Abstract An essential aspect of topological phases of matter is the existence of Wilson loop operators that keep the ground state subspace invariant. Here we present and implement an \it unbiased \rm numerical optimization scheme to systematically find the Wilson loop operators given a single ground state wave function of a gapped Hamiltonian on a disk. We then show how these Wilson loop operators can be cut and glued through further optimization to give operators that can create, move, and annihilate anyon excitations. We subsequently use these operators to determine the braiding statistics and topological twists of the anyons, yielding a way to fully extract topological order from a single wave function. We apply our method to the ground state of the perturbed toric code and doubled semion models with a magnetic field that is up to a half of the critical value. From a contemporary perspective, this can be thought of as a machine learning approach to discover emergent 1-form symmetries of a ground state wave function. From an application perspective, our approach can be relevant to find Wilson loop operators in current quantum simulators.
Abstract An essential aspect of topological phases of matter is the existence of Wilson loop operators that keep the ground state subspace invariant. Here we present and implement an \it unbiased \rm numerical optimization scheme to systematically find the Wilson loop operators given a single ground state wave function of a gapped Hamiltonian on a disk. We then show how these Wilson loop operators can be cut and glued through further optimization to give operators that can create, move, and annihilate anyon excitations. We subsequently use these operators to determine the braiding statistics and topological twists of the anyons, yielding a way to fully extract topological order from a single wave function. We apply our method to the ground state of the perturbed toric code and doubled semion models with a magnetic field that is up to a half of the critical value. From a contemporary perspective, this can be thought of as a machine learning approach to discover emergent 1-form symmetries of a ground state wave function. From an application perspective, our approach can be relevant to find Wilson loop operators in current quantum simulators.
The ground states encode the information of the topological phases of a 2-dimensional system, which makes them crucial in determining the associated topological quantum field theory (TQFT). Most numerical methods for detecting the TQFT relied on the use of minimum entanglement states (MESs), extracting the anyon mutual statistics and self statistics via overlaps and/or the entanglement spectra. The MESs are the eigenstates of the Wilson loop operators, and are labeled by the anyons corresponding to their eigenvalues. Here we revisit the definition of the Wilson loop operators and MESs. We derive the modular transformation of the ground states purely from the Wilson loop algebra, and as a result, the modular S and T matrices naturally show up in the overlap of MESs. Importantly, we show that due to the phase degree of freedom of the Wilson loop operators, the MES-anyon assignment is not unique. This ambiguity obstructs our attempt to detect the topological order, that is, there exist different TQFTs that cannot be distinguished solely by the overlap of MESs. In this paper, we provide the upper limit of the information one may obtain from the overlap of MESs without other additional structure. Finally, we show that if the phase is enriched by rotational symmetry, there may be additional TQFT information that can be extracted from overlap of MESs.
Local non-unitary deformations of topologically ordered wavefunctions can drive transitions into peculiar states that challenge modern perspectives on gapped quantum matter. The strongly deformed toric code offers a curious case, hosting m anyon condensation alongside perimeter-law scaling of Wilson loops charged under an exact 1-form symmetry---properties that typically do not coexist in gapped ground states. Nevertheless, we rigorously construct local gapped parent Hamiltonians for these strongly deformed toric code states. The Hamiltonians we construct are not strictly finite-range, but contain sums of Wilson loop operators whose coefficients decay exponentially in their diameter. If one adopts standard locality bounds used to define gapped phases---which allow for such exponentially decaying terms---our construction shows that these states realize a trivial gapped phase. Within this locality class, we demonstrate that perimeter-law scaling of Wilson loops does not imply a spontaneously broken 1-form symmetry, and from a dual perspective, that long-range ferromagnetic order and perimeter-law disorder parameter correlations can coexist in a 2D gapped ground state. We evade a recent no-go theorem [Sahay et al., arXiv:2503.01977] by relaxing its assumptions in a manner that we quantify as benign in the thermodynamic limit. More broadly, our results highlight that stronger notions of locality are necessary for prohibiting these counterintuitive properties within a gapped phase.
In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to Z d qudits, including instances where d is a nonprime number. This capability allows the identification of topological orders that differ from the Z d toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for Z p qudits (with p being a prime) are equivalent to finite copies of Z p toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order. Published by the American Physical Society 2024
We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in previous works [A. Francuz et al., Phys. Rev. B 101, 041108(R) (2020) and A. Francuz and J. Dziarmaga, Phys. Rev. B 102, 235112 (2020)] we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of matrix product states to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological F symbols of the underlying fusion category, which unequivocally identifies the topological order of the ground state. We bring the F symbols to the canonical gauge, and also compute the Drinfeld center of this unitary fusion category to extract the topological S and T matrices encoding mutual statistics and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, Kitaev model, double semion, and twisted quantum double of Z3, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of S3 and Rep(S3) string-net models.
Abstract How do we detect topological phases from a single ground state? Entanglement entropy and spectrum have long been the standard tools — but the reduced density matrix (RDM) itself contains far more information. We show that the RDM of a 2D topologically ordered system, expressed at the entanglement cut, realizes a 1D mixed-state phase. For the Z 2 toric code phase, it is a 1D Z 2 strong-to-weak spontaneous symmetry breaking (SW-SSB) phase, where deconfinement of anyons manifests as the short-range correlation of both Z 2 charge and Z 2 domain-wall in the RDM. The bulk e - m duality translates into a Kramers–Wannier self-duality of the SW-SSB phase. Extending the framework to gapped Z 2 spin liquids, the global spin-rotation symmetry manifests as an additional weak symmetry for the 1D RDM. Spin-1/2 spinons result in a cusp on the disorder parameter of spin-rotation at θ = π , providing a direct, ground-state signature of symmetry fractionalization. We verify this prediction analytically using the matrix product density operator formalism and numerically for the kagome-lattice resonating valence bond state. The proposed observable requires only a single ground-state wavefunction, making it amenable to quantum simulation platforms.
Topological phases are gapped quantum phases of matter classified beyond the paradigm of Landau's symmetry breaking theory.In particular, topologically ordered phases are topological phases with topological order.Unlike the conventional symmetry-breaking order described by local order parameter, topological order is characterized by the excitations.For example, in (2+1)D topologically ordered phases, there are particle-like excitations called anyons.In contrast, in (3+1)D topologically ordered phases, there are both particle-like and loop-like excitations.These excitations exhibit braiding statistics, meaning that carrying one excitation adiabatically around another transform the ground state by a phase (or generally a unitary transformation).The braiding statistics of these excitations serves as a crucial ingredient in characterizing the topological order in these phases.One important aspect is to understand the braiding statistics in topologically ordered phases.While the anyonic braiding statistics in (2+1)D topologically ordered phases is well established, the braiding statistics in (3+1)D topologically ordered phases is far from clear.In (3+1)D, the most well-understood braiding statistics is the particle-loop braiding statistics, where a particle winds around a loop excitation acquires an Aharonov Bohm phase.In the thesis, we introduce a new braiding statistics, which we called the Borromean-Rings braiding statistics.It is a particleloop-loop braiding statistics where a particle travels around two unlinked loops acquires a braiding phase.Unlike the particle-loop braiding, the particle trajectory in the Borromean-Rings braiding is not linked with any of the two loops.Another important aspect is to probe the topological order in topologically ordered phases.One usual way to probe the topological order is to examine the properties of the phases on various orientable surfaces.For example, for (2+1)D topologically ordered phases, the number of degenerate ground states on Torus counts the number of anyons.Also, ground state transformation under twisting the torus encodes the braiding data.In the thesis, we attempt to probe the topological order by putting the phases on non-orientable surfaces.We find that the ground state degeneracy and the ground state transformation reveals a special kind of anyons, called the parity symmetric anyons, which show a lost of orientation in their braiding statistics.They in turn distinguish different topological order.Even without topological order, there can be non-trivial topological phases in the presence of global symmetry.These topological phases are called the symmetry protected topological phases.For example, (3+1)D fermionic matter with time reversal symmetry and charge conservation symmetry has a non-trivial phase, namely the topological I would like to give my sincere thanks to my advisor Shinsei Ryu.Throughout my PhD study, he taught me a lot about condensed matter physics, particularly on the forefront subjects in topological phases.Under his supervision, I worked on different abstract yet inspiring topics on topological phases.His guide is always clear with mathematical interpretation and physical insight.His advice not only guided me though the research, but also shaped my views and thinking.From my advisor, I learnt the way to convey difficult ideas