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

arXiv:1502.03192

Measuring Modular Matrices by Shearing Lattices

Yi‐Zhuang You, Meng Cheng

Published Feb 11, 2015• 5 citations
arXivHTMLPDF

Abstract

A topologically ordered phase on a torus possesses degenerate ground states that transform nontrivially under the modular transformations of the torus, generated by Dehn twists. Representation of modular transformations on the ground states (modular matrices) characterizes the topological order. We show that the modular matrices can be numerically measured as the non-Abelian Berry phase of adiabatic deformations of the lattice model placed on a torus. We apply this method to the example of a gauged $p_x+i p_y$ superconductor, and show that the result is consistent with the topological quantum field theory descriptions.

DOI
10.48550/arxiv.1502.03192

Related papers

Modular transformations and topological orders in two dimensions

Liu, Fangzhou, Zhenghan Wang, Yi‐Zhuang You +1 more

The string-net approach by Levin and Wen and the local unitary transformation approach by Chen, Gu and Wen provided ways to systematically label non-chiral topological orders in 2D. In those approaches, different topologically ordered many-body wave functions were characterized by different fixed-point tensors. Though extremely powerful, the resulting fixed-point tensors were mathematical abstractions and thus lacked a physical interpretation. As a result it was hard to judge if two different fixed-point tensors described the same quantum phase or not. We want to improve that approach by giving a more physical description of the topological orders. We find that the non-Abelian Berry's phases, $T$- and $S$-matrices, of the topological protected degenerate ground states on a torus give rise to a more physical description of topological orders. Using the Verlinde conjecture, we can even choose the canonical basis for the $T$- and $S$-matrices. It is conjectured that the $T$ and $S$-matrices form a complete and one-to-one characterization of non-chiral topological orders and can replace the fixed-point tensor description to give us a more physical label for topological orders. As a result, all the topological properties can be obtained from the $T$- and $S$-matrices, such as number of quasiparticle types (from the dimension of $T$ or $S$), the quasiparticle statistics (from the diagonal elements of $T$), the quantum dimensions of quasiparticles (from the first row of $S$), \etc.

arXiv:1303.0829•Mar 4, 2013•10 citations

Twisted gauge theory model of topological phases in three dimensions

Yidun Wan, Juven Wang, Huan He

We propose an exactly solvable lattice Hamiltonian model of topological phases in $3+1$ dimensions, based on a generic finite group $G$ and a 4-cocycle $\ensuremath{\omega}$ over $G$. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasiexcitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the $SL(3,\mathbb{Z})$ generators as the modular $S$ and $T$ matrices of the ground states, which yield a set of topological quantum numbers classified by $\ensuremath{\omega}$ and quantities derived from $\ensuremath{\omega}$. Our model fulfills a Hamiltonian extension of the $(3+1)$-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group $G$. This work is presented to be accessible for a wide range of physicists and mathematicians.

arXiv:1409.3216•Physical Review B•Jul 1, 2015•126 citations

THE GROUND STATE STRUCTURE AND MODULAR TRANSFORMATIONS OF FRACTIONAL QUANTUM HALL STATES ON A TORUS

Esko Keski-Vakkuri, Xiao-Gang Wen

The structure of ground states of generic FQH states on a torus is studied by using both effective theory and electron wavefunction. The relation between the effective theory and the wavefunction becomes transparent when one considers the ground state structure. We find that the non-abelian Berry’s phases of the abelian Hall states generated by twisting the mass matrix are identical to the modular transformation matrix for the characters of Gaussian conformal field theory. We also show that the Haldane-Rezayi spin singlet state has a ten fold ground state degeneracy on a torus which indicates such a state is a non-abelian Hall state.

arXiv:hep-th/9303155•International Journal of Modern Physics B•Nov 15, 1993•111 citations

Torus Berry Data Determine All-Genus Abelian Topological Orders

Daniel Galviz

We show that for Abelian Chern-Simons topological orders, torus Berry matrices determine the all-genus extended TQFT. We identify the topological part of the projective Berry holonomy under metric deformations with the mapping-class-group representation of the Abelian Chern-Simons TQFT and prove that the normalized torus data reconstruct its finite quadratic module (G,q). Recent work showed that (G,q) classifies the extended theory up to symmetric monoidal natural isomorphism, and that the associated Abelian defect extension is determined by the pointed modular category C(G,q) arising from the same finite quadratic-module data. Therefore genus-one Berry data determine not only the all-genus bulk theory but also its associated defect structures without choosing a K-matrix presentation. We also prove that, for normalized character row errors $δ$<21.96%, nearest-row decoding recovers the Abelian fusion algebra independently of the number of anyons. The result applies to Abelian fractional quantum Hall and spin-liquid phases described by even-lattice Chern-Simons theories.

arXiv:2608.20330•Aug 20, 2026•0 citations

Topological Characterization of Fractional Quantum Hall Ground States from Microscopic Hamiltonians

Michael P. Zaletel, Roger S. K. Mong, Frank Pollmann

We show how to numerically calculate several quantities that characterize topological order starting from a microscopic fractional quantum Hall Hamiltonian. To find the set of degenerate ground states, we employ the infinite density matrix renormalization group method based on the matrix-product state representation of fractional quantum Hall states on an infinite cylinder. To study localized quasiparticles of a chosen topological charge, we use pairs of degenerate ground states as boundary conditions for the infinite density matrix renormalization group. We then show that the wave function obtained on the infinite cylinder geometry can be adapted to a torus of arbitrary modular parameter, which allows us to explicitly calculate the non-Abelian Berry connection associated with the modular T transformation. As a result, the quantum dimensions, topological spins, quasiparticle charges, chiral central charge, and Hall viscosity of the phase can be obtained using data contained entirely in the entanglement spectrum of an infinite cylinder.

arXiv:1211.3733•Physical Review Letters•Jun 4, 2013•227 citations

General procedure for determining braiding and statistics of anyons using entanglement interferometry

Yi Zhang, Tarun Grover, Ashvin Vishwanath

Recently, it was argued that the braiding and statistics of anyons in a two-dimensional topological phase can be extracted by studying the quantum entanglement of the degenerate ground states on the torus. This construction either required a lattice symmetry (such as $\ensuremath{\pi}/2$ rotation) or tacitly assumed that the minimum entanglement states (MESs) for two different bipartitions can be uniquely assigned quasiparticle labels. Here we describe a procedure to obtain the modular $\mathcal{S}$ matrix, which encodes the braiding statistics of anyons, which does not require making any of these assumptions. Our strategy is to coherently compare MESs of three independent entanglement bipartitions of the torus, which leads to a unique modular $\mathcal{S}$. This procedure also puts strong constraints on the modular $\mathcal{T}$ and $\mathcal{U}$ matrices without requiring any symmetries, and in certain special cases, completely determines it. Our method applies equally to Abelian and non-Abelian topological phases.

arXiv:1412.0677•Physical Review B•Jan 26, 2015•25 citations
A topologically ordered phase on a torus possesses degenerate ground states that transform nontrivially under the modular transformations of the torus, generated by Dehn twists. Representation of modular transformations on the ground states (modular matrices) characterizes the topological order. We show that the modular matrices can be numerically measured as the non-Abelian Berry phase of adiabatic deformations of the lattice model placed on a torus. We apply this method to the example of a gauged px+ipyp_x+i p_ypx​+ipy​ superconductor, and show that the result is consistent with the topological quantum field theory descriptions.
The string-net approach by Levin and Wen and the local unitary transformation approach by Chen, Gu and Wen provided ways to systematically label non-chiral topological orders in 2D. In those approaches, different topologically ordered many-body wave functions were characterized by different fixed-point tensors. Though extremely powerful, the resulting fixed-point tensors were mathematical abstractions and thus lacked a physical interpretation. As a result it was hard to judge if two different fixed-point tensors described the same quantum phase or not. We want to improve that approach by giving a more physical description of the topological orders. We find that the non-Abelian Berry's phases, TTT- and SSS-matrices, of the topological protected degenerate ground states on a torus give rise to a more physical description of topological orders. Using the Verlinde conjecture, we can even choose the canonical basis for the TTT- and SSS-matrices. It is conjectured that the TTT and SSS-matrices form a complete and one-to-one characterization of non-chiral topological orders and can replace the fixed-point tensor description to give us a more physical label for topological orders. As a result, all the topological properties can be obtained from the TTT- and SSS-matrices, such as number of quasiparticle types (from the dimension of TTT or SSS), the quasiparticle statistics (from the diagonal elements of TTT), the quantum dimensions of quasiparticles (from the first row of SSS), \etc.
We propose an exactly solvable lattice Hamiltonian model of topological phases in 3+13+13+1 dimensions, based on a generic finite group GGG and a 4-cocycle $\ensuremath{\omega}$ over GGG. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasiexcitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the SL(3,Z)SL(3,\mathbb{Z})SL(3,Z) generators as the modular SSS and TTT matrices of the ground states, which yield a set of topological quantum numbers classified by $\ensuremath{\omega}$ and quantities derived from $\ensuremath{\omega}$. Our model fulfills a Hamiltonian extension of the (3+1)(3+1)(3+1)-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group GGG. This work is presented to be accessible for a wide range of physicists and mathematicians.
The structure of ground states of generic FQH states on a torus is studied by using both effective theory and electron wavefunction. The relation between the effective theory and the wavefunction becomes transparent when one considers the ground state structure. We find that the non-abelian Berry’s phases of the abelian Hall states generated by twisting the mass matrix are identical to the modular transformation matrix for the characters of Gaussian conformal field theory. We also show that the Haldane-Rezayi spin singlet state has a ten fold ground state degeneracy on a torus which indicates such a state is a non-abelian Hall state.
We show that for Abelian Chern-Simons topological orders, torus Berry matrices determine the all-genus extended TQFT. We identify the topological part of the projective Berry holonomy under metric deformations with the mapping-class-group representation of the Abelian Chern-Simons TQFT and prove that the normalized torus data reconstruct its finite quadratic module (G,q). Recent work showed that (G,q) classifies the extended theory up to symmetric monoidal natural isomorphism, and that the associated Abelian defect extension is determined by the pointed modular category C(G,q) arising from the same finite quadratic-module data. Therefore genus-one Berry data determine not only the all-genus bulk theory but also its associated defect structures without choosing a K-matrix presentation. We also prove that, for normalized character row errors δδδ<21.96%, nearest-row decoding recovers the Abelian fusion algebra independently of the number of anyons. The result applies to Abelian fractional quantum Hall and spin-liquid phases described by even-lattice Chern-Simons theories.
We show how to numerically calculate several quantities that characterize topological order starting from a microscopic fractional quantum Hall Hamiltonian. To find the set of degenerate ground states, we employ the infinite density matrix renormalization group method based on the matrix-product state representation of fractional quantum Hall states on an infinite cylinder. To study localized quasiparticles of a chosen topological charge, we use pairs of degenerate ground states as boundary conditions for the infinite density matrix renormalization group. We then show that the wave function obtained on the infinite cylinder geometry can be adapted to a torus of arbitrary modular parameter, which allows us to explicitly calculate the non-Abelian Berry connection associated with the modular T transformation. As a result, the quantum dimensions, topological spins, quasiparticle charges, chiral central charge, and Hall viscosity of the phase can be obtained using data contained entirely in the entanglement spectrum of an infinite cylinder.
Recently, it was argued that the braiding and statistics of anyons in a two-dimensional topological phase can be extracted by studying the quantum entanglement of the degenerate ground states on the torus. This construction either required a lattice symmetry (such as $\ensuremath{\pi}/2$ rotation) or tacitly assumed that the minimum entanglement states (MESs) for two different bipartitions can be uniquely assigned quasiparticle labels. Here we describe a procedure to obtain the modular S\mathcal{S}S matrix, which encodes the braiding statistics of anyons, which does not require making any of these assumptions. Our strategy is to coherently compare MESs of three independent entanglement bipartitions of the torus, which leads to a unique modular S\mathcal{S}S. This procedure also puts strong constraints on the modular T\mathcal{T}T and U\mathcal{U}U matrices without requiring any symmetries, and in certain special cases, completely determines it. Our method applies equally to Abelian and non-Abelian topological phases.