Weyl fermions are two-component chiral in $(3+1)$ dimensions. When coupled to a gauge field, the fermion is known have an axial anomaly, which means current conservation of left-handed and right-handed cannot be preserved separately. Recently, been proposed condensed-matter systems named ``Weyl semimetals.'' In this paper we propose semimetal phase magnetically doped topological insulators, study anomaly system. We that magnetic fluctuation system plays role ``chiral field'' minimally...

We propose a superfluid phase of ``many-fracton system'' in which charge and total dipole moments are conserved quantities. In this work, both microscopic model long-wavelength effective theory analyzed. start with second quantized formulate the coherent-state path-integral representation. With repulsive interactions positive chemical potential, we calculate various properties resulting state make comparison conventional superfluid. deduce highly nonlinear Euler-Lagrange equation as well two...

As a hallmark of pure quantum effect, entanglement has provided unconventional routes to characterize condensed matter systems. Here, from the perspective entanglement, we disclose exotic physics in non-Hermitian quasicrystals. We study class experimentally realizable models for quasicrystal chains, which asymmetric hopping and complex potential coexist. diagnose global phase diagram by means both real-space momentum-space partitions. By measuring entropy, numerically determine...

Bosonic topological insulators (BTI) in three dimensions are symmetry-protected phases (SPT) protected by time-reversal and boson number conservation {symmetries}. BTI were first proposed classified the group cohomology theory which suggests two distinct root states, each carrying a $\mathbb{Z}_2$ index. Soon after, surface anomalous orders to identify different states of BTI, even leads new state beyond classification. In this paper, we propose universal physical mechanism via...

Recently, there is a considerable study on gapped symmetric phases of bosons that do not break any symmetry. Even without symmetry breaking, the can still be in many exotic new states matter, such as symmetry-protected topological (SPT) phases, which are short-range entangled and symmetry-enriched (SET) long-range entangled. It well known noninteracting fermionic insulators SPT protected by time-reversal U(1) fermion number conservation In this paper, we construct three-dimensional with...

Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT cannot be adiabatically connected to the trivial disordered state(or atomic insulator) as long certain symmetry $G$ is unbroken. At low energies, most of two-dimensional SPTs with Abelian can described quantum field theory (TQFT) multi-component Chern-Simons type. However, in contrast fractional Hall effect where TQFT give rise interesting bulk anyons, for only...

Gapped fracton phases of matter generalize the concept topological order and broaden our fundamental understanding entanglement in quantum many-body systems. However, their analytical or numerical description beyond exactly solvable models remains a formidable challenge. Here we employ an exact 3D tensor-network approach that allows us to study Z_{N} generalization prototypical X cube model its phase transitions between distinct states via fully tractable wave function deformations. We map...

A large class of symmetry-protected topological phases (SPT) in boson / spin systems have been recently predicted by the group cohomology theory. In this work, we consider SPT states at least with charge symmetry (U(1) or Z$_N$) $S^z$ rotation 2D, 3D, and surface 3D. If both are U(1), apply external electromagnetic field `spin gauge field' to study response. For examples (i.e. U$_c$(1)$\rtimes$Z$^T_2$, U$_s$(1)$\times$Z$^T_2$, U$_c$(1)$\times$[U$_s$(1)$\rtimes$Z$_2$]; subscripts $c$ $s$...

Quantum entanglement plays a ubiquitous role in theoretical physics, from the characterization of novel phases matter to understanding efficacy numerical algorithms. As such, there have been extensive studies on spectrum (ES) free-fermion systems, particularly relation between its spectral flow and topological charge pumping. However, far less has studied about spacing adjacent eigenenergies, which affects truncation error computations involving matrix product states or projected entangled...

We propose a general approach to construct symmetry protected topological (SPT) states (i.e., the short-range entangled with symmetry) in 2D spin/boson systems on lattice. In our approach, we fractionalize spins/bosons into different fermions, which occupy nontrivial Chern bands. After Gutzwiller projection of free fermion state obtained by filling bands, can obtain SPT particular, constructed U(1) spin-1 model, SO(3) boson system bosons and spinless bosons, SU(2) spin-1/2 system. By...

While winding a particlelike excitation around looplike yields the celebrated Aharonov-Bohm phase, we find distinctive braiding phase in absence of such mutual winding. In this Letter, propose an exotic particle-loop-loop process, dubbed Borromean rings braiding. particle moves two unlinked loops, that its trajectory and loops form or more general Brunnian links. As does not wind with any resulting is fundamentally different from phase. We derive explicit expression for terms underlying...

Motivated by symmetry-protected topological phases (SPTs) with both spatial symmetry (e.g., lattice rotation) and internal spin rotation), we propose a class of exotic terms, which generalize the well-known Wen-Zee terms quantum Hall systems [X.-G. Wen A. Zee, Phys. Rev. Lett. 69, 953 (1992)]. These generalized are expressed as wedge product connection usual gauge fields (1-form or higher) in various dimensions. In order to probe SPT orders, externally insert twists such domain walls...

Topological phases of matter are usually realized in deconfined gauge theories. In this context, confined with strongly fluctuating fields seem to be irrelevant the physics topological phases. For example, low-energy theory two-dimensional (2D) toric code model (i.e., phase ${\mathbb{Z}}_{2}$ theory) is a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ Chern-Simons which charges $e$ and $m$ particles) gapped, while topologically trivial. paper, we point out route constructing...

We study the instabilities of a particle-hole symmetric Weyl metal with both electron and hole Fermi surfaces (FS) around points. For repulsive interaction, we find that leading instability is towards longitudinal spin-density-wave order (SDW$_z$). Besides, there exist three degenerate subleading instabilities: charge-density-wave (CDW) two transverse (SDW$_{x,y}$) instabilities. an attractive interaction are pair-density-wave orders (PDW) which pair FS's separately. Both PDW SDW$_z$...

Excitations in (3+1)-dimensional [(3+1)D] topologically ordered phases have very rich structures. (3+1)D topological support both pointlike and stringlike excitations, particular the loop (closed string) excitations may admit knotted linked In this work, we ask following question: How do different types of contribute to entanglement entropy or, alternatively, can use detect structure further obtain information underlying order? We are mainly interested order that be realized Dijkgraaf-Witten...

Fracton topological order hosts fractionalized point-like excitations (e.g., fractons) that have restricted mobility. In this article, we explore even more bizarre realization of fracton phases admit spatially extended with restriction on both mobility and deformability. First, present exactly solvable lattice quantum frustrated spin models study their ground states excited analytically. We construct a family tree in which parent descendent share excitation DNA. Second, the help solvability...

Gutzwiller projection is a way to construct many-body wave functions that could carry topological order or symmetry-protected (SPT) order. However, an important issue determine whether not given Gutzwiller-projected function (GWF) carries nontrivial SPT order, and which carried by the function. In this paper, we numerically study in spin $S=1$ GWF on kagome lattice. Using standard Monte Carlo method, directly confirm has (1) gapped bulk with short-range correlations, (2) trivial via...

We propose an exact equivalence between the entanglement spectra of two completely different free-fermion systems at zero temperature. This follows from a position-momentum duality where physical roles occupied band and real space projectors are exchanged. examine consequences this in multi-band models as example also physically motivate spectrum partitioned two-band topological insulator with that bilayer Fermi gas interlayer partition. has interesting relation Wannier Spectrum...

In this paper, we propose a generalization of the $S$-duality four-dimensional quantum electrodynamics ($\text{QED}_4$) to $\text{QED}_4$ with fractionally charged excitations, fractional $S$-duality. Such can be obtained by gauging $\text{U(1)}$ symmetry topologically ordered state charges. When time-reversal is imposed, axion angle ($\theta$) take nontrivial but still invariant value $\pi/t^2$ ($t\in\mathbb{Z}$). Here, $1/t$ specifies minimal electric charge carried bulk excitations....

While two-dimensional symmetry-enriched topological phases $(\text{SET}\mathrm{s})$ have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry ${G}_{s}$ on gauge theories (denoted by $\text{GT})$ with group ${G}_{g}$. The resulting symmetric dubbed ``symmetry-enriched theories'' $(\text{SEG})$, which may be served as low-energy effective quantum spin liquids. focus $\text{SEG}\mathrm{s}$...

We calculate the topological part of electromagnetic response bosonic integer quantum Hall (BIQH) phases in odd (space-time) dimensions, and insulator (BTI) chiral semimetal (BCSM) even dimensions. To do this, we use nonlinear sigma model (NLSM) description symmetry-protected (SPT) phases, method gauged Wess-Zumino (WZ) actions. find surprising result that for BIQH states dimension $2m\ensuremath{-}1$ $(m=1,2,\ensuremath{\cdots})$, bulk to an field ${A}_{\ensuremath{\mu}}$ is characterized...

Braiding and fusion rules of topological excitations are indispensable invariants in quantum computation orders. While two dimensions (2D) always particlelike anyons, those three (3D) incorporate not only particles but also loops, spatially nonlocal objects, making it novel challenging to study higher dimensions. 2D have been well understood from bulk Chern-Simons field theory edge conformal theory, is yet be thoroughly explored for 3D higher-dimensional theory. Here, we perform a...

A bstract Topological orders are a prominent paradigm for describing quantum many-body systems without symmetry-breaking orders. We present topological field theoretical (TQFT) study on in five-dimensional spacetime (5D) which excitations include not only point-like particles , but also two types of spatially extended objects: closed string-like loops and two-dimensional membranes . Especially, have been rarely explored the literature By introducing higher-form gauge fields, we construct...

As an unconventional realization of topological orders with exotic interplay topology and geometry, fracton (topological) feature subextensive ground state degeneracy subdimensional excitations that are movable only within a certain subspace. It has been known in the exactly solvable three-dimensional X-cube model universally represents type-I orders, mobility constraints on originate from absence spatially deformable string-like operators. To unveil this paper, we study dynamical signature...