Topological orders and associated topological protected excitations satisfying non-Abelian statistics have been widely explored in various platforms. The Z3 parafermions are regarded as the most natural generation of the Majorana fermions to realize these topological orders. Here we investigate the topological phase and emergent Z2 spin phases in an extended parafermion chain. This model exhibits rich variety of phases, including not only topological ferromagnetic phase, which supports non-Abelian anyon excitation, but also spin-fluid, dimer and chiral phases from the emergent Z2 spin model. We generalize the measurement tools in Z2 spin models to fully characterize these phases in the extended parafermion model and map out the corresponding phase diagram. Surprisingly, we find that all the phase boundaries finally merge to a single supercritical point. In regarding of the rather generality of emergent phenomena in parafermion models, this approach opens a wide range of intriguing applications in investigating the exotic phases in other parafermion models.Topological orders have been one of the major concerns in modern physics due to their potential realization of non-Abelian anyons for topological quantum computation [1,2]. Along this line, the Majorana zero modes have been realized in experiments in semiconductor/topological insulator and superconductor hybrid structures [3][4][5][6][7][8]. This approach may be directly generalized to Z k parafermion[9-13] (with k = 2 for Majorana fermions) with k-fold ground state degeneracy, following the pioneering work by Fendley [14][15][16]. In these phases, the Z 3 parafermion model is most intriguing due to its potential construction of Fibonacci anyons [17][18][19][20][21][22] for universal quantum computation. Recently, these parafermions are proposed to be constructed in semiconductor and fractional quantum Hall state hybrid structures [17,20,21,23,24].In this work, we explore the emergent phenomena in the parafermion models. We consider an extended parafermion Z 3 model, which is mapped to a Z 3 clock model with next nearest neighboring (NNN) interaction. In the presence of strong Zeeman field, the clock model can be projected to a conventional Z 2 spin model, giving rise to emergent spin-fluid (SF), dimer and chiral phases. This model exhibits rich variety of phases, which are characterized using various tools directly generalized from Z 2 spin models. We map out the whole phase diagram and find that the topological ferromagnetic parafermion (FP) phase is greatly enhanced in the presence of ferromagnetic interaction between NNN sites. Strikingly, all the phase boundaries finally merge to a single supercritical (SC) point. This approach lays foundation for understanding exotic phases in other parafermion models.Model and Hamiltonian. We consider the following extended Z 3 parafermion chains,wherewith ω = e i2π/3 and α j are parafermions satisfying α 3 j = 1, α † j = α 2 j and α i α j = α j α i ω sgn(i−j) , J and h correspond to the pairings -0.5 0.5 Dimer SF Topolog...

The Quantum Approximate Optimization Algorithm (QAOA), which is a variational quantum algorithm, aims to give sub-optimal solutions of combinatorial optimization problems. It is widely believed that QAOA has the potential to demonstrate application-level quantum advantages in the noisy intermediate-scale quantum(NISQ) processors with shallow circuit depth. Since the core of QAOA is the computation of expectation values of the problem Hamiltonian, an important practical question is whether we can find an efficient classical algorithm to solve quantum mean value in the case of general shallow quantum circuits. Here, we present a novel graph decomposition based classical algorithm that scales linearly with the number of qubits for the shallow QAOA circuits in most optimization problems except for complete graph case. Numerical tests in Max-cut, graph coloring and Sherrington-Kirkpatrick model problems, compared to the state-of-the-art method, shows orders of magnitude performance improvement. Our results are not only important for the exploration of quantum advantages with QAOA, but also useful for the benchmarking of NISQ processors.

We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by generalizing the isotropic Majumdar–Ghosh model to the fully anisotropic region. We demonstrate that this phase can be realized even in models when antiferromagnetic interaction along one of the three directions. This model also supports three different ferromagnetic (FM) phases, denoted as x-FM, y-FM and z-FM, polarized along the three directions. The boundaries between the exact dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry breaking or $${\mathbb {Z}}_2$$ Z 2 symmetry breaking leads to exact dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge $$c=1$$ c = 1 for free fermions. We characterize the properties of these boundaries using entanglement entropy, excitation gap, and long-range spin–spin correlation functions. These results are relevant to a large number of one dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in realistic materials with magnetic field along different directions and show that the anisotropy may resolve the disagreement between theory and experiments based on isotropic spin-spin interactions.

The Haldane phase represents one of the most important symmetry protected states in modern physics. This state can be realized using spin-1 and spin-1 2 Heisenberg models and bosonic particles. Here we explore the emergent Haldane phase in an alternating bond Z3 parafermion chain, which is different from the previous proposals from fundamental statistics and symmetries. We show that this emergent phase can also be characterized by a modified long-range string order, as well as four-fold degeneracy in the ground state energies and entanglement spectra. This phase is protected by both the charge conjugate and parity symmetry, and the edge modes are shown to satisfy parafermionic statistics, in which braiding of the two edge modes yields a 2π 3 phase. This model also supports rich phases, including topological ferromagnetic parafermion (FP) phase, trivial paramagnetic parafermion phase, classical dimer phase and gapless phase. The boundaries of the FP phase are shown to be gapless and critical with central charge c = 4/5. Even in the topological FP phase, it is also characterized by the long-range string order, thus we observe a drop of string order across the phase boundary between the FP phase and Haldane phase. These phenomena are quite general and this work opens a new way for finding exotic topological phases in Z k parafermion models. *