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Register a new userGapped domain walls between 2+1D topologically ordered states
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The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. In this paper, we use those representations to determine the possible gapped boundaries of a 2+1D topological order, as well as the domain walls between two topological orders. We find that mapping-class-group representations for both genus-1 and genus-2 surfaces are needed to determine the gapped domain walls and boundaries. Our systematic theory is based on the fixed-point partition functions for the walls (or the boundaries), which completely characterize the gapped domain walls (or the boundaries). The mapping-class-group representations give rise to conditions that must be satisfied by the fixed-point partition functions, which leads to a systematic theory. Such conditions can be viewed as bulk topological order determining the (non-invertible) gravitational anomaly at the domain wall, and our theory can be viewed as finding all types of the gapped domain wall given a (non-invertible) gravitational anomaly. We also developed a systematic theory of gapped domain walls (boundaries) based on the structure coefficients of condensable algebras.
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We develop a theory of gapped domain wall between topologically ordered systems in two spatial dimensions. We find a new type of superselection sector -- referred to as the parton sector -- that subdivides the known superselection sectors localized on gapped domain walls. Moreover, we introduce and study the properties of composite superselection sectors that are made out of the parton sectors. We explain a systematic method to define these sectors, their fusion spaces, and their fusion rules, by deriving nontrivial identities relating their quantum dimensions and fusion multiplicities. We propose a set of axioms regarding the ground state entanglement entropy of systems that can host gapped domain walls, generalizing the bulk axioms proposed in [B. Shi, K. Kato, and I. H. Kim, Ann. Phys. 418, 168164 (2020)]. Similar to our analysis in the bulk, we derive our main results by examining the self-consistency relations of an object called information convex set. As an application, we define an analog of topological entanglement entropy for gapped domain walls and derive its exact expression.
We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with emergent symmetries of the edge, which play a crucial role in the structure of its phase diagram. Using this symmetry principle, transitions between distinct gapped phases at the boundaries of Abelian states can be understood in terms of symmetry breaking transitions or transitions between symmetry protected topological phases. Yet more exotic phenomena occur when the bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the phase diagrams of the edges of a single and a double layer of the toric code, as well as those of domain walls in a single and double-layer Kitaev spin liquid (KSL). In the case of the KSL, we find that the presence of a non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a symmetry of the effective boundary theory. These examples illustrate a number of surprising phenomena, such as spontaneous duality-breaking, two-sector phase transitions, and unfreezing of marginal operators at a transition between different gapless phases.
In a certain regime of low carrier densities and strong correlations, electrons can crystallize into a periodic arrangement of charge known as Wigner crystal. Such phases are particularly interesting in one dimension (1D) as they display a variety of charge and spin ground states which may be harnessed in quantum devices as high-fidelity transmitters of spin information. Recent theoretical studies suggest that the strong Coulomb interactions in Mott insulators and other flat band systems, may provide an attractive higher temperature platform for Wigner crystallization, but due to materials and device constraints experimental realization has proven difficult. In this work we use scanning tunneling microscopy at liquid helium temperatures to directly image the formation of a 1D Wigner crystal in a Mott insulator, TaS$_2$. Charge density wave domain walls in TaS$_2$ create band bending and provide ideal conditions of low densities and strong interactions in 1D. STM spectroscopic maps show that once the lower Hubbard band crosses the Fermi energy, the charges rearrange to minimize Coulomb energy, forming zigzag patterns expected for a 1D Wigner crystal. The zigzag charge patterns show characteristic noise signatures signifying charge or spin fluctuations induced by the tunneling electrons, which is expected for this more fragile condensed state. The observation of a Wigner crystal at orders of magnitude higher temperatures enabled by the large Coulomb energy scales combined with the low density of electrons, makes TaS$_2$ a promising system for exploiting the charge and spin order in 1D Wigner crystals.
We study the Z2 topologically ordered surface state of three-dimensional bosonic SPT phases with the discrete symmetries G1 x G2. It has been argued that the topologically ordered surface state cannot be realized on a purely two-dimensional lattice model. We carefully examine the statement and show that the surface state should break G2 if the symmetry G1 is gauged. This manifests the conflict of the symmetry G1 and G2 on the surface of the three-dimensional SPT phase. Given that there is no such phenomena in the purely two-dimensional model, it signals that the symmetries are encoded anomalously on the surface of the three-dimensional SPT phases and that the surface state can never be realized on the purely two-dimensional models.
Recent low temperature heat capacity (C$_P$) measurements on polycrystalline samples of the pyrochlore antiferromagnet Tb$_{2+x}$Ti$_{2-x}$O$_{7+delta}$ have shown a strong sensitivity to the precise Tb concentration $x$, with a large anomaly exhibited for $x sim 0.005$ at $T_C sim 0.5$ K and no such anomaly and corresponding phase transition for $x le 0$. We have grown single crystal samples of Tb$_{2+x}$Ti$_{2-x}$O$_{7+delta}$, with approximate composition $x=-0.001, +0.0042$, and $+0.0147$, where the $x=0.0042$ single crystal exhibits a large C$_P$ anomaly at $T_C$=0.45 K, but neither the $x=-0.001$ nor the $x=+0.0147$ single crystals display any such anomaly. We present new time-of-flight neutron scattering measurements on the $x=-0.001$ and the $x=+0.0147$ samples which show strong $left(frac{1}{2},frac{1}{2},frac{1}{2}right)$ quasi-Bragg peaks at low temperatures characteristic of short range antiferromagnetic spin ice (AFSI) order at zero magnetic field but only under field-cooled conditions, as was previously observed in our $x = 0.0042$ single crystal. These results show that the strong $left(frac{1}{2},frac{1}{2},frac{1}{2}right)$ quasi-Bragg peaks and gapped AFSI state at low temperatures under field cooled conditions are robust features of Tb$_2$Ti$_2$O$_7$, and are not correlated with the presence or absence of the C$_P$ anomaly and phase transition at low temperatures. Further, these results show that the ordered state giving rise to the C$_P$ anomaly is confined to $0 leq x leq 0.01$ for Tb$_{2+x}$Ti$_{2-x}$O$_{7+delta}$, and is not obviously connected with conventional order of magnetic dipole degrees of freedom.
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