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Exact solutions in interacting dimerized Kitaev topological superconductors: Topological charge-density-wave and Schr\"odinger-cat states

Motohiko Ezawa

Published Jul 13, 2017• 0 citations
OpenAlex

Abstract

It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is also exactly solvable. In this extended model there are two topological indices associated with the sublattice symmetry and the particle-hole symmetry. We analytically determine the topological phase diagram containing seven distinct topological phases. There are two tetra-critical points, at which four distinct phases touch. It is intriguing that a topological charge-density-wave state and a topological Schr\odinger-cat state emerge for strong interactions. We confirm various topological phases by examining the presence of zero-energy edge states.

Related papers

Exact solutions and topological phase diagram in interacting dimerized Kitaev topological superconductors

Motohiko Ezawa

It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two-step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is shown also to be exactly solvable. We analytically determine the topological phase diagram containing seven distinct phases. It is argued that the system is topological when a fermionic many-body Majorana zero-energy edge state emerges. It is intriguing that there are two tetracritical points, at each of which four distinct phases touch.

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arXiv:2610.37761•Oct 27, 2026•0 citations
It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is also exactly solvable. In this extended model there are two topological indices associated with the sublattice symmetry and the particle-hole symmetry. We analytically determine the topological phase diagram containing seven distinct topological phases. There are two tetra-critical points, at which four distinct phases touch. It is intriguing that a topological charge-density-wave state and a topological Schr\odinger-cat state emerge for strong interactions. We confirm various topological phases by examining the presence of zero-energy edge states.
It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two-step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is shown also to be exactly solvable. We analytically determine the topological phase diagram containing seven distinct phases. It is argued that the system is topological when a fermionic many-body Majorana zero-energy edge state emerges. It is intriguing that there are two tetracritical points, at each of which four distinct phases touch.
We study analytically topological properties of a noninteracting modified dimerized Kitaev chain and an exactly solvable interacting dimerized Kitaev chain under open boundary conditions by analyzing two introduced edge correlation functions. The interacting dimerized Kitaev chain at the symmetry point $\mathrm{\ensuremath{\Delta}}=t$ and the chemical potential $\ensuremath{\mu}=0$ can be exactly solved by applying two Jordan-Wigner transformations and a spin rotation, which permits us to calculate the edge correlation functions analytically. We demonstrate that the two edge correlation functions can be used to characterize the trivial, Su-Schrieffer-Heeger-like topological and topological superconductor phases of both the noninteracting and interacting systems and give their phase diagrams.
We present a detailed study of the phase diagram of the Kitaev-Hubbard chain, that is the Kitaev chain in the presence of a nearest-neighbor density-density interaction, using both analytical techniques as well as DMRG. In the case of a moderate attractive interaction, the model has the same phases as the noninteracting chain, a trivial and a topological phase. For repulsive interactions, the phase diagram is more interesting. Apart from the previously observed topological, incommensurate, and charge density wave phases, we identify the ``excited state charge density wave'' phase. In this phase, the ground state resembles an excited state of an ordinary charge density phase, but is lower in energy due to the frustrated nature of the model. We find that the dynamical critical exponent takes the value $z\ensuremath{\simeq}1.8$. Interestingly, this phase only appears for even system sizes, and is sensitive to the chemical potential on the edges of the chain. For the topological phase, we present an argument that excludes the presence of a strong zero mode for a large part of the topological phase. For the remaining region, we study the time dependence of the edge magnetization (using the bosonic incarnation of the model). These results further expand the region where a strong zero mode does not occur.
The Kitaev chain model with a nearest neighbor interaction U is solved exactly at the symmetry point Δ=t and chemical potential μ=0 in an open boundary condition. By applying two Jordan-Wigner transformations and a spin rotation, such a symmetric interacting model is mapped onto a noninteracting fermion model, which can be diagonalized exactly. The solutions include a topologically nontrivial phase at |U|<t and a topologically trivial phase at |U|>t. The two phases are related by dualities. Quantum phase transitions in the model are studied with the help of the exact solution.
The p-wave superconductive wire with Hubbard interaction UUU is solved exactly at the symmetry point Δ=t\Delta=tΔ=t and chemical potential μ=0\mu=0μ=0 in an open boundary condition. By applying two Jordan-Wigner transformations and individuating non-local constant of motions such interacting model is mapped onto a noninteracting fermion model which can be diagonalized exactly. In the lowest energy subspace the mapping is onto a Kitaev chain. The system enters in a topologically trivial phase due to a second order quantum phase transition at U=±2tU=\pm 2tU=±2t. The features of the quantum phases are characterized with the help of the exact solution.
We study the interacting Kitaev chain with nearest-neighbor interactions and finite chemical potential using a combination of bosonization and density-matrix renormalization-group calculations. Focusing away from special exactly solvable lattice limits, we develop a continuum field-theoretic description of the generic finite-density interacting regime, where doping nonperturbatively modifies the competition between superconducting pairing and commensurate charge ordering. We show that the phase diagram consists of an intermediate incommensurate Luttinger liquid between the topological superconducting and commensurate charge-density-wave Mott phases, as well as a trivial band-insulating regime at large chemical potential. The commensurate--incommensurate and incommensurate--topological-superconducting phase boundaries are governed by Pokrovsky--Talapov and Berezinskii--Kosterlitz--Thouless criticality, respectively. We analyze the interacting topological superconducting regime and its boundary excitations. At the semiclassical level, and explicitly in the free-fermion boundary theory, the topological phase supports exponentially localized Majorana edge modes at interfaces between topological and trivial regions. We also construct a dual continuum fermionic representation in which this topological bulk sector is described instead by conventional symmetry breaking. This gives a continuum version of the SPT--SSB correspondence between different fermionic representations. Overall, our results provide an analytical description of one-dimensional interacting topological superconductors, from the competition between topology and commensurability and the resulting quantum criticality to the dual symmetry-breaking description of the topological phase.