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arXiv:2302.03707 · Physical review. B./Physical review. B

U(1) symmetry enriched toric code

Kai-Hsin Wu, Alexey Khudorozhkov, Guilherme Delfino, Dmitry Green, Claudio Chamon

Published Sep 27, 2023• 6 citations
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Abstract

We propose and study a generalization of Kitaev's ${\mathbb{Z}}_{2}$ toric code on a square lattice with an additional global $U(1)$ symmetry. Using quantum Monte Carlo simulation, we find strong evidence for a topologically ordered ground state manifold with indications of UV-IR mixing, i.e., the topological degeneracy of the ground state depends on the microscopic details of the lattice. Specifically, the ground state degeneracy depends on the lattice tilt relative to the directions of the torus cycles. In particular, we observe that while the usual compactification along the vertical or horizontal lines of the square lattice shows a twofold ground state degeneracy, compactifying the lattice at ${45}^{\ensuremath{\circ}}$ leads to a threefold degeneracy. In addition to its unusual topological properties, this system also exhibits Hilbert space fragmentation. Finally, we propose a candidate experimental realization of the model in an array of superconducting quantum wires.

DOI
10.1103/physrevb.108.115159

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We propose and study a generalization of Kitaev's Z2{\mathbb{Z}}_{2}Z2​ toric code on a square lattice with an additional global U(1)U(1)U(1) symmetry. Using quantum Monte Carlo simulation, we find strong evidence for a topologically ordered ground state manifold with indications of UV-IR mixing, i.e., the topological degeneracy of the ground state depends on the microscopic details of the lattice. Specifically, the ground state degeneracy depends on the lattice tilt relative to the directions of the torus cycles. In particular, we observe that while the usual compactification along the vertical or horizontal lines of the square lattice shows a twofold ground state degeneracy, compactifying the lattice at 45\ensuremath∘{45}^{\ensuremath{\circ}}45\ensuremath∘ leads to a threefold degeneracy. In addition to its unusual topological properties, this system also exhibits Hilbert space fragmentation. Finally, we propose a candidate experimental realization of the model in an array of superconducting quantum wires.
Some topological lattice models in two spatial dimensions exhibit intricate lattice size dependence in their ground state degeneracy (GSD). This and other features such as the position-dependent anyonic excitations are manifestations of UV/IR mixing. In the first part of this paper, we perform an exact calculation of the topological entanglement entropy (TEE) for a specific model, the rank-2 toric code. This analysis includes both contractible and non-contractible boundaries, with the minimum entropy states identified specifically for non-contractible boundaries. Our results show that TEE for a contractible boundary remains independent of lattice size, whereas TEE for non-contractible boundaries, similarly to the GSD, shows intricate lattice-size dependence. In the latter part of the paper we focus on the fact that the rank-2 toric code is an example of a translation symmetry-enriched topological phase, and show that viewing distinct lattice size as a consequence of different translation symmetry defects can explain both our TEE results and the GSD of the rank-2 toric code. Our work establishes the translation symmetry defect framework as a robust description of the UV/IR mixing in topological lattice models.
Absence of topological order in the U(1)U(1)U(1) checkerboard toric code
We investigate the U(1) U ( 1 ) checkerboard toric code which corresponds to the U(1) U ( 1 ) -symmetry enriched toric code with two distinct star sublattices. One can therefore tune from the limit of isolated stars to the uniform system. The uniform system has been conjectured to possess topological order based on quantum Monte Carlo simulations suggesting a non-trivial ground-state degeneracy depending on the compactification of the finite clusters. Here we show that these non-trivial properties can be naturally explained in the perturbative limit of isolated stars. Indeed, the compactification dependence of the ground-state degeneracy can be traced back to geometric constraints stemming from the plaquette operators. Further, the ground-state degeneracy is fully lifted in fourth-order degenerate perturbation theory giving rise to a non-topological phase with confined fracton excitations. These fractons are confined for small perturbations so that they cannot exist as single low-energy excitation in the thermodynamic limit but only as topologically trivially composite particles. However, the confinement scale is shown to be surprisingly large so that gaps are extremely small on finite clusters up to the uniform limit which is calculated explicitly by high-order series expansions. Our findings suggest that these gaps were not distinguished from finite-size effects by the recent quantum Monte Carlo simulation in the uniform limit. All our results therefore point towards the absence of topological order in the U(1) U ( 1 ) checkerboard toric code along the whole parameter axis.
Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.
We investigate how symmetry and topological order are coupled in the (2+12+12+1)--dimensional ZN{\mathbb{Z}}_{N}ZN​ rank-2 toric code for general NNN, which is an exactly solvable point in the Higgs phase of a symmetric rank-2 U(1)\text{U}(1)U(1) gauge theory. The symmetry-enriched topological order present has a nontrivial realization of square-lattice translation (and rotation and reflection) symmetry, where anyons on different lattice sites have different types and belong to different superselection sectors. We call such particles ``position-dependent excitations.'' As a result, in the rank-2 toric code anyons can hop by one lattice site in some directions while only by NNN lattice sites in others, reminiscent of fracton topological order in 3+13+13+1 dimensions. We find that while there are N2{N}^{2}N2 flavors of eee charges and 2N2N2N flavors of mmm fluxes, there are not NN2+2N{N}^{{N}^{2}+2N}NN2+2N anyon types. Instead, there are N6{N}^{6}N6 anyon types, and we can use Chern-Simons theory with six U(1)\text{U}(1)U(1) gauge fields to describe all of them. While the lattice translations permute anyon types, we find that such permutations cannot be expressed as transformations on the six U(1)\text{U}(1)U(1) gauge fields. Thus, the realization of translation symmetry in the U6(1){\text{U}}^{6}(1)U6(1) Chern-Simons theory is not known. Despite this, we find a way to calculate the translation-dependent properties of the theory. In particular, we find that the ground-state degeneracy on an ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ torus is N3gcd(Lx,N)gcd(Ly,N)gcd(Lx,Ly,N){N}^{3}gcd({L}_{x},N)gcd({L}_{y},N)gcd({L}_{x},{L}_{y},N)N3gcd(Lx​,N)gcd(Ly​,N)gcd(Lx​,Ly​,N), where gcdgcdgcd stands for ``greatest common divisor.'' We argue that this is a manifestation of UV/IR mixing which arises from the interplay between lattice symmetries and topological order.
FINDING: The toric code is a topological quantum error-correcting code whose logical operators are non-contractible loops on a torus, with the p4m wallpaper group (square lattice) providing the underlying spatial symmetry; its ground state degeneracy and logical operator algebra are governed by the first homology group of the torus, H1(T2)=Z2 H_1(T^2) = \mathbb{Z}^2 H1​(T2)=Z2. | MATH: Hamiltonian H=−∑vAv−∑pBp H = -\sum_v A_v - \sum_p B_p H=−∑v​Av​−∑p​Bp​, where Av=∏i∈vXi A_v = \prod_{i \in v} X_i Av​=∏i∈v​Xi​, Bp=∏i∈pZi B_p = \prod_{i \in p} Z_i Bp​=∏i∈p​Zi​. Logical operators: Xˉ1,Zˉ1 \bar{X}_1, \bar{Z}_1 Xˉ1​,Zˉ1​ along one non-contractible cycle, Xˉ2,Zˉ2 \bar{X}_2, \bar{Z}_2 Xˉ2​,Zˉ2​ along the other; they satisfy XˉiZˉj=(−1)δijZˉjXˉi \bar{X}_i \bar{Z}_j = (-1)^{\delta_{ij}} \bar{Z}_j \bar{X}_i Xˉi​Zˉj​=(−1)δij​Zˉj​Xˉi​. Ground state degeneracy = ∣H1(T2)∣=4 |H_1(T^2)| = 4 ∣H1​(T2)∣=4 (for Z2 \mathbb{Z}_2 Z2​ coefficients). The p4m group (order 8, generated by 90° rotation and reflections) acts on the lattice, and the code is invariant under this wallpaper group. | CONNECTION: The square lattice is the A1×A1 A_1 \times A_1 A1​×A1​ root lattice Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Kitaev's toric code is an exactly solvable model with Z2{\mathbb{Z}}_{2}Z2​-topological order, which has potential applications in quantum computation and error correction. However, a direct experimental realization remains an open challenge. Here, we propose a building block for Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories coupled to dynamical matter and demonstrate how it allows for an implementation of the toric-code ground state and its topological excitations. This is achieved by introducing separate matter excitations on individual plaquettes, whose motion induce the required plaquette terms. The proposed building block is realized in the second-order coupling regime and is well suited for implementations with superconducting qubits. Furthermore, we propose a pathway to prepare topologically nontrivial initial states during which a large gap on the order of the underlying coupling strength is present. This is verified by both analytical arguments and numerical studies. Moreover, we outline experimental signatures of the ground-state wave function and introduce a minimal braiding protocol. Detecting a $\ensuremath{\pi}$-phase shift between Ramsey fringes in this protocol reveals the anyonic excitations of the toric-code Hamiltonian in a system with only three triangular plaquettes. Our work paves the way for realizing non-Abelian anyons in analog quantum simulators.