At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase transitions separating them. While the field theory provides details on the properties of the system close to the phase transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of this work. We describe a lattice gauge theory of the Majorana toric code in terms of $\mathrm{U}(1)$ matter fields coupled to an emergent ${\mathbb{Z}}_{2}$ gauge field. Subsequently, we use a generalized Wilson-loop order parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott-insulator and the charge- $2e$ superconductor phases, while showing that the toric code phase disappears in the charge- $e$ superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase transitions separating them. While the field theory provides details on the properties of the system close to the phase transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of this work. We describe a lattice gauge theory of the Majorana toric code in terms of $\mathrm{U}(1)$ matter fields coupled to an emergent ${\mathbb{Z}}_{2}$ gauge field. Subsequently, we use a generalized Wilson-loop order parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott-insulator and the charge-$2e$ superconductor phases, while showing that the toric code phase disappears in the charge-$e$ superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.

arXiv:1906.05158Physical review. B./Physical review. BSep 6, 20195 citations

Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases

Meng Sun, Zongyuan Wang, Nathanan Tantivasadakarn +1 more

Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic $\mathbb Z_2$ topological order in $(2{+}1)$ dimensions, using $\mathbb Z_8$ Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for $\mathbb Z_D^F$ symmetries for $D$ even. Using this, we realize a stabilizer model for a nontrivial $\mathbb Z_8^F$ fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic $\mathbb Z_2$ topological order in $(2{+}1)$ dimensions, using $\mathbb Z_8$ Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for $\mathbb Z_D^F$ symmetries for $D$ even. Using this, we realize a stabilizer model for a nontrivial $\mathbb Z_8^F$ fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.

quant-pharXiv:2606.25048Jun 25, 20260 citations

Topological order, topological insulators, and the search for the Majorana fermion

Joel E. Moore

Bulletin of the American Physical SocietyMar 23, 20110 citations

Majorana edge states for Z2 topological orders of the Wen plaquette and toric code models

Jing Yu, Xing-Hai Zhang, Su-Peng Kou

In this paper, we study the symmetry protected Majorana edge states for the ${\mathbb{Z}}_{2}$ topological order of the Wen plaquette and toric code models and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at $k=0$ and $\ensuremath{\pi}$. For the edge states of the toric code model without translational symmetry, the edge modes become gapped.In this paper, we study the symmetry protected Majorana edge states for the ${\mathbb{Z}}_{2}$ topological order of the Wen plaquette and toric code models and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at $k=0$ and $\ensuremath{\pi}$. For the edge states of the toric code model without translational symmetry, the edge modes become gapped.

arXiv:1209.5460Physical Review BMay 3, 201312 citations

Topological Quantum Computation by Manipulating Toric Code

Su-Peng Kou

May 18, 20080 citations

\mathrm{U}(1)

matter fields coupled to an emergent

{\mathbb{Z}}_{2}

gauge field. Subsequently, we use a generalized Wilson-loop order parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott-insulator and the charge-

2e

superconductor phases, while showing that the toric code phase disappears in the charge-

e

superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.

\mathbb Z_2

topological order in

(2{+}1)

dimensions, using

\mathbb Z_8

Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for

\mathbb Z_D^F

symmetries for

D

even. Using this, we realize a stabilizer model for a nontrivial

\mathbb Z_8^F

fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.

In this paper, we study the symmetry protected Majorana edge states for the

{\mathbb{Z}}_{2}

topological order of the Wen plaquette and toric code models and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at

k=0

and $\ensuremath{\pi}$. For the edge states of the toric code model without translational symmetry, the edge modes become gapped.