Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Path-dependent Monopole Braiding and Anyonic Dipoles in a Higgsed Tensor Gauge Model

Yun-Tak Oh, Jintae Kim, Eun-Gook Moon, Jung Hoon Han

Published Oct 6, 2021• 0 citations
OpenAlex

Abstract

We propose two-dimensional spin models obtained by the U(1) rank-2 tensor gauge theory with scalar or vector charges on the L_x \times L_y square lattice with p degrees of freedom per orbital and three orbitals per unit cell. The resulting spin model is a stabilizer model consisting of three mutually commuting projectors which are, in turn, obtained by Higgsing the mutually commuting Gauss's law operators and the magnetic field operators in the underlying U(1) lattice gauge theory. The spin model thus obtained is exactly solvable, with the ground state degeneracy (GSD) given by log_p GSD = 2 + (1 +{\delta}_{L_x mod p},0)(1 +{\delta}_{L_y mod p},0) when p is a prime number. Two types of dipoles, pristine and emergent, are identified. Both the monopoles and the dipoles are free to move, with certain restrictions on monopoles to hop only by p lattice spacing. The monopole-monopole braiding phase depends on the separation of the x or y coordinates of the initial monopole positions, whereas the monopole-dipole braiding is anyonic and depends on the product of the monopole charge and the dipole moment. The braiding phases can be understood as the Aharonov-Bohm phases of some emergent vector potentials.

Related papers

Rank-2 toric code in two dimensions

Yun-Tak Oh, Jintae Kim, Eun-Gook Moon +1 more

We study a two-dimensional spin model obtained by ``Higgsing'' the rank-2 U(1) lattice gauge theory (LGT) with scalar or vector charges on the ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ square lattice under the periodic boundary condition (PBC). There are $p$ degrees of freedom per orbital and three orbitals per unit cell in the spin model. The resulting spin model is a stabilizer code consisting of three mutually commuting projectors that are, in turn, obtained by Higgsing the mutually commuting Gauss's law operators and the magnetic field operators in the underlying LGT. The spin model thus obtained is exactly solvable, with the ground state degeneracy (GSD) $D$ given by ${\text{log}}_{p}D=2+(1+{\ensuremath{\delta}}_{{L}_{x}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.16em}{0ex}}p,0})(1+{\ensuremath{\delta}}_{{L}_{y}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.16em}{0ex}}p,0})$ when $p$ is a prime number. Two types of dipole excitations, pristine and emergent, are identified. Both the monopoles and the dipoles are free to move, with restrictions on monopoles to hop only by $p$ lattice spacing along with certain directions. The monopole-monopole braiding phase depends on the separation of the $x$ or $y$ coordinates of the initial monopole positions, making it distinct from the ordinary anyon braiding statistics. The monopole-dipole braiding obeys the usual anyonic statistics. Despite the oddity, the monopole-monopole braiding phase can be understood as the Aharonov-Bohm phase of some emergent vector potentials.

arXiv:2110.02658•Physical review. B./Physical review. B•Jan 21, 2022•35 citations

Effective field theory of dipolar braiding statistics in two dimensions

Yun-Tak Oh, Jintae Kim, Jung Hoon Han

A rank-2 toric code (R2TC) Hamiltonian in two dimensions can be constructed as a Higgsed descendant of rank-2 U(1) lattice gauge theory. As noted by the authors recently [Y.-T. Oh, J. Kim, E.-G. Moon, and J. H. Han, Phys. Rev. B 105, 045128 (2022)] the quasiparticles in that model show unusual braiding statistics that depends on the initial locations of the particles which participate in the braiding. We show that this kind of statistical phase captures the total dipole moment of quasiparticles encompassed in the braiding, in contrast to the conventional anyonic braiding seeing the total charge. An Aharonov-Bohm interpretation of such dipolar braiding statistics is made in terms of emergent, rank-1 vector potentials that are built out of the underlying rank-2 gauge fields. Pertinent field theories of the quasiparticle dynamics in the R2TC are developed, and the accompanying conservation laws derived. A dipolar BF theory of the rank-2 gauge fields is constructed and shown to correctly capture the dipolar braiding statistics, in contrast to the conventional BF theory capturing the monopolar braiding statistics of anyons in the rank-1 toric code.

arXiv:2204.01279•Physical review. B./Physical review. B•Oct 26, 2022•23 citations

Aspects of ZN rank-2 gauge theory in (2+1) dimensions: Construction schemes, holonomies, and sublattice one-form symmetries

Yun-Tak Oh, Salvatore D. Pace, Jung Hoon Han +2 more

Rank-2 toric code (R2TC), a prototypical archetype of the discrete rank-2 symmetric gauge theory, has properties that differ from those of the standard toric code. Specifically, it features a blending of UV and IR in its ground state, restricted mobility of its quasiparticles, and variations in the braiding statistics of its quasiparticles based on their position. In this paper, we investigate various aspects of ${\mathbb{Z}}_{N}$ rank-2 gauge theory in $(2+1)$-dimensional spacetime. Firstly, we demonstrate that $U(1)$ rank-2 gauge theory can arise from $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ rank-1 gauge theory after condensing the gauge charges in a specific way. This construction scheme of $U(1)$ rank-2 gauge theory carries over to the ${\mathbb{Z}}_{N}$ case simply by Higgsing $U(1)$ to ${\mathbb{Z}}_{N}$, after which the resulting rank-2 gauge theory can be tuned to the R2TC. The holonomy operators of R2TC are readily identified using this scheme and are given clear physical interpretation as the pair creation/annihilation of various monopoles and dipoles. Explicit tensor network construction of the ground states of R2TC is given as two copies of the ground states of Kitaev's toric code that are ``sewn together'' according to the condensation scheme. In addition, through a similar anyon condensation protocol, we present a double semion version of rank-2 toric code whose flux excitations exhibit restricted mobility and semionic statistics. Finally, we identify the generalized discrete symmetries of the R2TC, which are much more complex than typical 1-form symmetries. They include conventional and unconventional 1-form symmetries, such as framed 1-form symmetries and what we call sublattice 1-form symmetries. Using these, we interpret the R2TC's unique properties (UV/IR mixing, position-dependent braiding, etc.) from the modern perspective of generalized spontaneous symmetry breaking and 't Hooft anomalies.

arXiv:2301.04706•Physical review. B./Physical review. B•Apr 28, 2023•22 citations

Hybrid rank-1 and rank-2 U(1) lattice gauge theory, the F3 model, and its effective field theory

Jintae Kim, Yun-Tak Oh, Jung Hoon Han

A number of exactly solvable spin models, including the Kitaev toric code in two and three dimensions and the X-cube model in three dimensions, can be related to their respective parent lattice gauge theories (LGTs) through the mathematical process of ``Higgsing.'' Field theories of the low-energy excitations of these spin models can be developed subsequently, building upon the symmetry of the parent LGTs. Recently, two of the present authors proposed a variant of the three-dimensional toric code which we now call the F3 model, whose elementary excitations consist of freeon and fluxon excitations of the three-dimensional toric code and fracton excitations of the X-cube model. In this work, we identify the parent LGT of the F3 model as the hybrid rank-1 and rank-2 U(1) LGT and develop the corresponding field theory. The resulting Lagrangian of the F3 model is that of a three-dimensional toric code with an extra term, which ties the dynamics of fractons to that of fluxons. The matter part of the effective action for the F3 model can be derived as well by carefully keeping track of the gauge invariance of the F3 model. Hydrodynamic equations of motion of the quasiparticles are derived which properly reflect their constrained dynamics. Finally, we present a tight-binding model for the quasiparticle motion in the F3 model.

arXiv:2203.14917•Physical review. B./Physical review. B•Oct 28, 2022•6 citations

Majorana Edge States and Braiding in an Exactly Solvable One-dimensional Spin Model

Zhao-Yang Dong, Jianxin Li

We derive an exactly solvable one-dimensional (1D) spin model from the three-band Hubbard model with a strong spin-orbit coupling by introducing U(1) gauge fields to the isospin states. We find that it has a topological nontrivial phase characterized by Majorana end modes which are protected by a new Z_2 topological invariant related to the parity of the lattice sites (odd or even number of sites) in the spin chain. With the protection of this Z_2 topological invariant, a novel braiding of two Majorana edge states in this strictly geometric 1D chain is realized. We also discuss the possible realization of the gauge fields.

arXiv:1412.8183•Dec 28, 2014•0 citations

Excitations in the higher-lattice gauge theory model for topological phases. I. Overview

Joe Huxford, Steven H. Simon

In this series of papers, we study a Hamiltonian model for (3+1)-dimensional topological phases introduced in [Bullivant et al., Phys. Rev. B 95, 155118 (2017)], based on a generalization of lattice gauge theory known as ``higher-lattice gauge theory.'' Higher-lattice gauge theory has so-called ``2-gauge fields'' describing the parallel transport of lines, in addition to ordinary 1-gauge fields which describe the parallel transport of points. In this series we explicitly construct the creation operators for the pointlike and looplike excitations supported by the model. We use these creation operators to examine the properties of the excitations, including their braiding statistics. These creation operators also reveal that some of the excitations are confined, costing energy to separate that grows linearly with the length of the creation operator used. This is discussed in the context of condensation-confinement transitions between different cases of this model. We also discuss the topological charges of the model and use explicit measurement operators to rederive a relationship between the number of charges measured by a 2-torus and the ground-state degeneracy of the model on the 3-torus. From these measurement operators, we can see that the ground-state degeneracy on the 3-torus is related to the number of types of linked looplike excitations. This first paper provides an accessible summary of our findings, with more detailed results and proofs to be presented in the other papers in the series.

arXiv:2202.08294•Physical review. B./Physical review. B•Dec 12, 2023•5 citations
We propose two-dimensional spin models obtained by the U(1) rank-2 tensor gauge theory with scalar or vector charges on the L_x \times L_y square lattice with p degrees of freedom per orbital and three orbitals per unit cell. The resulting spin model is a stabilizer model consisting of three mutually commuting projectors which are, in turn, obtained by Higgsing the mutually commuting Gauss's law operators and the magnetic field operators in the underlying U(1) lattice gauge theory. The spin model thus obtained is exactly solvable, with the ground state degeneracy (GSD) given by log_p GSD = 2 + (1 +{\delta}{L_x mod p},0)(1 +{\delta}{L_y mod p},0) when p is a prime number. Two types of dipoles, pristine and emergent, are identified. Both the monopoles and the dipoles are free to move, with certain restrictions on monopoles to hop only by p lattice spacing. The monopole-monopole braiding phase depends on the separation of the x or y coordinates of the initial monopole positions, whereas the monopole-dipole braiding is anyonic and depends on the product of the monopole charge and the dipole moment. The braiding phases can be understood as the Aharonov-Bohm phases of some emergent vector potentials.
We study a two-dimensional spin model obtained by ``Higgsing'' the rank-2 U(1) lattice gauge theory (LGT) with scalar or vector charges on the ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ square lattice under the periodic boundary condition (PBC). There are ppp degrees of freedom per orbital and three orbitals per unit cell in the spin model. The resulting spin model is a stabilizer code consisting of three mutually commuting projectors that are, in turn, obtained by Higgsing the mutually commuting Gauss's law operators and the magnetic field operators in the underlying LGT. The spin model thus obtained is exactly solvable, with the ground state degeneracy (GSD) DDD given by ${\text{log}}_{p}D=2+(1+{\ensuremath{\delta}}_{{L}_{x}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.16em}{0ex}}p,0})(1+{\ensuremath{\delta}}_{{L}_{y}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.16em}{0ex}}p,0})$ when ppp is a prime number. Two types of dipole excitations, pristine and emergent, are identified. Both the monopoles and the dipoles are free to move, with restrictions on monopoles to hop only by ppp lattice spacing along with certain directions. The monopole-monopole braiding phase depends on the separation of the xxx or yyy coordinates of the initial monopole positions, making it distinct from the ordinary anyon braiding statistics. The monopole-dipole braiding obeys the usual anyonic statistics. Despite the oddity, the monopole-monopole braiding phase can be understood as the Aharonov-Bohm phase of some emergent vector potentials.
A rank-2 toric code (R2TC) Hamiltonian in two dimensions can be constructed as a Higgsed descendant of rank-2 U(1) lattice gauge theory. As noted by the authors recently [Y.-T. Oh, J. Kim, E.-G. Moon, and J. H. Han, Phys. Rev. B 105, 045128 (2022)] the quasiparticles in that model show unusual braiding statistics that depends on the initial locations of the particles which participate in the braiding. We show that this kind of statistical phase captures the total dipole moment of quasiparticles encompassed in the braiding, in contrast to the conventional anyonic braiding seeing the total charge. An Aharonov-Bohm interpretation of such dipolar braiding statistics is made in terms of emergent, rank-1 vector potentials that are built out of the underlying rank-2 gauge fields. Pertinent field theories of the quasiparticle dynamics in the R2TC are developed, and the accompanying conservation laws derived. A dipolar BF theory of the rank-2 gauge fields is constructed and shown to correctly capture the dipolar braiding statistics, in contrast to the conventional BF theory capturing the monopolar braiding statistics of anyons in the rank-1 toric code.
Rank-2 toric code (R2TC), a prototypical archetype of the discrete rank-2 symmetric gauge theory, has properties that differ from those of the standard toric code. Specifically, it features a blending of UV and IR in its ground state, restricted mobility of its quasiparticles, and variations in the braiding statistics of its quasiparticles based on their position. In this paper, we investigate various aspects of ZN{\mathbb{Z}}_{N}ZN​ rank-2 gauge theory in (2+1)(2+1)(2+1)-dimensional spacetime. Firstly, we demonstrate that U(1)U(1)U(1) rank-2 gauge theory can arise from $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ rank-1 gauge theory after condensing the gauge charges in a specific way. This construction scheme of U(1)U(1)U(1) rank-2 gauge theory carries over to the ZN{\mathbb{Z}}_{N}ZN​ case simply by Higgsing U(1)U(1)U(1) to ZN{\mathbb{Z}}_{N}ZN​, after which the resulting rank-2 gauge theory can be tuned to the R2TC. The holonomy operators of R2TC are readily identified using this scheme and are given clear physical interpretation as the pair creation/annihilation of various monopoles and dipoles. Explicit tensor network construction of the ground states of R2TC is given as two copies of the ground states of Kitaev's toric code that are ``sewn together'' according to the condensation scheme. In addition, through a similar anyon condensation protocol, we present a double semion version of rank-2 toric code whose flux excitations exhibit restricted mobility and semionic statistics. Finally, we identify the generalized discrete symmetries of the R2TC, which are much more complex than typical 1-form symmetries. They include conventional and unconventional 1-form symmetries, such as framed 1-form symmetries and what we call sublattice 1-form symmetries. Using these, we interpret the R2TC's unique properties (UV/IR mixing, position-dependent braiding, etc.) from the modern perspective of generalized spontaneous symmetry breaking and 't Hooft anomalies.
A number of exactly solvable spin models, including the Kitaev toric code in two and three dimensions and the X-cube model in three dimensions, can be related to their respective parent lattice gauge theories (LGTs) through the mathematical process of ``Higgsing.'' Field theories of the low-energy excitations of these spin models can be developed subsequently, building upon the symmetry of the parent LGTs. Recently, two of the present authors proposed a variant of the three-dimensional toric code which we now call the F3 model, whose elementary excitations consist of freeon and fluxon excitations of the three-dimensional toric code and fracton excitations of the X-cube model. In this work, we identify the parent LGT of the F3 model as the hybrid rank-1 and rank-2 U(1) LGT and develop the corresponding field theory. The resulting Lagrangian of the F3 model is that of a three-dimensional toric code with an extra term, which ties the dynamics of fractons to that of fluxons. The matter part of the effective action for the F3 model can be derived as well by carefully keeping track of the gauge invariance of the F3 model. Hydrodynamic equations of motion of the quasiparticles are derived which properly reflect their constrained dynamics. Finally, we present a tight-binding model for the quasiparticle motion in the F3 model.
We derive an exactly solvable one-dimensional (1D) spin model from the three-band Hubbard model with a strong spin-orbit coupling by introducing U(1) gauge fields to the isospin states. We find that it has a topological nontrivial phase characterized by Majorana end modes which are protected by a new Z_2 topological invariant related to the parity of the lattice sites (odd or even number of sites) in the spin chain. With the protection of this Z_2 topological invariant, a novel braiding of two Majorana edge states in this strictly geometric 1D chain is realized. We also discuss the possible realization of the gauge fields.
In this series of papers, we study a Hamiltonian model for (3+1)-dimensional topological phases introduced in [Bullivant et al., Phys. Rev. B 95, 155118 (2017)], based on a generalization of lattice gauge theory known as ``higher-lattice gauge theory.'' Higher-lattice gauge theory has so-called ``2-gauge fields'' describing the parallel transport of lines, in addition to ordinary 1-gauge fields which describe the parallel transport of points. In this series we explicitly construct the creation operators for the pointlike and looplike excitations supported by the model. We use these creation operators to examine the properties of the excitations, including their braiding statistics. These creation operators also reveal that some of the excitations are confined, costing energy to separate that grows linearly with the length of the creation operator used. This is discussed in the context of condensation-confinement transitions between different cases of this model. We also discuss the topological charges of the model and use explicit measurement operators to rederive a relationship between the number of charges measured by a 2-torus and the ground-state degeneracy of the model on the 3-torus. From these measurement operators, we can see that the ground-state degeneracy on the 3-torus is related to the number of types of linked looplike excitations. This first paper provides an accessible summary of our findings, with more detailed results and proofs to be presented in the other papers in the series.