Related papers: Spin Leonard pairs and the zero diagonal space
A spin system is a sequence of self-adjoint unitary operators $U_1,U_2,...$ acting on a Hilbert space $H$ which either commute or anticommute, $U_iU_j=\pm U_jU_i$ for all $i,j$; it is is called irreducible when $\{U_1,U_2,...\}$ is an…
A tridiagonal pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on the eigenspaces of the other in a block-tridiagonal fashion. We consider a tridiagonal pair $(A, A^*)$ of…
It is known that if $(A,B)$ is a Leonard pair, then the linear transformations $A$, $B$ satisfy the Askey-Wilson relations A^2 B - b A B A + B A^2 - g (A B+B A) - r B = h A^2 + w A + e I, B^2 A - b B A B + A B^2 - h (A B+B A) - s A = g B^2…
We classify line nodes in superconductors with strong spin-orbit interactions and time-reversal symmetry, where the latter may include non-primitive translations in the magnetic Brillouin zone to account for coexistence with…
A vector space S of linear operators between finite-dimensional vector spaces U and V is called locally linearly dependent (in abbreviate form: LLD) when every vector x in U is annihilated by a non-zero operator in S. By a duality argument,…
A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. In the present paper we give an elementary…
The well known Haldane map from spin chains into the $O(3)$ non linear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, exactly in the…
Let $L_{K}(A)$ be the free Lie algebra on a finite alphabet $A$ over a commutative ring $K$ with unity. For a word $u$ in the free monoid $A^{*}$ let $\tilde{u}$ denote its reversal. Two words in $A^{*}$ are called twin (resp. anti-twin) if…
Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schr\"odinger family of…
For w-legged antiferromagnetic spin-1/2 Heisenberg ladders, a long-range spin-pairing order can be identified which enables the separation of the space spanned by finite-range (covalent) valence-bond configurations into w+1 subspaces. Since…
Let $\mathbb{K}$ denote an algebraically closed field. Let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on $V$ we mean an ordered triple of linear transformations in ${\rm End}(V)$ such…
We give an explicit bijective correspondence between between nonzero pairs of complex numbers, which we regard as spinors or spin vectors, and horospheres in 3-dimensional hyperbolic space decorated with certain spinorial directions. This…
The leading order finite size effects due to spin, namely that of the cubic and quartic in spin interactions, are derived for the first time for generic compact binaries via the effective field theory for gravitating spinning objects. These…
We construct, for spin $0,1,2$ tensor fields on S$^d$, a set of ladder operators that connect the distinct UIRs of SO$(d+1)$. This is achieved by relying on the conformal Killing vectors of S$^d$. For the case of spinning fields, the ladder…
Two matrix vector spaces $V,W\subset \mathbb C^{n\times n}$ are said to be equivalent if $SVR=W$ for some nonsingular $S$ and $R$. These spaces are congruent if $R=S^T$. We prove that if all matrices in $V$ and $W$ are symmetric, or all…
We introduce and discuss an Erd\H{o}s-Ko-Rado basis for the underlying vector space of a Leonard system $\Phi = (A; A^*; \{E_i\}_{i=0}^d ; \{E_i^* \}_{i=0}^d)$ that satisfies a mild condition on the eigenvalues of $A$ and $A^*$. We describe…
Motivated by the conjectures formulated in 2003 by Tun\c{c}el et al., we study interlacing properties of the eigenvalues of $A\otimes B + B\otimes A$ for pairs of $n$-by-$n$ matrices $A, B$. We prove that for every pair of symmetric…
Different types of order are discussed in the context of strongly correlated transition metal oxides, involving pure compounds and $3d^{3}-4d^{4}$ and $3d^{2}-4d^{4}$ hybrids. Apart from standard, long-range spin and orbital orders we…
Let \K denote an algebraically closed field and let V denote a vector space over \K with finite positive dimension. We consider an ordered pair of linear transformations A:V\to V,A*:V \to V that satisfy the following conditions:(i)Each of…
Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\cat{M}_{lc}(n,d)$ denote the moduli space of pairs $(E,D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic…