In this work, we analyze the macroscopic observables in the 1+1D gapped phases with Abelian onsite symmetries and show that the spacetime observables for each gapped phase form a clear structure that can be mathematically described by enriched fusion categories, which uncovers the behavior of nonlocal excitations that were blurry in traditional Landau paradigm. These categorical descriptions not only generate the known classification results for symmetry preserving and breaking phases, but also unifies lattice dualities in a broader picture. After analyzing the general lattice model together with their boundaries, we give explicit examples including nontrivial SPT phase, where nontrivial boundaries can be given directly through our classification. Using enriched categorical descriptions, the lattice dualities and their gapped phases are unified under a holographic duality between an 2d. topological order with gapped 1d boundaries and 1+1D gapped quantum liquids with a categorical symmetry, which shed light on a unified definition of all quantum phases.
We investigate the phase diagram of a quantum spin-1 chain whose Hamiltonian is invariant under a global onsite $A_4$, translation and lattice inversion symmetries. We detect different gapped phases characterized by SPT order and symmetry breaking using matrix product state order parameters. We observe a rich variety of phases of matter characterized by a combination of symmetry breaking and symmetry fractionalization and also the interplay between the onsite and spatial symmetries. Examples of continuous phase transitions directly between topologically nontrivial SPT phases are also observed.
In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category. We also study the categorical descriptions of the boundaries of these m…
(2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·派尔斯理论物理中心）；Center for Quantum Mathematics at IMADA, Southern Denmark University（南丹麦大学IMADA量子数学中心）
We discuss quantum many-body systems with lattice translation and discrete onsite symmetries. We point out that, under a boundary condition twisted by a symmetry operation, there is an exact degeneracy of ground states if the unit cell forms a projective representation of the onsite discrete symmetry. Based on the quantum transfer matrix formalism, we show that, if the system is gapped, the ground-state degeneracy under the twisted boundary condition also implies a ground-state (quasi-)degeneracy under the periodic boundary conditions. This gives a compelling evidence for the recently proposed…
We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category $\calC$ as in the Levin-Wen model, whereas the boundary is associated with a module category over $\calC$. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictio…
We study properties of normal, superconducting (SC) and CDW states for an attractive Hubbard model on the square lattice, using a variational Monte Carlo method. In trial wave functions, we introduce an interspinon binding factor, indispensable to induce a spin-gap transition in the normal state, in addition to the onsite attractive and intersite repulsive factors. It is found that, in the normal state, as the interaction strength $|U|/t$ increases, a first-order spin-gap transition arises at $|U_{\rm c}|\sim W$ ($W$: band width) from a Fermi liquid to a spin-gapped state, which is conductive …
We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a local fusion $n$-category. We find another way to describe algebraic higher symmetry by restricting to symmetric sub Hilbert space where symmetry transformations all become trivial. In this case, algebraic higher symmetry can be fully characterized by a non-invertible gravitational anomaly (i.e. an topological order in one higher dimension). Thus we also ref…
This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protec…
We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.
We study the concept of "categorical symmetry" introduced recently, which in the most basic sense refers to a pair of dual symmetries, such as the Ising symmetries of the $1d$ quantum Ising model and its self-dual counterpart. In this manuscript we study discrete categorical symmetry at higher dimensional critical points and gapless phases. At these selected gapless states of matter, we can evaluate the behavior of categorical symmetries analytically. We analyze the categorical symmetry at the following examples of criticality: (1) Lifshit critical point of a $(2+1)d$ quantum Ising system; (2)…
We introduce an unconventional regularization of condensed matter effective field theory by a non-local bulk topological regulator in one higher dimension, which is equivalent to a bulk-interface-bulk formulation. We show its necessity motivated by a (0+1)-dimensional system. Although the system can be strictly defined on the lattice in its own dimensions, the bulk regulator is important to derive its correct physical observables, which cannot be captured by any local regulator. We explicitly obtain a constraint on the thermal and electric integer Hall conductances of half-filled translation i…
We study the gauging of a global U(1) symmetry in a gapped system in (2+1)d. The gauging procedure has been well-understood for a finite global symmetry group, which leads to a new gapped phase with emergent gauge structure and can be described algebraically using the mathematical framework of modular tensor category (MTC). We develop a categorical description of U(1) gauging in an MTC, taking into account the dynamics of U(1) gauge field absent in the finite group case. When the ungauged system has a non-zero Hall conductance, the gauged theory remains gapped and we determine the complete set…
Bulk-boundary correspondence of (1+1)D symmetric gapped phases
The Chinese University of Hong Kong（香港中文大学）；The State Key Laboratory of Quantum Information Technologies and Materials, The Chinese University of Hong Kong（香港中文大学量子信息技术与材料国家重点实验室）
For a zero-temperature Landau symmetry breaking transition in $n$-dimensional space that completely breaks a finite symmetry $G$, the critical point at the transition has the symmetry $G$. In this paper, we show that the critical point also has a dual symmetry - a $(n-1)$-symmetry described by a higher group when $G$ is Abelian or an algebraic $(n-1)$-symmetry beyond higher group when $G$ is non-Abelian. In fact, any $G$-symmetric system can be viewed as a boundary of $G$-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk $G$-gauge theory gives ri…
Strong, electromagnetic, and weak forces were unified in the Standard Model (SM) with spontaneous gauge symmetry breaking. These forces were further conjectured to be unified in a simple Lie group gauge interaction in the Grand Unification (GUT). In this work, we propose a theory beyond the SM and GUT by adding new gapped Topological Phase Sectors consistent with the nonperturbative global anomaly cancellation and cobordism constraints (especially from the baryon minus lepton number ${\bf B}-{\bf L}$, the electroweak hypercharge $Y$, and the mixed gauge-gravitational anomaly). Gapped Topologic…
Gapped Boundaries of Kitaev's Quantum Double Models: A Lattice Realization of Anyon Condensation from Lagrangian Algebras
Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学量子科学与工程研究院）；International Quantum Academy（国际量子学院）；Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学广东省量子科学与工程重点实验室）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）；Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China（中国科学技术大学微尺度物质科学国家研究中心物理科学学院）
We predict two topological superconducting phases in microscopic models arising from the Berry phase associated with the valley degree of freedom in gapped Dirac honeycomb systems. The first one is a topological helical spin-triplet superconductor with a nonzero center-of-mass momentum that does not break time-reversal symmetry. We also find a topological chiral-triplet superconductor with Chern number $\pm 1$ with equal-spin-pairing in one valley and opposite-spin-triplet pairing in the other valley. Our results are obtained for the Kane-Mele model in which we have explored the effect of thre…
The spectra and role in the spin dynamical properties of bound states of elementary magnetic excitations named Bethe strings that occur in some integrable spin and electronic one-dimensional models have recently been identified and realized in several materials by experiments. Corresponding theoretical studies have usually relied on the one-dimensional spin-1/2 Heisenberg antiferromagnet in a magnetic field. At the isotropic point, it describes the large onsite repulsion U limit of the spin degrees of freedom of the one-dimensional fermionic Hubbard model with one electron per site in a magnet…
We consider a single-component gas of dipolar bosons confined in a one-dimensional optical lattice, where the dipoles are aligned such that the long-ranged dipolar interactions are maximally repulsive. In the limit of zero inter-site hopping and sufficiently large on-site interaction, the phase diagram is a complete devil's staircase for filling fractions between 0 and 1: every commensurate state at a rational filling is stable over a finite interval in chemical potential, and for every chemical potential the system is in a gapped commensurate phase. We perturb away from this limit in two expe…
In the ionic Hubbard model, the onsite repulsion $U$, which drives a Mott insulator and the ionic potential $V$, which drives a band insulator, compete with each other to open up a window of charge fluctuations when $U \sim V$. We study this model on square and cubic lattices in the limit of large $U$ and $V$, with $V\sim U$. Using an effective Hamiltonian and a slave boson approach with both doublons and holes, we find that the system undergoes a phase transition as a function of $V$ from an antiferromagnetic Mott insulator to a paramagnetic insulator with strong singlet correlations, which i…
We study the effects of electron-electron interactions on the charge excitation spectrum of the spinful Su-Schrieffer-Heeger (SSH) model, a prototype of a 1D bulk obstructed topological insulator. In view of recent progress in the fabrication of dopant-based quantum simulators we focus on experimentally detectable signatures of interacting topology in finite lattices. To this end we use Lanczos-based exact diagonalization to calculate the single-particle spectral function in real space which generalizes the local density of states to interacting systems. Its spatial and spectral resolution all…
Lattice Models for Phases and Transitions with Non-Invertible Symmetries
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔学院，尼尔斯·玻尔研究所）
