Microscopic models for Kitaev's sixteenfold way of anyon theories

Chulliparambil, Sreejith; Seifert, Urban F. P.; Vojta, Matthias; Janssen, Lukas; Tu, Hong-Hao

doi:10.1103/physrevb.102.201111

Cited by 36 publications

(17 citation statements)

References 40 publications

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“…This additional Z 2 symmetry generated by (−1) N1 combines with SO(2n) to form O(2n) group. Recently, a family of exactly solvable generalizations of Kitaev's Z 2 spin liquid was introduced [46,47], that realizes all Spin(ν) 1 topological phases for any integer ν ≥ 1, and notably the O(ν) symmetry is realized exactly in the lattice model.…”

Section: Spin(2n)1mentioning

confidence: 99%

Topological Disorder Parameter

Chen,

Tu,

Meng

et al. 2022

Preprint

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We introduce a many-body topological invariant, called the topological disorder parameter (TDP), to characterize gapped quantum phases with global internal symmetry in (2+1)d. TDP is defined as the constant correction that appears in the ground state expectation value of a partial symmetry transformation applied to a connected spatial region M , the absolute value of which scales generically as exp(−αl+γ) where l is the perimeter of M and γ is the TDP. Motivated by a topological quantum field theory interpretation of the operator, we show that e γ can be related to the quantum dimension of the symmetry defect, and provide a general formula for γ when the entanglement Hamiltonian of the topological phase can be described by a (1+1)d conformal field theory (CFT). A special case of TDP is equivalent to the topological Rényi entanglement entropy when the symmetry is the cyclic permutation of the replica of the gapped phase. We then investigate several examples of lattice models of topological phases, both analytically and numerically, in particular when the assumption of having a CFT edge theory is not satisfied. We also consider an example of partial translation symmetry in Wen's plaquette model and show that the result can be understood using the edge CFT. Our results establish a new tool to detect quantum topological order.

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Section: Spin(2n)1mentioning

confidence: 99%

Topological Disorder Parameter

Chen,

Tu,

Meng

et al. 2022

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“…[22,24] in the weakly-interacting limit. Similarly, for 𝑁 = 2, the 𝜋-flux model describes the ground-state flux sector of a generalized Kitaev spin-orbital liquid on the square lattice [23,25], perturbed by an additional Ising spin-spin interaction [32]. In these spin-orbital realizations of the SO(2) and SO(3) Majorana-Hubbard Hamiltonians, the hopping term corresponds to a generalized Kitaev spin-orbital exchange coupling [25], while the interaction terms map to Heisenberg and Ising spin-spin interactions, respectively [32].…”

Section: Modelsmentioning

confidence: 99%

“…The model is exactly solvable using a parton decomposition, in which the spin Hamiltonian is mapped to a tight-binding Hamiltonian of Majorana fermions hopping in the background of a static Z 2 gauge field. This con-struction has recently been extended to other tricoordinated lattices [14][15][16][17][18][19][20][21], as well as to systems with larger local Hilbert spaces [22][23][24][25]. In the latter cases, instead of a single Majorana fermion, an 𝑁-component vector of Majorana fermions emerges at each lattice site.…”

Section: Introductionmentioning

confidence: 96%

Phase diagrams of SO($N$) Majorana-Hubbard models: Dimerization, internal symmetry breaking, and fluctuation-induced first-order transitions

Janssen,

Seifert

2021

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We study the zero-temperature phase diagrams of Majorana-Hubbard models with SO(𝑁) symmetry on twodimensional honeycomb and 𝜋-flux square lattices, using mean-field and renormalization group approaches. The models can be understood as real counterparts of the SU(𝑁) Hubbard-Heisenberg models, and may be realized in Abrikosov vortex phases of topological superconductors, or in fractionalized phases of strongly-frustrated spin-orbital magnets. In the weakly-interacting limit, the models feature stable and fully symmetric Majorana semimetal phases. Increasing the interaction strength beyond a finite threshold for large 𝑁, we find a direct transition towards dimerized phases, which can be understood as staggered valence bond solid orders, in which part of the lattice symmetry is spontaneously broken and the Majorana fermions acquire a mass gap. For small to intermediate 𝑁, on the other hand, phases with spontaneously broken SO(𝑁) symmetry, which can be understood as generalized Néel antiferromagnets, may be stabilized. These antiferromagnetic phases feature fully gapped fermion spectra for even 𝑁, but gapless Majorana modes for odd 𝑁. While the transitions between Majorana semimetal and dimerized phases are strongly first order, the transitions between Majorana semimetal and antiferromagnetic phases are continuous for small 𝑁 ≤ 3 and weakly first order for intermediate 𝑁 ≥ 4. The weakly-first-order nature of the latter transitions arises from fixed-point annihilation in the corresponding effective field theory, which contains a real symmetric tensorial order parameter coupled to the gapless Majorana degrees of freedom, realizing interesting examples of fluctuation-induced first-order transitions.

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“…SO(6) Majorana representation. The largest symmetry allowed for spins S=3/2 is SU(4) SO( 6), thus, one can construct an SO(6) Majorana representation for spin-3/2's [39][40][41][42][43][44][45][46][47][48][49][50][51][52]. We introduce three gauge Majorana fermions η a i and three itinerant Majorana fermions θ a i (a = x, y, z), to obtain…”

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confidence: 99%

Unveiling the S=3/2 Kitaev Honeycomb Spin Liquids

Jin,

Natori,

Pollmann

et al. 2021

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The S=3/2 Kitaev honeycomb model (KHM) has defied an analytical as well as numerical understanding because it is not exactly soluble like its S=1/2 brethren and in contrast to other spin-S Kitaev models numerical methods are plagued by a massive pile up of low energy states. Here, we uncover the phase diagram of the S=3/2 KHM and find gapped and gapless quantum spin liquids (QSLs) generally coexisting with spin quadrupolar orders. Employing an SO(6) Majorana fermion representation of spin-3/2's, we find an exact representation of the conserved plaquette fluxes in terms of static Z 2 gauge fields akin to the S=1/2 KHM which enables us to treat the remaining interacting matter fermion sector in a parton mean-field theory. The latter provides an explanation for the extensive near degeneracy of low energy states in the gapless phase via the appearance of almost flat Majorana bands close to zero energy. Our parton description is in remarkable quantitative agreement with numerical simulations using the density matrix renormalization group method, and is furthermore corroborated by the addition of a single ion anisotropy which continuously connects the gapless Dirac QSL of our model with that of the S=1/2 KHM. We discuss the implications of our findings for materials realization of higher S=3/2 KHMs and the stability of the QSL phase with respect to additional interactions.

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Microscopic models for Kitaev's sixteenfold way of anyon theories

Cited by 36 publications

References 40 publications

Topological Disorder Parameter

Topological Disorder Parameter

Phase diagrams of SO($N$) Majorana-Hubbard models: Dimerization, internal symmetry breaking, and fluctuation-induced first-order transitions

Unveiling the S=3/2 Kitaev Honeycomb Spin Liquids

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