Temperley–Lieb, Hecke normalization and quantum Potts chains

Let $Q\in \{2,3,4\}$ and let $L\ge 1$ be an integer. The finite Potts-clock space is ${\mathcal{H}}_{Q,L}:=({\mathbb{C}}^Q{)}^{\otimes L}$ of dimension $Q^L$, with the tensor-product Hermitian inner product taken linear in its first argument and computational orthonormal basis $\mathrm{\mid }{a}_1,\dots ,a_L⟩:=\mathrm{\mid }{a}_1⟩\otimes \cdots \otimes \mathrm{\mid }{a}_L⟩,a_j\in \{0,\dots ,Q-1\}\mathrm{.}$ On ${\mathbb{C}}^Q$, this inner product is $⟨x,y⟩={\sum }_{a=0}^{Q-1}{x}_a\overline{y_a}$, and on elementary tensors it is $⟨u_1\otimes \cdots \otimes u_L,v_1\otimes \cdots \otimes v_L⟩={\prod }_{j=1}^L⟨u_j,v_j⟩$. For a unit vector $u$, the notation $\mathrm{\mid }u⟩⟨u\mathrm{\mid }$ denotes the endomorphism $v\mapsto ⟨v,u⟩u$. On one site use the clock and shift convention $R_Q\mathrm{\mid }a⟩=e^{2\pi ia\mathrm{/}Q}\mathrm{\mid }a⟩,M_Q\mathrm{\mid }a⟩=\mathrm{\mid }a-1\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}(\mathrm{m}\mathrm{o}\mathrm{d}Q)⟩\mathrm{.}$ Set $\mathrm{\mid }+⟩:=Q^{-1\mathrm{/}2}{\sum }_{a=0}^{Q-1}\mathrm{\mid }a⟩$. For $1\le j\le L$, let $P_j$ be $\mathrm{\mid }+⟩⟨+\mathrm{\mid }$ on factor $j$ and identity on every other factor. For $1\le j<L$, let $D_{j,j+1}:={\sum }_{a=0}^{Q-1}\mathrm{\mid }aa⟩⟨aa\mathrm{\mid }$ act on factors $j,j+1$ and as identity on the others. The open-chain local operators are $e_{2j-1}:=\sqrt{Q}{P}_j(1\le j\le L),e_{2j}:=\sqrt{Q}{D}_{j,j+1}(1\le j<L)\mathrm{.}$ Equivalently, these are the exact clock-shift operators of the open Potts representation: $e_{2j-1}=Q^{-1\mathrm{/}2}{\sum }_{r=0}^{Q-1}{M}_{Q,j}^r$ and $e_{2j}=Q^{-1\mathrm{/}2}{\sum }_{r=0}^{Q-1}{R}_{Q,j}^r{R}_{Q,j+1}^{-r}$. Thus there are $2L-1$ operators and no bond from site $L$ back to site $1$.

For real coefficients $J_1,\dots ,J_{2L-1}$, define the weighted open-chain matrix $H_{Q,L}(\mathbf{J}):=-{\sum }_{i=1}^{2L-1}{J}_i{e}_i\mathrm{.}$ It is an endomorphism of ${\mathcal{H}}_{Q,L}$. For an endomorphism $T$ of this inner-product space, $T^{\ast }$ is its adjoint, characterized by $⟨Tv,w⟩=⟨v,T^{\ast }w⟩$; $T$ is self-adjoint (Hermitian) exactly when $T=T^{\ast }$. An endomorphism is normal exactly when $T{T}^{\ast }=T^{\ast }T$, so every self-adjoint endomorphism is normal. For a subspace $W\subseteq {\mathcal{H}}_{Q,L}$, the orthogonal projection onto $W$ sends $w+z$ to $w$ in the unique decomposition ${\mathcal{H}}_{Q,L}=W\oplus W^{\perp }$, with $w\in W$ and $z\in W^{\perp }$. Viewed as an endomorphism, it satisfies ${\mathrm{\Pi }}_W^2={\mathrm{\Pi }}_W={\mathrm{\Pi }}_W^{\ast }$; the adjoint identity follows from the orthogonality of the two summands. A self-adjoint endomorphism $A$ is positive definite if $⟨Av,v⟩>0$ for every nonzero $v$, and is positive semidefinite exactly when $⟨Av,v⟩\ge 0$ for every $v$; these quadratic forms are real because $A=A^{\ast }$. The trace $\mathrm{tr}(T)$ of a finite-dimensional endomorphism is the basis-independent sum of diagonal entries of its matrix.

For self-adjoint $H_{Q,L}$ and real $\beta$, a scalar $\lambda$ is an eigenvalue of $H_{Q,L}$ when $\ker (H_{Q,L}-\lambda I)\mathrm{\ne }\{0\}$, and this kernel is its eigenspace. Let $\mathrm{\Lambda }(H_{Q,L})$ be the finite set of distinct eigenvalues. Every $\lambda \in \mathrm{\Lambda }(H_{Q,L})$ is real: for a unit eigenvector $v$, $\lambda =⟨H_{Q,L}v,v⟩=⟨v,H_{Q,L}v⟩=\bar{\lambda }$. Let ${\mathrm{\Pi }}_{\lambda }$ be the orthogonal projection onto the $\lambda$-eigenspace as defined above. A self-adjoint endomorphism is normal, and every normal endomorphism of a finite-dimensional complex inner-product space has an orthonormal eigenbasis; therefore ${\sum }_{\lambda \in \mathrm{\Lambda }(H_{Q,L})}{\mathrm{\Pi }}_{\lambda }=I$. Define $B_{\beta }(H_{Q,L}):={\sum }_{\lambda \in \mathrm{\Lambda }(H_{Q,L})}{e}^{-\beta \lambda }{\mathrm{\Pi }}_{\lambda },Z_{\beta }:=\mathrm{tr}{B}_{\beta }(H_{Q,L}),{\rho }_{\beta }:=Z_{\beta }^{-1}{B}_{\beta }(H_{Q,L})$ whenever $Z_{\beta }>0$. Here a density matrix means a self-adjoint positive semidefinite endomorphism of trace one. The A-page Gibbs lemma establishes that the displayed $H_{Q,L}$ is self-adjoint and that $Z_{\beta }>0$ for every finite real $\beta$.

Open Potts clock-shift Temperley–Lieb generators

The Q = 2, 3, 4 open Potts matrices satisfy Temperley–Lieb

Real and complex inner product spaces, with the inner product linear in the first argument

Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space

The orthogonal projection $P_{W}v$ is the $W$-component in $V=W\oplus W^{\perp }$

Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely

The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$

The basis-independent trace of an endomorphism of a finite-dimensional vector space

For $Q\in \{2,3,4\}$, $L\ge 1$, real coefficients $J_i$, and finite real $\beta$, the matrix $H_{Q,L}(\mathbf{J})$ defined on the Potts-clock chain is self-adjoint. With the spectral definitions $B_{\beta }$, $Z_{\beta }$, and ${\rho }_{\beta }$ from that definition, $Z_{\beta }>0$ and ${\rho }_{\beta }$ is a positive semidefinite matrix of trace one. This is a finite-dimensional mathematical fact; it makes no thermodynamic-limit or empirical claim.

Finite Hermitian Potts-clock chain

Let $Q\in \{2,3,4\}$, $L\ge 1$, $\delta :=\sqrt{Q}$, and let $q\in \mathbb{C}\setminus \{0\}$ satisfy $q+q^{-1}=\delta$. Use the $2L-1$ open-chain operators $e_i$ defined for this Potts-clock chain, and put $g_i:=qI-e_i$ for $1\le i\le 2L-1$. Then $u_i\mapsto e_i$ represents the rank-$2L$ Temperley-Lieb algebra $P_{2L}(\mathbb{C},\delta )$, and $(g_i-qI)(g_i+q^{-1}I)=0,g_i{g}_{i+1}{g}_i=g_{i+1}{g}_i{g}_{i+1},g_i{g}_j=g_j{g}_i(\mathrm{\mid }i-j\mathrm{\mid }>1)\mathrm{.}$ These are the type-A Hecke relations in the normalized generator convention after the coefficient specialization $v\mapsto q^{-1}$. The specified clock-shift model selects $q=e^{i\pi \mathrm{/}4},e^{i\pi \mathrm{/}6},1$ for $Q=2,3,4$, respectively. At $Q=4$, $q=1$ and the same operators $g_i=I-e_i$ satisfy the symmetric-group Coxeter relations, so they define an action of $S_{2L}$. No faithfulness, semisimplicity, or unitarity is asserted.

Finite Hermitian Potts-clock chain

Open Potts clock-shift Temperley–Lieb generators

The abstract Temperley–Lieb generator presentation

The normalized type-A Hecke algebra and its bar involution

The symmetric group has the Coxeter presentation

The local matrices $e_i$ are Hermitian projectors scaled by $\sqrt{Q}$, but this does not make $g_i=qI-e_i$ a star representation when $q$ is non-real. The $Q=2,3$ braid and Hecke relations above are algebraic statements only; no semisimplicity, faithfulness, integrability, or unitary deformation follows from them. The $Q=4$ symmetric-group action is a separate finite representation, not a claim about the dimension or equivalence of any other Potts or spin representation.

The first diagram uses the Temperley-Lieb rectangle convention: boundary points are read left to right, composition stacks bottom to top, and each closed loop is assigned the scalar $\delta$. The second diagram reads braid words bottom to top. At each crossing, the right-to-left strand is the over-strand: its white underlay opens a gap in the other strand, then its black centerline is drawn across that gap. Both diagrams illustrate conventions only; the proofs above are algebraic.

Let $N\ge 1$ and let ${\mathcal{H}}_N=({\mathbb{C}}^2{)}^{\otimes N}$ have the standard tensor-product Hermitian inner product, linear in its first argument, with $\mathrm{\mid }0⟩,\mathrm{\mid }1⟩$ orthonormal on each factor. For $1\le i<N$, let $s_i$ be the swap of tensor factors $i,i+1$, let $\mathrm{\mid }\omega ⟩=(\mathrm{\mid }01⟩-\mathrm{\mid }10⟩)\mathrm{/}\sqrt{2}$, and let $P_i=I^{\otimes (i-1)}\otimes \mathrm{\mid }\omega ⟩⟨\omega \mathrm{\mid }\otimes I^{\otimes (N-i-1)}$ be the projection onto this singlet on those two factors. Put $e_i:=I-s_i$. Then these are the loop-weight-two Temperley-Lieb generators. In the Hecke normalization $g_i(q):=qI-e_i$, at $q=1$ one has $g_i(1)=s_i$. For arbitrary real coefficients ${\mathcal{J}}_i$, the finite exchange sum $X_N(\mathcal{J}):={\sum }_{i=1}^{N-1}{\mathcal{J}}_i{s}_i$ satisfies $X_N(\mathcal{J})=\left({\sum }_{i=1}^{N-1}{\mathcal{J}}_i\right)I-{\sum }_{i=1}^{N-1}{\mathcal{J}}_i{e}_i\mathrm{.}$ The equality $g_i(1)=s_i$ is internal to this spin-half representation; it does not identify its state space with the distinct $Q=4$ Potts representation.

Spin-half swaps realize the Temperley–Lieb algebra at loop weight two

Potts local algebra and braid quotient

No nonzero-$q$ deformation is identified with this Hermitian swap representation. For the Potts $Q=2,3$ values, $q$ is non-real and the relation $g_i=qI-e_i$ alone supplies no star-preserving or unitary representation.