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Motivated by the order fractionalization in Kitaev-Kondo model, we propose an exactly solvable spin-charge ladder model to study the order fractionalization with discrete symmetry. The spin-charge ladder is composed of a spin chain and a superconducting wire coupled via an Ising-type interaction, and we obtain the exact solution in the flat band limit. The exact solution reveals the $Z_{2}$ order fractionalization with dual symmetry breaking and intertwined order parameters. We investigate the topological phase transition of the spin-charge ladder via the spectral chiral index, and identify the correlated topological superconductor (TSC*) phase with gapped $Z_{2}$ Kondo flux excitations. We demonstrate Majorana spinons generated odd frequency pairing in the superconducting wire. We also discuss the order fractionalization in the perspective of $Z_{2}$ lattice gauge theory.

The physical nature of doped Mott-insulator has been intensively studied for more than three decades. It is well known that the single band Hubbard model or $t$-$J$ model on the bipartite lattice is the simplest model to describe a doped Mott insulator. Unfortunately, the key mechanism of superconductivity in these toy models is still under debate. In this paper, we propose a new mechanism for the $d+id$-wave superconductivity (SC) that occurs in the small-doping region of the honeycomb lattice $t$-$J$ model based on the Grassmann tensor product state numerical simulation and spin-charge separation formulation. Moreover, in the presence of anti-ferromagnetic order, a continuum effective field theory for holons is developed near half-filling. It reveals the competition between repulsive and attractive holon interactions induced by spinon fluctuations and gauge fluctuations, respectively. At a large value of $t/J$, the repulsive interaction dominates, leading to the non-Fermi liquid like behavior; while in a moderate range of $t/J$, the attractive interaction dominates, leading to the SC order. Possible experimental detection of spin-charge separation phenomena is also discussed., Comment: 18 pages, 7 figures

A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev's sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with $\nu$ species of Majorana fermions coupled to a static $\mathbb{Z}_2$ gauge field. Here each specie of Majorana fermions gives rise to an energy band that carries a Chern number $\mathcal{C}=1$, yielding a total Chern number $\mathcal{C}=\nu$. It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when $\nu$ is an odd (even) number. These exactly solvable models can be achieved by quantum simulations., Comment: 20 pages, 9 figures

The Majorana lattice gauge theory purely composed of Majorana fermions on square lattice is studied throughly. The ground state is obtained exactly and exhibits the coexistence of symmetry breaking and topological order. The $Z_2$ symmetry breaking of matter fields leads to the intertwined antiferromagnetic spin order and $\eta$-pairing order. The topological order is reflected in the $Z_2$ quantum spin liquid ground state of gauge fields. The Majorana lattice gauge theory, alternatively can be viewed as interacting Majorana fermion model, is possibly realized on a Majorana-zero-mode lattice.

Starting from the Landau's kinetic equation, we show that an electronic liquid in $d=2,3$ dimensions depicted by a Landau type effective theory will become incompressible on condition that the Landau parameters satisfy either (i) $1+F_{1}^{s}/d=0$ or (ii) $F_{0}^{s}\to{}+\infty$. The condition (i) is the Pomeranchuk instability in the current channel and suggests a quantum spin liquid (QSL) state with a spinon Fermi surface; while the condition (ii) means that the strong repulsion in the charge channel and leads to a conventional charge and thermal insulator. In the collisionless regime ($\omega\tau\gg{}1$) and the hydrodynamic regime ($\omega\tau\ll{}1$), the zero and first sound modes have been studied and classified by symmetries, including the longitudinal and transverse modes in $d=2,3$ and the higher angular momentum modes in $d=3$. The sufficient (and/or necessary) conditions of these collective modes have been revealed. It has been demonstrated that some of these collective modes will behave in quite different manners under two incompressibility conditions (i) or (ii). Possible nematic QSL states and a hierarchy structure for gapless QSL states have been proposed in $d=3$., Comment: 12+21 pages

Current Knowledge-Grounded Dialogue Generation (KDG) models specialize in producing rational and factual responses. However, to establish long-term relationships with users, the KDG model needs the capability to generate responses in a desired style or attribute. Thus, we study a new problem: Stylized Knowledge-Grounded Dialogue Generation (SKDG). It presents two challenges: (1) How to train a SKDG model where no triples are available. (2) How to cohere with context and preserve the knowledge when generating a stylized response. In this paper, we propose a novel disentangled template rewriting (DTR) method which generates responses via combing disentangled style templates (from monolingual stylized corpus) and content templates (from KDG corpus). The entire framework is end-to-end differentiable and learned without supervision. Extensive experiments on two benchmarks indicate that DTR achieves a significant improvement on all evaluation metrics compared with previous state-of-the-art stylized dialogue generation methods. Besides, DTR achieves comparable performance with the state-of-the-art KDG methods in standard KDG evaluation setting., Comment: Accepted to NAACL 2022 Main Conference

Though recent end-to-end neural models have shown promising progress on Conversational Recommender System (CRS), two key challenges still remain. First, the recommended items cannot be always incorporated into the generated replies precisely and appropriately. Second, only the items mentioned in the training corpus have a chance to be recommended in the conversation. To tackle these challenges, we introduce a novel framework called NTRD for recommender dialogue system that decouples the dialogue generation from the item recommendation. NTRD has two key components, i.e., response template generator and item selector. The former adopts an encoder-decoder model to generate a response template with slot locations tied to target items, while the latter fills in slot locations with the proper items using a sufficient attention mechanism. Our approach combines the strengths of both classical slot filling approaches (that are generally controllable) and modern neural NLG approaches (that are generally more natural and accurate). Extensive experiments on the benchmark ReDial show our NTRD significantly outperforms the previous state-of-the-art methods. Besides, our approach has the unique advantage to produce novel items that do not appear in the training set of dialogue corpus. The code is available at \url{https://github.com/jokieleung/NTRD}., Comment: EMNLP 2021 long paper, code link: https://github.com/jokieleung/NTRD

We propose a Haldane-BCS-Hubbard model on a honeycomb lattice, which is composed of two copies of the Haldane model of the quantum anomalous Hall effect, an equal-spin pairing term and an onsite Hubbard interaction term. For any interaction strength, this model is exactly solvable along the symmetric line where the hopping and pairing amplitudes are equal to each other. The ground state of the Haldane-BCS-Hubbard model is a topological superconducting state at weak interaction with two chiral Majorana edge states. A strong interaction drives the system across a topological quantum phase transition to a topologically trivial superconductor. A $\mathbb{Z}_{2}$ symmetry of the Hamiltonian, which is a composition of the bond-centered inversion and a gauge transformation, is spontaneously broken by the interaction, resulting a finite antiferromagnetic order in the $y$-direction., Comment: 9 pages, 7 figures

The largest allowed symmetry in a spin-1 quantum system is an $SU(3)$ symmetry rather than the $SO(3)$ spin rotation. In this work, we reveal some $SU(2)$ symmetries as subgroups of $SU(3)$ that, to the best of our knowledge, have not previously been recognized. Then, we construct $SU(2)$ symmetric Hamiltonians and explore the ground-state phase diagram in accordance with the $SU(3)\supset SU(2)\times U(1)$ symmetry hierarchy. It is natural to treat the eight generators of the $SU(3)$ symmetry on an equal footing; this approach is called the eight-fold way. We find that the spin spectral functions and spin quadrupole spectral functions share the same structure, provided that the elementary excitations are flavor waves at low energies, which serves as a clue to the eight-fold way. An emergent $S=1/2$ quantum spin liquid is proposed to coexist with gapful spin nematic order in one of the ground states. In analogy to quantum chromodynamics, we find the gap relation for hydrodynamic modes in quantum spin-orbital liquid states, which is nothing but the Gell-Mann-Okubo formula., Comment: 8 pages of main text, 10 pages of appendices

We construct a class of exactly solvable generalized Kitaev spin-$1/2$ models in arbitrary dimensions, which is beyond the category of quantum compass models. The Jordan-Wigner transformation is employed to prove the exact solvability. An exactly solvable quantum spin-$1/2$ models can be mapped to a gas of free Majorana fermions coupled to static $Z_2$ gauge fields. We classify these exactly solvable models according to their parent models. Any model belonging to this class can be generated by one of the parent models. For illustration, a two dimensional ($2D$) tetragon-octagon model and a three dimensional ($3D$) $xy$ bond model are studied., Comment: 11 pages, 10 figures

We propose an exactly solvable quantum spin-1/2 model with time reversal invariance on a two dimensional brick-wall lattice, where each unit cell consists of three sites. We find that the ground states are algebraic quantum spin liquid states. The spinon excitations are gapless and the energy dispersion is linear around two Dirac points. The ground states are of three-fold topological degeneracy on a torus. Breaking the time reversal symmetry opens a bulk energy gap and the $Z_2$ vortices obey non-Abelian statistics., Comment: 5+2 pages, 4 figures

We study analytically a modified dimerized Kitaev chain model under both periodic and open boundary conditions and determine its phase diagram by calculating the topological number and edge correlation functions of Majorana fermions. We explicitly give analytical solutions for eigenstates of the model under open boundary conditions, which permit us to calculate two introduced edge correlation functions of Majorana fermions analytically. Our results indicate that the two edge correlation functions can be used to characterize the trivial, Su-Schrieffer-Heeger-like topological and topological superconductor phases. Furthermore, we find the existence of two different coupling patterns of Majorana fermions in topological superconductor phase characterized by different edge correlation functions, which can also be distinguished by investigating the survival probability of an edge Majorana under sudden quenching of parameters., Comment: 8 pages, 5 figures

We study the interacting dimerized Kitaev chain at the symmetry point $\Delta=t$ and the chemical potential $\mu=0$ under open boundary conditions, which can be exactly solved by applying two Jordan-Wigner transformations and a spin rotation. By using exact analytic methods, we calculate two edge correlation functions of Majorana fermions and demonstrate that they can be used to distinguish different topological phases and characterize the topological phase transitions of the interacting system. According to the thermodynamic limit values of these two edge correlation functions, we give the phase diagram of the interacting system which includes three different topological phases: the trivial, the topological superconductor and the Su-Schrieffer-Heeger-like topological phase and we further distinguish the trivial phase by obtaining the local density distribution numerically., Comment: 7 pages, 4 figures