interaction on a square lattice. The finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of second order. Two distinct types of quantum phase transitions are witnessed by fidelity susceptibility in Kitaev-Heisenbe…
We propose here a counterdiabatic (CD) strategy for fast preparation of a state with long-range topological order by magnetic field tuning of an initial separable state. For concreteness, we consider the ground state of the honeycomb Kitaev model whose long-range topological order together with the anyonic excitations make it an interesting candidate for fault-tolerant universal quantum computation and storage. The required CD perturbation is found to be local, having the form of the off-diagonal exchange interactions reminiscent of trigonal deformations in Kitaev Hamiltonians. We show that th…
The Kitaev honeycomb model is a paradigm of exactly-solvable models, showing non-trivial physical properties such as topological quantum order, abelian and non-abelian anyons, and chirality. Its solution is one of the most beautiful examples of the interplay of different mathematical techniques in condensed matter physics. In this paper, we show how to derive a tensor network (TN) description of the eigenstates of this spin-1/2 model in the thermodynamic limit, and in particular for its ground state. In our setting, eigenstates are naturally encoded by an exact 3d TN structure made of fermioni…
Matrix product states, a key ingredient of numerical algorithms widely employed in the simulation of quantum spin chains, provide an intriguing tool for quantum phase transition engineering. At critical values of the control parameters on which their constituent matrices depend, singularities in the expectation values of certain observables can appear, in spite of the analyticity of the ground state energy. For this class of generalized quantum phase transitions we test the validity of the recently introduced fidelity approach, where the overlap modulus of ground states corresponding to slight…
We have applied the real space quantum renormalization group approach to study the topological quantum phase transition in the one-dimensional chain of a spinless p-wave superconductor. We investigate the behavior of local compressibility and ground-state fidelity of the Kitaev chain. We show that the topological phase transition is signaled by the maximum of local compressibility at the quantum critical point tuned by the chemical potential. Moreover, a sudden drop of the ground-state fidelity and the divergence of fidelity susceptibility at the topological quantum critical point have been us…
Dynamical phase transitions (DPT) are characterized by nonanalytical time evolution of the dynamical free energy. For general 2-band systems in one and two dimensions (eg. SSH model, Kitaev-chain, Haldane model, p+ip superconductor, etc.), we show that the time evolution of the dynamical free energy is crucially affected by the ground state topology of both the initial and final Hamiltonians, implying DPTs when the topology is changed under the quench. Similarly to edge states in topological insulators, DPTs can be classified as being topologically protected or not. In 1D systems the number of…
Reduced fidelities for free fermions out of equilibrium: From dynamical quantum phase transitions to Mpemba effect
Laboratoire d’Annecy de Physique Théorique (LAPTh), CNRS, Université Savoie Mont Blanc（萨瓦-蒙布朗大学）；Dipartimento di Fisica, Università di Pisa, and INFN Sezione di Pisa（比萨大学）
Decoding Equilibrium and Dynamical Criticality in the 2D Topological Order
University of Science and Technology Beijing（北京科技大学）；Beijing Normal University（北京师范大学）；China University of Petroleum-Beijing（中国石油大学（北京））；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
In order to investigate the quantum phase transition in the one-dimensional quantum compass model, we numerically calculate non-local string correlations, entanglement entropy, and fidelity per lattice site by using the infinite matrix product state representation with the infinite time evolving block decimation method. In the whole range of the interaction parameters, we find that the four distinct string orders characterize the four different Haldane phases and the topological quantum phase transition occurs between the Haldane phases. The critical exponents of the string order parameters β=…
We study the exactly solvable 1D model: the dimerized $XY$ chain with uniform and staggered transverse fields, equivalent upon fermionization to the noninteracting dimerized Kitaev-Majorana chain with modulation. The model has three known gapped phases with local and nonlocal (string) orders, along with the gapless incommensurate (IC) phase in the $U(1)$ limit. The criticality is controlled by the properties of zeros of model's partition function, analytically continued onto the complex wave numbers. In the ground state they become complex zeros of the spectrum of the Hamiltonian. The analysis…
