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arXiv:1703.01800 · Physical review. B./Physical review. B

Matrix product states for topological phases with parafermions

Wen-Tao Xu, Guang-Ming Zhang

Published May 10, 2017• 16 citations
arXivHTMLPDF

Abstract

In the Fock representation, we propose a framework to construct the generalized matrix product states (MPS) for topological phases with ${\mathbb{Z}}_{p}$ parafermions. Unlike the ${\mathbb{Z}}_{2}$ Majorana fermions, the ${\mathbb{Z}}_{p}$ parafermions form intrinsically interacting systems. Here we explicitly construct two topologically distinct classes of irreducible ${\mathbb{Z}}_{3}$ parafermionic MPS wave functions, characterized by one or two parafermionic zero modes at each end of an open chain. Their corresponding parent Hamiltonians are found as the fixed point models of the single ${\mathbb{Z}}_{3}$ parafermion chain and two-coupled parafermion chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry. Our results thus pave the road to investigate all possible topological phases with ${\mathbb{Z}}_{p}$ parafermions within the matrix product representation in one dimension.

DOI
10.1103/physrevb.95.195122

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•Jan 10, 2018•0 citations
In the Fock representation, we propose a framework to construct the generalized matrix product states (MPS) for topological phases with Zp{\mathbb{Z}}_{p}Zp​ parafermions. Unlike the Z2{\mathbb{Z}}_{2}Z2​ Majorana fermions, the Zp{\mathbb{Z}}_{p}Zp​ parafermions form intrinsically interacting systems. Here we explicitly construct two topologically distinct classes of irreducible Z3{\mathbb{Z}}_{3}Z3​ parafermionic MPS wave functions, characterized by one or two parafermionic zero modes at each end of an open chain. Their corresponding parent Hamiltonians are found as the fixed point models of the single Z3{\mathbb{Z}}_{3}Z3​ parafermion chain and two-coupled parafermion chains with Z3\ifmmode×\else\texttimes\fiZ3{\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}Z3​\ifmmode×\else\texttimes\fiZ3​ symmetry. Our results thus pave the road to investigate all possible topological phases with Zp{\mathbb{Z}}_{p}Zp​ parafermions within the matrix product representation in one dimension.
In the Fock representation, we construct matrix product states (MPS) for one-dimensional gapped phases for Zp{\mathbb{Z}}_{p}Zp​ parafermions. From the analysis of irreducibility of MPS, we classify all possible gapped phases of Zp{\mathbb{Z}}_{p}Zp​ parafermions without extra symmetry other than Zp{\mathbb{Z}}_{p}Zp​ charge symmetry, including topological phases, spontaneous symmetry breaking phases, and a trivial phase. For all phases, we find the irreducible forms of local matrices of MPS, which span different kinds of graded algebras. The topological phases are characterized by the nontrivial simple Zp{\mathbb{Z}}_{p}Zp​ graded algebras with the characteristic graded centers, yielding the degeneracies of the full transfer matrix spectra uniquely. However, the spontaneous symmetry breaking phases correspond to the trivial semisimple Zp/n{\mathbb{Z}}_{p/n}Zp/n​ graded algebras, which can be further reduced to the trivial simple Zp/n{\mathbb{Z}}_{p/n}Zp/n​ graded algebras, where nnn is the divisor of ppp. So the present results provide the complete classification of the parafermionic gapped phases and deepen our understanding of topological phases in one dimension.
Under an appropriate symmetric bulk bipartition in a one-dimensional symmetry protected topological phase with the Affleck-Kennedy-Lieb-Tasaki matrix product state wave function for the odd integer spin chains, a bulk critical entanglement spectrum can be obtained, describing the excitation spectrum of the critical point separating the topological phase from the trivial state with the same symmetry. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry-breaking phase transitions. Recently, the framework of matrix product states for topological phases with Majorana fermions/parafermions has been established. Here we first generalize these fixed-point matrix product states with the zero correlation length to the more generic ground-state wave functions with a finite correlation length for the general one-dimensional interacting Majorana fermion/parafermion systems. Then we employ the previous method to decode quantum criticality to the interacting Majorana fermion/parafermion matrix product states. The obtained quantum critical spectra are described by the conformal field theories with central charge $c\ensuremath{\le}1$, characterizing the quantum critical theories separating the fermionic/parafermionic topological phase from the trivial phases with the same symmetry.
Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this Letter, we present a nontrivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wave functions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the nonapproximate study of topological one-dimensional parafermionic chains with spatial inversion and time-reversal symmetry in the absence of strong edge modes.
Parafermion modes are non-Abelian anyons which were introduced as ZN{\mathbb{Z}}_{N}ZN​ generalizations of Z2{\mathbb{Z}}_{2}Z2​ Majorana states. In particular, Z3{\mathbb{Z}}_{3}Z3​ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on Z3{\mathbb{Z}}_{3}Z3​ parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of Z3{\mathbb{Z}}_{3}Z3​ parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t\ensuremath{-}J$ model with a Kitaev-like chain supporting free Z3{\mathbb{Z}}_{3}Z3​ parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing Z3{\mathbb{Z}}_{3}Z3​ parafermions in strongly interacting electronic systems.
We develop the formalism of fermionic matrix product states (fMPS) and show how irreducible fMPS fall in two different classes, related to the different types of simple Z2{\mathbb{Z}}_{2}Z2​ graded algebras, which are physically distinguished by the absence or presence of Majorana edge modes. The local structure of fMPS with Majorana edge modes also implies that there is always a twofold degeneracy in the entanglement spectrum. Using the fMPS formalism, we make explicit the correspondence between the Z8{\mathbb{Z}}_{8}Z8​ classification of time-reversal-invariant spinless superconductors and the modulo 8 periodicity in the representation theory of real Clifford algebras. Studying fMPS with general onsite unitary and antiunitary symmetries allows us to define invariants that label symmetry-protected phases of interacting fermions. The behavior of these invariants under stacking of fMPS is derived, which reveals the group structure of such interacting phases. We also consider spatial symmetries and show how the invariant phase factor in the partition function of reflection-symmetric phases on an unorientable manifold appears in the fMPS framework.
Topological orders and associated topological protected excitations satisfying Abelian or non-Abelian statistics have been widely explored in various platforms. In recent years, the Majorana zero modes with these features have been realized in experiments based on semiconductor and superconducting hybrid structures, opening an exciting avenue for realization of fault-tolerant topological quantum computations. The Z3\mathbb{Z}_3Z3​ parafermions are regarded as the most natural generation of the Majorana fermions for realizing of more intriguing topological orders. In this work, we investigate the topological phase and emergent fermions in an extended parafermion chain using exact numerical methods. This model exhibits rich variety of phases, including threefold degenerate topological ferromagnetic phase, trivial paramagnetic phase, XX phase, dimer phase, chiral phase and commensurate phase. We present a smooth connection between this model and the extended spin-12{1\over 2}21​ XX model, by which we can generalize their measurements, such as the order parameters and long-range spin-spin correlations {\it etc}, to fully characterize the properties of these phases and map out their corresponding phase diagram. Strikingly, we find that all the phase boundaries finally merge to a single supercritical point, determined by the extended three-state Potts model. This approach opens a wide range of intriguing applications in investigating the phases and emergent phenomena in other Zk\mathbb{Z}_kZk​ parafermion models, and may be also illuminating for their experimental realizations.