Finite Symmetry Research Articles

Finite Symmetries, Orientations of Bases, Extended CPT for Gravity and the Standard Model

The author suggested three 8-dimensional vector spaces with different multiplications and number of generators for SI with the SU(3) 8 gluon GellMann matrices λk, the signed quaternion SU(2) generators (Pauli matrices σj, j = 1,2,3 and id) of the weak interaction WI and newly added is in the MINT-Wigris project the octonians which have seven Gleason measuring frames GF, not only the weak Pauli spin GF. Developing a unified theory for the four basi interactions the doubling of quaternion numbers to octonians is necessary. The use of the GF real cross product was necessary to double up the complex numbers to quaternions. As projections the λk matrices are 3-dimensional blown up σj matrices with a row and column of coordinates 0 added. The blown up σ3 matrices are linearly independent and give only 2 not 3 gluons as geometrical invariants. The 3 WI invariants are the weak bosons W+, W- and Z° and one more invariant photon for EMI for the electromagnetic interaction with the U(1) symmetry, a circle.

Lattice models that realize Zn -1 symmetry-protected topological states for even n

Higher symmetries can emerge at low energies in a topologically ordered state with no symmetry, when some topological excitations have very high-energy scales while other topological excitations have low energies. The low-energy properties of topological orders in this limit, with the emergent higher symmetries, may be described by higher symmetry-protected topological order. This motivates us, as a simplest example, to study a lattice model of ${\mathbb{Z}}_{n}$-1 symmetry-protected topological (1-SPT) states in $3+1$ dimensions for even $n$. We write an exactly solvable lattice model and study its boundary transformation. On the boundary, we show the existence of anyons with nontrivial self-statistics. For the $n=2$ case, where the bulk classification is given by an integer $m$ mod 4, we show that the boundary can be gapped with double-semion topological order for $m=1$ and toric code for $m=2$. The bulk ground-state wave-function amplitude is given in terms of the linking numbers of loops in the dual lattice. Our construction can be generalized to arbitrary 1-SPT protected by finite unitary symmetry.

Nonlocal symmetries and group invariant solutions for the coupled variable-coefficient Newell-Whitehead system

Starting from the Lax pairs, the nonlocal symmetries of the coupled variable-coefficient Newell-Whitehead system are obtained. By introducing an appropriate auxiliary dependent variable, the nonlocal symmetries are localized to Lie point symmetries and the coupled variable-coefficient Newell-Whitehead system is extended to an enlarged system with the auxiliary variable. Then the finite symmetry transformation for the prolonged system is found by solving the initial-value problems. Furthermore, by applying symmetry reduction method to the enlarged system, two kinds of the group invariant solutions are given.

Reflection groups and 3d mathcal{N} > 6 SCFTs

We point out that the moduli spaces of all known 3d mathcal{N} = 8 and mathcal{N} = 6 SCFTs, after suitable gaugings of finite symmetry groups, have the form ℂ4r/Γ where Γ is a real or complex reflection group depending on whether the theory is mathcal{N} = 8 or mathcal{N} = 6, respectively. Real reflection groups are either dihedral groups, Weyl groups, or two sporadic cases H3,4 Since the BLG theories and the maximally supersymmetric Yang-Mills theories correspond to dihedral and Weyl groups, it is strongly suggested that there are two yet-to­be-discovered 3d mathcal{N} = 8 theories for H3,4. We also show that all known mathcal{N} = 6 theories correspond to complex reflection groups collectively known as G(k, x, N). Along the way, we demonstrate that two ABJM theories (SU(N)k x SU(N)-k)/ℤN and (U(N)k x U(N)-k) /ℤk are actually equivalent.

Controllable symmetry breaking solutions for a nonlocal Boussinesq system

The generalized Boussinesq equation is a useful model to describe the water wave. In this paper, with the coupled Alice-Bob (AB) systems, the nonlocal Boussinesq system can be obtained via the parity and time reversal symmetry reduction. By introducing an extended Bäcklund transformation, the symmetry breaking rogue wave, symmetry breaking soliton and symmetry breaking breather solutions for a nonlocal Boussinesq system are obtained through the derived Hirota bilinear form. The residual symmetry and finite symmetry transformation of the nonlocal AB-Boussinesq system are also studied.

Residual symmetry, nth Bäcklund transformation, and soliton‐cnoidal wave interaction solution for the combined modified KdV–negative‐order modified KdV equation

This paper is concerned with the nth Bäcklund transformation (BT) related to multiple residual symmetries and soliton‐cnoidal wave interaction solution for the combined modified KdV–negative‐order modified KdV (mKdV‐nmKdV) equation. The residual symmetry derived from the truncated Painlevé expansion can be extended to the multiple residual symmetries, which can be localized to Lie point symmetries by prolonging the combined mKdV‐nmKdV equation to a larger system. The corresponding finite symmetry transformation, ie, nth BT, is presented in determinant form. As a result, new multiple singular soliton solutions can be obtained from known ones. We prove that the combined mKdV‐nmKdV equation is integrable, possessing the second‐order Lax pair and consistent Riccati expansion (CRE) property. Furthermore, we derive the exact soliton and soliton‐cnoidal wave interaction solutions by applying the nonauto‐BT obtained from the CRE method.

3d one-form mixed anomaly and entanglement entropy

We study mixed anomaly between G1 and G2 of one-form finite symmetry G1 × G2 in 3d Chern-Simons theories. We assign a quantum entanglement structure to two linked G-symmetry lines (Wilson loops) and compute the entanglement entropy S[G]. We find a measure of the mixed anomaly by computing S[G1 × G2] − S[G1] − S[G2].

Automorphisms of generic gradient vector fields with prescribed finite symmetries

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $\Gamma$, and let $f$ be a $\Gamma$-invariant Morse function on $M$. We prove that the space of $\Gamma$-invariant Riemannian metrics on $M$ contains a residual subset ${\mathcal M\mathrm{et}}\_f$ with the following property. Let $g\in\mathcal{M}\mathrm{et}\_f$ and let $\nabla^gf$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\phi\in \mathrm {Diff}(M)$ preserving $\nabla^gf$ there exists some $t\in\mathbb R$ and some $\gamma\in\Gamma$ such that for every $x\in M$ we have $\phi(x)=\gamma,\Phi\_t^g(x)$, where $\Phi\_t^g$ is the time-$t$ flow of the vector field $\nabla^gf$.

On 2-form gauge models of topological phases

We explore 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space B2G of the symmetry group G, and they are classified by cohomology classes of B2G. For finite symmetry groups, 2-form topological theories have a natural lattice interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d that is exactly solvable. This construction relies on the introduction of a cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified with the simplicial cocycles of B2G as provided by the so-called W -construction of Eilenberg-MacLane spaces. We show algebraically and geometrically how a 2-form 4-cocycle reduces to the associator and the braiding isomorphisms of a premodular category of G-graded vector spaces. This is used to show the correspondence between our 2-form gauge model and the Walker-Wang model.

Classification of 3+1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions

In this paper, we classify EF topological orders for 3+1D bosonic systems where some emergent pointlike excitations are fermions. (1) We argue that all 3+1D bosonic topological orders have gappable boundary. (2) All the pointlike excitations in EF topological orders are described by the representations of $G_f=Z_2^f\leftthreetimes_{e_2} G_b$ -- a $Z_2^f$ central extension of a finite group $G_b$ characterized by $e_2\in H^2(G_b,Z_2)$. (3) We find that the EF topological orders are classified by 2+1D anomalous topological orders $\mathcal{A}_b^3$ on their unique canonical boundary. Here $\mathcal{A}_b^3$ is a unitary fusion 2-category with simple objects labeled by $\hat G_b=Z_2^m\leftthreetimes G_b$. $\mathcal{A}_b^3$ also has one invertible fermionic 1-morphism for each object as well as quantum-dimension-$\sqrt 2$ 1-morphisms that connect two objects $g$ and $gm$, where $g\in \hat G_b$ and $m$ is the generator of $Z_2^m$. (4) When $\hat G_b$ is the trivial $Z_2^m$ extension, the EF topological orders are called EF1 topological orders, which is classified by simple data $(G_b,e_2,n_3,\nu_4)$. (5) When $\hat G_b$ is a non-trivial $Z_2^m$ extension, the EF topological orders are called EF2 topological orders, where some intersections of three stringlike excitations must carry Majorana zero modes. (6) Every EF2 topological order with $G_f=Z_2^f\leftthreetimes G_b$ can be associated with a EF1 topological order with $G_f=Z_2^f\leftthreetimes \hat G_b$. (7) We find that all EF topological orders correspond to gauged 3+1D fermionic symmetry protected topological (SPT) orders with a finite unitary symmetry group. (8) We further propose that the general classification of 3+1D topological orders with finite unitary symmetries for bosonic and fermionic systems can be obtained by gauging or partially gauging the finite symmetry group of 3+1D SPT phases of bosonic and fermionic systems.

Orbifolds of n–dimensional defect TQFTs

We introduce the notion of $n$-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or open-closed TQFTs are special cases of defect TQFTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of "generalised orbifolds" for any $n$-dimensional defect TQFT: Given a defect TQFT $\mathcal{Z}$, one obtains a new TQFT $\mathcal{Z}_{\mathcal{A}}$ by decorating the Poincar\'e duals of triangulated bordisms with certain algebraic data $\mathcal{A}$ and then evaluating with $\mathcal{Z}$. The orbifold datum $\mathcal{A}$ is constrained by demanding invariance under $n$-dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any $n$. After developing the general theory, we focus on the case $n=3$.

The fourth-order symmetry energy of nuclear matter and symmetry energy coefficients of finite nuclei using extended semi-empirical mass formula

An extended nuclear mass formula has been used by considering the bulk, surface and coulomb contributions to the nuclear mass. In this mass formula, the fourth-order symmetry energy coefficient [Formula: see text] of finite nuclei and fourth-order symmetry energy [Formula: see text] of nuclear matter (NM) are related explicitly to the characteristic parameters of NM equation of state (EOS) using finite range effective interaction. The calculations are carried out with Yukawa form of exchange interaction having the same range but with different strengths for interaction between like and unlike nucleon. In this extended mass formula, by approximating [Formula: see text] to a constant [Formula: see text] an explicit relation between [Formula: see text] and fourth-order symmetry energy [Formula: see text] is obtained, which provides the possibility to extract information on [Formula: see text].

Formula omitted]-th Bäcklund transformation and soliton-cnoidal wave interaction solution to the combined KdV–negative-order KdV equation

The truncated Painlevé expansion is used to obtain the residual symmetry of the combined KdV–negative-order KdV (KdV–nKdV) equation, which can be localized to the Lie point symmetry. The multiple residual symmetries are established and localized in the prolonged system by defining more new quantities, and the corresponding finite symmetry transformation which is nth Bäcklund transformation is given by determinant form. In addition, we derive a second order Lax pair by introducing a transformation. Finally, we apply the consistent Riccati expansion (CRE) method to prove that the combined KdV–nKdV equation is CRE solvable and construct exact interaction solution between the soliton and the cnoidal periodic wave.

Higher-order CPM Constructions

We define a higher-order generalisation of the CPM construction based on arbitrary finite abelian group symmetries of symmetric monoidal categories. We show that our new construction is functorial, and that its closure under iteration can be characterised by seeing the construction as an algebra for an appropriate monad. We provide several examples of the construction, connecting to previous work on the CPM construction and on categorical probabilistic theories, as well as upcoming work on higher-order interference and hyper-decoherence.

Classification of subsystem symmetry-protected topological phases

We consider symmetry-protected topological (SPT) phases in 2D protected by linear subsystem symmetries, i.e. those that act along rigid lines. There is a distinction between a "strong" subsystem SPT phase, and a "weak" one, which is composed of decoupled 1D SPTs with global symmetries. We propose a natural definition for strong equivalence of such phases, in terms of a linearly-symmetric local unitary transformation, under which a weak subsystem SPT is equivalent to the trivial phase. This leads to a number of distinct equivalence classes of strong subsystem SPTs, which we show are in one-to-one correspondence with elements of the group $\mathcal{C}[G_s] = \mathcal{H}^{2}[G_s^2,U(1)]/(\mathcal{H}^{2}[G_s,U(1)])^3$, where $G_s$ is the finite abelian onsite symmetry group. We also show that strong subsystem SPTs by our classification necessarily exhibit a spurious topological entanglement entropy on a cylinder.

Nonlocal symmetries and the nth finite symmetry transformation or AKNS system

In this paper, by introduction of pseudopotentials, the nonlocal symmetry is obtained for the Ablowitz–Kaup–Newell–Segur system, which is used to describe many physical phenomena in different applications. Together with some auxiliary variables, this kind of nonlocal symmetry can be localized to Lie point symmetry and the corresponding once finite symmetry transformation is calculated for both the original system and the prolonged system. Furthermore, the nth finite symmetry transformation represented in terms of determinant and exact solutions are derived.

Unwinding short-range entanglement

Symmetry-Protected Topological (SPT) phases are gapped phases of quantum matter protected by global symmetries that cannot be adiabatically deformed to a trivial phase without breaking symmetry. In this work, we show that, for several SPT phases that are short range entangled (SRE), enlarging symmetries may effectively achieve the consequences of explicitly breaking symmetries. In other words, we demonstrate that non-trivial SPT phases can be unwound to trivial ones by symmetry extension- through a path where the Hilbert space is enlarged and the Hamiltonian is invariant under an extended symmetry group applying the idea of Wang, Wen and Witten in arXiv:1705.06728. We show examples of both bosonic and fermionic SPT phases in 1+1 dimensions, including Haldane's bosonic spin chain and layers of Kitaev's fermionic Majorana chains. By adding degrees of freedom into the boundary/bulk, we can lift the zero mode degeneracy, or unwind the whole system. Furthermore, based on properties of Schur cover, we sketch a general picture of unwinding applicable to any 1+1 D bosonic SPT phase protected by on-site finite symmetry. Altogether we show that SRE states can be unwound by symmetry breaking, inversion and symmetry extension.

Nonadiabatic Hydrogen Dissociation on Copper Nanoclusters.

Copper surfaces exhibit high catalytic selectivity but have poor hydrogen dissociation kinetics; therefore, we consider icosahedral Cu13 nanoclusters to understand how nanoscale structure might improve catalytic prospects. We find that the spin state is a surprisingly important design consideration. Cu13 clusters have large magnetic moments due to finite size and symmetry effects and exhibit magnetization-dependent catalytic behavior. The most favorable transition state for hydrogen dissociation has a lower activation energy than that on single-crystal copper surfaces but requires a magnetization switch from 5 to 3 μB. Without this switch, the activation energy is higher than that on single-crystal surfaces. Weak spin-orbit coupling hinders this switch, decreasing the kinetic rate of hydrogen dissociation by a factor of 16. We consider strategies to facilitate magnetization switches through optical excitations, substitution, charge states, and co-catalysts; these considerations demonstrate how control of magnetic properties could improve catalytic performance.

Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$ symmetry along the boundary becomes on-site when viewed as an extended $H$ symmetry, via a suitable group extension $1\to K\to H\to G\to1$. Namely, a non-perturbative global (gauge/gravitational) anomaly in $G$ becomes anomaly-free in $H$. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, always existent for any SPT state in any spacetime dimension $d \geq 2$ of any finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an $H$-symmetry extended boundary of bulk $d \geq 2$, or more naturally by a topological emergent $K$-gauge theory with a global symmetry $G$ on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry $G$, described by representations of $H$. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped interfaces obtained via symmetry breaking (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension. More generally, applying our approach to SPT, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.

Localized excitations and interactional solutions for the reduced Maxwell-Bloch equations

Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell–Bloch equations. The nonlocal symmetries of the reduced Maxwell–Bloch equations are obtained by the truncated Painlevé expansion approach and the Möbious invariant property. The nonlocal symmetries are localized to a prolonged system by introducing suitable auxiliary dependent variables. The extended system can be closed and a novel Lie point symmetry system is constructed. By solving the initial value problems, a new type of finite symmetry transformations is obtained to derive periodic waves, Ma breathers and breathers travelling on the background of periodic line waves. Then rich exact interactional solutions are derived between solitary waves and other waves including cnoidal waves, rational waves, Painlevé waves, and periodic waves through similarity reductions. In particular, several new types of localized excitations including rogue waves are found, which stem from the arbitrary function generated in the process of similarity reduction. By computer numerical simulation, the dynamics of these localized excitations and interactional solutions are discussed, which exhibit meaningful structures.