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Duality places an important constraint on the renormalization group flows and the phase diagrams. For self-dual theories, the self-duality can be promoted as a symmetry, this leads to the multi-criticalities. This work investigates a description of the deconfined quantum criticality, the $N_f=2$ QED$_3$, as an example of self-dual theories and its multi-critical behavior under perturbative deformations. The multi-criticality is described by the theory with Gross-Neveu couplings and falls in a different universality class than the standard deconfined quantum criticality. We systematically calculate the scaling dimensions of various operators in the 3d quantum electrodynamics with the Chern-Simons term and Gross-Neveu couplings by the large-$N$ renormalization group analysis. Specifically, we find certain non-relativistic four-fermion interactions corresponding to the dimer-dimer interactions in the lattice model will drive the deconfined quantum criticality to the first-order transition, consistent with previous numerical studies., 10+14 pages, 6 figures

Random measurements have been shown to induce a phase transition in an extended quantum system evolving under chaotic unitary dynamics, when the strength of measurements exceeds a threshold value. Below this threshold, a steady state with a sub-thermal volume law entanglement emerges, which is resistant to the disentangling action of measurements, suggesting a connection to quantum error-correcting codes. Here we quantify these notions by identifying a universal, subleading logarithmic contribution to the volume law entanglement entropy: $S^{(2)}(A)=\kappa L_A+\frac{3}{2}\log L_A$ which bounds the mutual information between a qudit inside region $A$ and the rest of the system. Specifically, we find the power law decay of the mutual information $I(\{x\}:\bar{A})\propto x^{-3/2}$ with distance $x$ from the region's boundary, which implies that measuring a qudit deep inside $A$ will have negligible effect on the entanglement of $A$. We obtain these results by mapping the entanglement dynamics to the imaginary time evolution of an Ising model, to which we can apply field-theoretic and matrix-product-state techniques. Finally, exploiting the error-correction viewpoint, we assume that the volume-law state is an encoding of a Page state in a quantum error-correcting code to obtain a bound on the critical measurement strength $p_{c}$ as a function of the qudit dimension $d$: $p_{c}\log[(d^{2}-1)({p_{c}^{-1}-1})]\le \log[(1-p_{c})d]$. The bound is saturated at $p_c(d\rightarrow\infty)=1/2$ and provides a reasonable estimate for the qubit transition: $p_c(d=2) \le 0.1893$., Comment: 26 pages, 10 figures

The unified mathematical theory of gapped and gapless edges of two-dimensional (2d) topological orders was developed by two of the authors. According to this theory, the critical point of a purely edge topological phase transition of a 2d topological order can be mathematically characterized by an enriched fusion category. In this work, we provide a physical proof of this fact in a concrete example: the 2d ${\mathbb{Z}}_{2}$ topological order. In particular, we construct an enriched fusion category, which describes a gappable nonchiral gapless edge of the 2d ${\mathbb{Z}}_{2}$ topological order. Then, we use an explicit lattice model construction to realize a topological phase transition between the two well-known gapped edges of the 2d ${\mathbb{Z}}_{2}$ topological order, and show that all the ingredients of the above enriched fusion category can be realized explicitly in this lattice model.

We study the time evolution of quantum entanglement for a specific class of quantum dynamics, namely the locally scrambled quantum dynamics, where each step of the unitary evolution is drawn from a random ensemble that is invariant under local (on-site) basis transformations. In this case, the average entanglement entropy follows Markovian dynamics that the entanglement property of the future state can be predicted solely based on the entanglement properties of the current state and the unitary operator at each step. We introduce the entanglement feature formulation to concisely organize the entanglement entropies over all subsystems into a many-body wave function, which allows us to describe the entanglement dynamics using an imaginary-time Schr\"odinger equation, such that various tools developed in quantum many-body physics can be applied. The framework enables us to investigate a variety of random quantum dynamics beyond Haar random circuits and Brownian circuits. We perform numerical simulations for these models and demonstrate the validity and prediction power of the entanglement feature approach., Comment: 20 pages, 14 figures

The deconfined quantum critical point (DQCP) was originally proposed as a continuous transition between two spontaneous symmetry breaking phases in 2D spin-1/2 systems. While great efforts have been spent on the DQCP for 2D systems, both theoretically and numerically, ambiguities among the nature of the transition are still not completely clarified. Here we shift the focus to a recently proposed 1D incarnation of DQCP in a spin-1/2 chain. By solving it with the variational matrix product state in the thermodynamic limit, a continuous transition between a valence-bond solid phase and a ferromagnetic phase is discovered. The scaling dimensions of various operators are calculated and compared with those from field theoretical description. At the critical point, two emergent O(2) symmetries are revealed, and the associated conserved current operators with exact integer scaling dimensions are determined with scrutiny. Our findings provide the low-dimensional analog of DQCP where unbiased numerical results are in perfect agreement with the controlled field theoretical predictions and have extended the realm of the unconventional phase transition as well as its identification with the advanced numerical methodology., 9 pages + references + appendix, 9 figures

Deconfined quantum critical points govern continuous quantum phase transitions at which fractionalized (deconfined) degrees of freedom emerge. Here we study dynamical signatures of the fractionalized excitations in a quantum magnet (the easy-plane J-Q model) that realize a deconfined quantum critical point with emergent O(4) symmetry. By means of large-scale quantum Monte Carlo simulations and stochastic analytic continuation of imaginary-time correlation functions, we obtain the dynamic spin structure factors in the $S^{x}$ and $S^{z}$ channels. In both channels, we observe broad continua that originate from the deconfined excitations. We further identify several distinct spectral features of the deconfined quantum critical point, including the lower edge of the continuum and its form factor on moving through the Brillouin Zone. We provide field-theoretical and lattice model calculations that explain the overall shapes of the computed spectra, which highlight the importance of interactions and gauge fluctuations to explaining the spectral-weight distribution. We make further comparisons with the conventional Landau O(2) transition in a different quantum magnet, at which no signatures of fractionalization are observed. The distinctive spectral signatures of the deconfined quantum critical point suggest the feasibility of its experimental detection in neutron scattering and nuclear magnetic resonance experiments., 13 pages, 8 figures and 2 appendices

Symmetry protected topological (SPT) phases in free fermion and interacting bosonic systems have been classified, but the physical phenomena of interacting fermionic SPT phases have not been fully explored. Here, employing large-scale quantum Monte Carlo simulation, we investigate the edge physics of a bilayer Kane-Mele-Hubbard model with zigzag ribbon geometry. Our unbiased numerical results show that the fermion edge modes are gapped out by interaction, while the bosonic edge modes remain gapless at the $(1+1)d$ boundary, before the bulk quantum phase transition to a topologically trivial phase. Therefore, finite fermion gaps both in the bulk and on the edge, together with the robust gapless bosonic edge modes, prove that our system becomes an emergent bosonic SPT phase at low energy, which is, for the first time, directly observed in an interacting fermion lattice model., Comment: 5 pages, 5 figures in the main text; 5 pages, 8 figures in the supplemental material

In the last few years a lot of exotic and anomalous topological phases were constructed by proliferating the vortexlike topological defects on the surface of the $3d$ topological insulator (TI) [Fidkowski et al., Phys. Rev. X 3, 041016 (2013); Chen et al., Phys. Rev. B 89, 165132 (2014); Bonderson et al., J. Stat. Mech. (2013) P09016; Wang et al., Phys. Rev. B 88, 115137 (2013); Metlitski et al., Phys. Rev. B 92, 125111 (2015)]. In this work, rather than considering topological phases at the boundary, we will study quantum critical points driven by vortexlike topological defects. In general, we will discuss a $(2+1)d$ quantum phase transition described by the following field theory: $\mathcal{L}=\overline{\ensuremath{\psi}}{\ensuremath{\gamma}}_{\ensuremath{\mu}}({\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{-}i{a}_{\ensuremath{\mu}})\ensuremath{\psi}+|({\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{-}ik{a}_{\ensuremath{\mu}}){\ensuremath{\phi}|}^{2}+{r|\ensuremath{\phi}|}^{2}+g{|\ensuremath{\phi}|}^{4}$, with tuning parameter $r$, arbitrary integer $k$, Dirac fermion $\ensuremath{\psi}$, and complex scalar bosonic field $\ensuremath{\phi}$, which both couple to the same $(2+1)d$ dynamical noncompact U(1) gauge field ${a}_{\ensuremath{\mu}}$. The physical meaning of these quantities/fields will be explained in the text. Making use of the new duality formalism developed in [Metlitski et al., Phys. Rev. B 93, 245151 (2016); Wang et al., Phys. Rev. X 5, 041031 (2015); Wang et al., Phys. Rev. B 93, 085110 (2016); D. T. Son, Phys. Rev. X 5, 031027 (2015)], we demonstrate that this quantum critical point has a quasi-self-dual nature. And at this quantum critical point, various universal quantities such as the electrical conductivity and scaling dimension of gauge-invariant operators, can be calculated systematically through a $1/{k}^{2}$ expansion, based on the observation that the limit $k\ensuremath{\rightarrow}+\ensuremath{\infty}$ corresponds to an ordinary $3d XY$ transition.

For a gapped disordered many-body system with both internal and translation symmetry, one can define the corresponding weak and strong symmetry protected topological (SPT) phases. A strong SPT phase is protected by the internal symmetry $G$ only while a weak SPT phase, fabricated by alignment of a strong SPT state in a lower dimension, requires additional discrete translation symmetry protection. In this paper, we construct a phase transition between weak and strong SPT phase in a strongly interacting boson system. The starting point of our construction is the superconducting Dirac fermions with pair density wave (PDW) order in $2d$. We first demonstrate that the nodal line of the PDW contains a $1d$ boson SPT phase. We further show that melting the PDW stripe and condensing the nodal line provoke the transition from weak to strong SPT phase in $2d$. The phase transition theory contains an O(4) nonlinear-$\ensuremath{\sigma} \mathrm{model}\phantom{\rule{0.28em}{0ex}}(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ with topological $\mathrm{\ensuremath{\Theta}}$ term emerging from the proliferation of domain walls bound to an SPT chain. A similar scheme also applies to weak-strong SPT transition in other dimensions and predicts possible phase transition from $2d$ to $3d$ topological order.

It is expected that the interplay between nontrivial band topology and strong electron correlation will lead to very rich physics. Thus a controlled study of the competition between topology and correlation is of great interest. Here, employing large-scale quantum Monte Carlo simulations, we provide a concrete example of the Kane-Mele-Hubbard model on an AA-stacking bilayer honeycomb lattice with interlayer antiferromagnetic interaction. Our simulation identified several different phases: a quantum spin Hall insulator (QSH), an $xy\text{-plane}$ antiferromagnetic Mott insulator, and an interlayer dimer-singlet insulator. Most importantly, a bona fide topological phase transition between the QSH and the dimer-singlet insulators, purely driven by the interlayer antiferromagnetic interaction, is found. At the transition, the spin and charge gap of the system close while the single-particle excitations remain gapped, which means that this transition has no mean-field analog and it can be viewed as a transition between bosonic symmetry-protected topological (SPT) states. At one special point, this transition is described by a $(2+1)d\phantom{\rule{4pt}{0ex}}O(4)$ nonlinear sigma model with exact $SO(4)$ symmetry and a topological term at exactly $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$. The relevance of this work towards more general interacting SPT states is discussed.

Using determinant quantum Monte Carlo simulations, we demonstrate that an extended Hubbard model on a bilayer honeycomb lattice has two novel quantum phase transitions. The first is a quantum phase transition between the weakly interacting gapless Dirac fermion phase and a strongly interacting fully gapped and symmetric trivial phase, which cannot be described by the standard Gross-Neveu model. The second is a quantum critical point between a quantum spin Hall insulator with spin ${S}^{z}$ conservation and the previously mentioned strongly interacting fully gapped phase. At the latter quantum critical point the single-particle excitations remain gapped, while spin and charge gaps both close. We argue that the first quantum phase transition is related to the ${\mathbb{Z}}_{16}$ classification of the topological superconductor $^{3}\mathrm{He}\text{\ensuremath{-}}B$ phase with interactions, while the second quantum phase transition is a topological phase transition described by a bosonic $O(4)$ nonlinear sigma model field theory with a $\ensuremath{\Theta}$ term.

We demonstrate that the anyon statistics and three-loop statistics of various 2d and 3d topological phases can be derived using semiclassical nonlinear Sigma model field theories with a topological $\Theta$-term. In our formalism, the braiding statistics has a natural geometric meaning: The braiding process of anyons or loops leads to a nontrivial field configuration in the space-time, which will contribute a braiding phase factor due to the $\Theta$-term., Comment: 4 pages, 3 figures

Topological degeneracy is the degeneracy of the ground states in a many-body system in the large-system-size limit. Topological degeneracy cannot be lifted by any local perturbation of the Hamiltonian. The topological degeneracies on closed manifolds have been used to discover/define topological order in many-body systems, which contain excitations with fractional statistics. In this paper, we study a new type of topological degeneracy induced by condensing anyons along a line in two-dimensional topological ordered states. Such topological degeneracy can be viewed as carried by each end of the line defect, which is a generalization of Majorana zero modes. The topological degeneracy can be used as a quantum memory. The ends of line defects carry projective non-Abelian statistics even though they are produced by the condensation of Abelian anyons, and braiding them allows us to perform fault tolerant quantum computations.