We study a tight binding model of $\mathbb{Z}_3$-Fock parafermions with single-particle and pair-hopping terms. The phase diagram has four different phases: a gapped phase, a gapless phase with central charge $c=2$, and two gapless phases with central charge $c=1$. We characterise each phase by analysing the energy gap, entanglement entropy and different correlation functions. The numerical simulations are complemented by analytical arguments.
Parafermions that generalize (Majorana or usual) fermions appear as interacting quasi-particles because of their nature. Although attempts to develop models with free (non-interacting) parafermions have been undertaken, existing proposals require unphysical conditions such as realizing purely non-Hermitian systems. Here we present a way for the realization of free Fock parafermions in the tight-binding model with controlled dissipation of a very simple form. Introducing dissipation transforms an originally non-integrable quantum model to an exactly solvable classical one.
We study the effects of disorder in a one-dimensional model of $\mathbb{Z}_{3}$ Fock parafermions which can be viewed as a generalization of the prototypical Kitaev chain. Exact diagonalization is employed to determine level statistics, participation ratios, and the dynamics of domain walls. This allows us to identify ergodic as well as finite-size localized phases. In order to distinguish Anderson from many-body localization, we calculate the time evolution of the entanglement entropy in random initial states using tensor networks. We demonstrate that a purely quadratic parafermion model does…
We study the effects of a random magnetic field on a one-dimensional (1D) spin-1 chain with {\it correlated} nearest-neighbor $XY$ interaction. We show that this spin model can be exactly mapped onto the 1D disordered tight-binding model of ${\mathbb Z}_3$ Fock parafermions (FPFs), exotic anyonic quasiparticles that generalize usual spinless fermions. Thus, we have a peculiar case of a disordered Hamiltonian that, despite being bilinear in the creation and annihilation operators, exhibits a many-body localization (MBL) transition owing to the nontrivial statistics of FPFs. This is in sharp con…
Parafermions are emergent excitations which generalize Majorana fermions and are potentially relevant to topological quantum computation. Using the concept of Fock parafermions, we present a mapping between lattice $\mathbb{Z}_4$ parafermions and lattice spin-$1/2$ fermions which preserves the locality of operators with $\mathbb{Z}_4$ symmetry. Based on this mapping, we construct an exactly solvable, local, and interacting one-dimensional fermionic Hamiltonian which hosts zero-energy modes obeying parafermionic algebra. We numerically show that this parafermionic phase remains stable in a wide…
Parafermions are emergent quasi-particles which generalize Majorana fermions and possess intriguing anyonic properties. The theoretical investigation of effective models hosting them is gaining considerable importance in view of present-day condensed-matter realizations where they have been predicted to appear. Here we study the simplest number-conserving model of particle-like Fock parafermions, namely a one-dimensional tight-binding model. By means of numerical simulations based on exact diagonalization and on the density-matrix renormalization group, we prove that this quadratic model is no…
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 $\ma…
Parafermions are a natural generalization of Majorana fermions. We consider a breathing Kagome lattice with complex hoppings by imposing $\mathbb{Z}_{3}$ clock symmetry in the complex energy plane. It is a non-Hermitian generalization of the second-order topological insulator characterized by the emergence of topological corner states. We demonstrate that the topological corner states are parafermions in the present $\mathbb{Z}_{3}$ clock-symmetric model. It is also shown that the model is realized in electric circuits properly designed, where the parafermion corner states are observed by impe…
Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this paper we present a non-trivial 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 wavefunctions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for th…
In this work we numerically study critical phases in translation-invariant $\mathbb{Z}_N$ parafermion chains with both nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a $\mathbb{Z}_N$ spin model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation and translation invariance ensures that the spin model is always self-dual. We first study the low-energy spectrum of chains with only nearest-neighbor coupling, which are mapped onto standard self-dual $\mathbb{Z}_N$ clock models. For $3\leq N\leq 6$ we match the numerical results to the known c…
Systems with topologically protected ground-state degeneracies are currently of great interest due to their potential applications in quantum computing. In practise this degeneracy is never exact, and the magnitude of the ground-state degeneracy splitting imposes constraints on the timescales over which information is topologically protected. In this Letter we use an instanton approach to evaluate the splitting of topological ground-state degeneracy in quasi-1D systems with parafermion zero modes, in the specific case where parafermions are realized by inducing a superconducting gap in pairs o…
Parafermions are exotic quasiparticles with non-Abelian fractional statistics that could be exploited to realize universal topological quantum computing. Here, we study the scrambling of quantum information in one-dimensional parafermionic chains, with a focus on $\mathbb{Z}_3$ parafermions in particular. We use the generalized out-of-time-ordered correlators (OTOCs) as a measure of the information scrambling and introduce an efficient method based on matrix product operators to compute them. With this method, we compute the OTOCs for $\mathbb{Z}_3$ parafermions chains up to $200$ sites for th…
Recent concrete proposals suggest it is possible to engineer a two-dimensional bulk phase supporting non-Abelian Fibonacci anyons out of Abelian fractional quantum Hall systems. The low-energy degrees of freedom of such setups can be modeled as $\mathbb{Z}_3$ parafermions "hopping" on a two-dimensional lattice. We use the density matrix renormalization group to study a model of this type interpolating between the decoupled-chain, triangular-lattice, and square-lattice limits. The results show clear evidence of the Fibonacci phase over a wide region of the phase diagram, most notably including …
In the Fock representation, we construct matrix product states (MPS) for one-dimensional gapped phases for $\mathbb{Z}_{p}$ parafermions. From the analysis of irreducibility of MPS, we classify all possible gapped phases of $\mathbb{Z}_{p}$ parafermions without extra symmetry other than $\mathbb{Z}%_{p}$ 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 non-trivial simple $\m…
It is shown that certain fractionally-charged quasiparticles can be modeled on \(D-\)dimensional lattices in terms of unconventional yet simple Fock algebras of creation and annihilation operators. These unconventional Fock algebras are derived from the usual fermionic algebra by taking roots (the square root, cubic root, etc.) of the usual fermionic creation and annihilation operators. If the fermions carry non-Abelian charges, then this approach fractionalizes the Abelian charges only. In particular, the \(m\)th-root of a spinful fermion carries charge \(e/m\) and spin \(1/2\). Just like tak…
The interacting Kane-Mele model with a long-range hopping is studied using analytical method. The original Kane-Mele model is defined on a honeycomb lattice. In the work, we introduce a four-lattice-constant range hopping and the on-site Hubbard interaction into the model and keep its lattice structure unchanged. From the single-particle energy spectrum, we obtain the critical strength of the long-range hopping $t_L$ at which the topological transition occurs in the non-interacting limit of the model and our results show that it is independent of the spin-orbit coupling. After introducing the …
We report the emergence or broadening of exotic magnetic metallic phases upon explicit breaking of $SU(2)$ symmetry by introduction of mass imbalance in a variant of Hubbard model, known as the ionic Hubbard model in the presence of frustration at half-filling on a square lattice. The ionic Hubbard model has in addition to hopping($\sim t$) and onsite coulomb repulsion ($\sim U$), a staggered ionic potential ($\sim Δ$) which breaks translational symmetry of the underlying lattice. In the low to intermediate ranges of $U$ and $Δ$, we use unrestricted Hartree-Fock theory to construct the phase d…
The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG) can be decomposed into a TBG-like flat band Hamiltonian and a high-velocity Dirac fermion Hamiltonian. We use Hartree-Fock mean field approach to study the projected Coulomb interacting Hamiltonian of TSTG developed in Călugăru et al. [Phys. Rev. B 103, 195411 (2021)] at integer fillings $ν=-3, -2, -1$ and $0$ measured from charge neutrality. We study the phase diagram with $w_0/w_1$, the ratio of $AA$ and $AB$ interlayer hoppings, and the displacement field, which introduces an interlayer potential $U$ and hybridize…
The role of the multiorbital effects on the emergence of frustrated electronic orders on the triangular lattice at half filling is investigated through an extended spinless fermion Hubbard model. By using two complementary approaches, unrestricted Hartree-Fock and exact diagonalizations, we unravel a very rich phase diagram controlled by the strength of both local and off-site Coulomb interactions and by the interorbital hopping anisotropy ratio $t'/t$. Three robust unconventional electronic phases, a pinball liquid, an inverse pinball liquid, and a large-unit-cell $\sqrt{12} \times \sqrt{12}$…
We study the correlated insulating phases of twisted bilayer graphene (TBG) in the absence of lattice strain at integer filling $ν=\pm3$. Using the self-consistent Hartree-Fock method on a particle-hole symmetric model and allowing translation symmetry breaking terms, we obtain the phase diagram with respect to the ratio of $AA$ interlayer hopping $(w_0)$ and $AB$ interlayer hopping $(w_1)$. When the interlayer hopping ratio is close to the chiral limit ($w_0/w_1 \lesssim 0.5$), a quantum anomalous Hall state with Chern number $ν_c = \pm 1$ can be observed consistent with previous studies. Aro…
We employ the Hartree-Fock approximation to identify the magnetic ground state of the Hubbard model on a frustrated square lattice. We investigate the phase diagram as a function of the Coulomb repulsion's strength $U$, and the ratio $t'/t$ between the nearest and next nearest neighbor hoppings $t$ and $t'$. At half-filling and for a sufficiently large $U$, an antiferromagnetic chiral spin density wave order with nonzero spin chirality emerges as the ground state in a wide regime of the phase diagram near $t'/t=1/\sqrt{2}$, where the Fermi surface is well-nested for both $(π,π)$ and $(π,0)/(0,…
Charge order appears to be an ubiquitous phenomenon in doped Mott insulators, which is currently under intense experimental and theoretical investigations particularly in the high $T_c$ cuprates. This phenomenon is conventionally understood in terms of Hartree-Fock type mean field theory. Here we demonstrate a mechanism for charge modulation which is rooted in the many-particle quantum physics arising in the strong coupling limit. Specifically, we consider the problem of a single hole in a bipartite $t-J$ ladder. As a remnant of the fermion signs, the hopping hole picks up subtle phases pendin…
Experimental realizations of topologically ordered states of matter, such as fractional quantum Hall states, with cold atoms are now within reach. In particular, optical lattices provide a promising platform for the realization and characterization of such states, where novel detection schemes enable an unprecedented microscopic understanding. Here we show that the central charge can be directly measured in current cold atom experiments using the number entropy as a proxy for the entanglement entropy. We perform density-matrix renormalization-group simulations of Hubbard-interacting bosons on …
