Hidden Twisted Sectors and Exponential Degeneracy in Root-of-Unity XXZ Heisenberg Chains

TL;DR

This work classifies the degenerate subspace of the 1D periodic XXZ spin-1/2 chain at roots of unity by marrying Bethe Ansatz insights with affine Temperley-Lieb (aTL) representation theory. The authors construct exact intertwiners between aTL modules, revealing hidden twisted-boundary sectors that mediate degeneracy and tying the product-state degeneracy to L(\mathfrak{sl}_2) highest-weight multiplets whose Drinfeld-polynomial degree is r = N/\ell. They prove exponential lower bounds g \ge 2^{N/\ell}\ell for commensurate chains (q^N=1) and analogous bounds for incommensurate cases, with numerical evidence showing saturation up to N ≤ 20. The results illuminate how the Bethe Ansatz, aTL morphisms, and twisted boundary conditions jointly explain long-lived product states and suggest avenues for generalization to higher dimensions and potential quantum-sensing applications.

Abstract

Recently, product states have been identified as simple-structured eigenstates of XXZ Heisenberg spin models in arbitrary dimensions, occurring at anisotropy values corresponding to certain roots of unity. Yet, the product states typically only span parts of a larger degenerate eigenspace. Here, we classify this eigenspace in the one-dimensional periodic XXZ chain at all roots of unity $q$, where $q^2$ is an $\ell$-th primitive root of unity. For commensurate chain lengths $N$ with $q^N=1$, we prove that the minimal degeneracy is $2^{N/\ell}\ell$ using the representation theory of the affine Temperley-Lieb (aTL) algebra. For the incommensurate case, we derive analogous exponential lower bounds of $2^{2\lfloor\frac{N}{2\ell}\rfloor+1}$ if $N$ is even and $2^{2\lfloor \frac{N}{2\ell}+\frac{1}{2}\rfloor}$ if $N$ is odd and $q^\ell=1$. Our proof employs the morphisms between aTL modules discovered by Pinet and Saint-Aubin and emphasizes the importance of exact sequences and hidden twisted boundary condition sectors that mediate the degeneracy. In the case of commensurate chain lengths, we connect to the Fabricius-McCoy string construction of all Bethe roots of the degenerate subspace, which previously uncovered parts of our results. We corroborate our results numerically and demonstrate that the lower bound is saturated for chain lengths $N\leq20$. Our work demonstrates for a concrete system how the interplay of the Bethe ansatz, aTL representation theory, and twisted boundary conditions explains degeneracy connected to long-lived product states, stimulating research towards generalization to higher dimensions. Exponential degeneracy could boost applications of spin chains as quantum sensors.

Figures (7)

• Figure 1: (\ref{['fig:descendant_tower_a']}) The descendant tower structure of the FM construction reveals a hypothesis for the structure and degeneracy of $\mathcal{D}$. Visualization adapted from Ref. miao_q_2021. For $M=0$, the primitive state is $\ket{\downarrow\dots\downarrow}$, and there are 3 FM strings, resulting in a degeneracy of 8 states. For $M=1,2$, at each sector there are two roots at infinity, and there are 2 FM strings, resulting in a degeneracy of $4\times3=12$ states. In total, there are 24 degenerate states, of which 20 are projected product states and 4 are additional non-product states. The dashed line indicates "the equator" where the spin-flip symmetry connects sectors above and below the line, which becomes manifest in Fig. \ref{['fig:diagram_twolines_b']} where we include the hidden sectors with non-zero twists. The left tower corresponds to the sequence in Fig. \ref{['fig:diagram_oneline']}, while the right two towers are interlinked and correspond to the sequences in Fig. \ref{['fig:diagram_twolines']}. (\ref{['fig:descendant_tower_b']}) Listing of the Bethe roots. $\alpha_{FM}$ indicates 3-strings of the form $\{\alpha,\alpha-i\pi/3,\alpha+i\pi/3\}$ with string center $\alpha$.
• Figure 2: aTL intertwiners reveal how sectors with different $S_\text{tot}^z$ and twists are connected by intertwiners $F^\pm$ starting from the fully polarized sectors with integer power twists of $q$, relating to the FM construction in Fig. \ref{['fig:descendant_tower']} with $N=9$, $q=e^{\frac{2}{3}\pi i}$, $\ell=3$. For brevity, we denote $\mathcal{H}_{N;t,v}^\pm \equiv (N;t,v)^\pm$. Reversing the $F^\pm$ arrows yields $E^\mp$. (\ref{['fig:diagram_oneline']}) The exact sequence of sectors where $\ell\mid d$, connected by $F^+$. The state in the fully polarized sector $(9;9,1)^+$ is a highest weight vector in the $L(\mathfrak{sl}_2)$ algebra deguchi_xxz_2007, inducing a degeneracy of $2^{N/\ell}=8$. This corresponds to the left tower in Fig. \ref{['fig:descendant_tower_a']}. (\ref{['fig:diagram_twolines']}) Sequences of sectors that start with non-zero twists. Horizontal arrows indicate the $F^-$ intertwiners between $\mathcal{H}^-$ spaces, and diagonal arrows indicate the $F^+$ intertwiners between $\mathcal{H}^+$ spaces. The boxed red sectors highlight the "hidden" sectors with non-zero twists and $\ell\mid d$ which explain the total degeneracy of the zero twist sectors (black). For each horizontal sequence, the red sectors have a total degeneracy of $2^{N/\ell}=8$, resulting in a total degeneracy across all periodic sectors of $16$. (\ref{['fig:diagram_twolines_b']}) The aTL algebra perspective reveals that the right two descendant towers in Fig. \ref{['fig:descendant_tower_a']} are interconnected via hidden towers with nonzero twists $q$ and twist $-q$ (red). Blue solid arrows represent $F^-$ intertwiners (horizontal arrows in \ref{['fig:diagram_twolines']}); cyan dotted arrows represent $F^+$ intertwiners (diagonal arrows in \ref{['fig:diagram_twolines']}). The hidden towers account for the degeneracies of $8$ per periodic tower. With the hidden twisted sectors included, the spin-flip symmetry become manifest: the four-tower structure is symmetric about the equator (dashed line at $M=4.5$), with each sector at $M$ having a mirror partner at $9-M$ connected by the same intertwiner pattern.
• Figure 3: The general sectors connected by intertwiners $F^{\pm}$ for $q^\ell=1$, $0<r\leq\ell/2$, and $r\in\mathbb{N}$. For different $N$, as long as they admit these sectors, they are part of the sequences that truncate at the appropriate sectors. In general, all odd $N$ are in one set of sequences and all even $N$ are in another set of sequences for each $\ell$. (\ref{['fig:diagram_oddl_a']}) The sequence that start with the periodic boundary sector ($w=1$). (\ref{['fig:diagram_oddl_b']}) Sequences starting with non-zero twists, which include hidden half-sequences of twisted sectors (boxed in red, $w=q^{\pm r}$) that mediate connections between periodic boundary sectors.
• Figure 4: The general sectors connected by intertwiners $F^{\pm}$ for $q^\ell=-1$, even $N$, $0<r\leq\ell/2$, and $r\in\mathbb{N}$. Solid red boxes indicate hidden twisted boundary sectors ($w=\pm q^{\pm r}$). Dashed blue boxes indicate sectors whose twists acquire a factor of $-1$ compared to the $q^\ell=1$ case (Fig. \ref{['fig:diagram_oddl']}). (\ref{['fig:diagram_ql-1_evenN_a']}) The sequence that starts with periodic boundary sector ($w=1$). Some sectors with $w=-1$ appear. (\ref{['fig:diagram_ql-1_evenN_b']}) The sequence that starts with anti-periodic boundary sector ($w=-1$). This parallels \ref{['fig:diagram_ql-1_evenN_a']} with all twists multiplied by $-1$. (\ref{['fig:diagram_ql-1_evenN_c']}) Sequences starting with non-zero twists that include hidden half-sequences of twisted sectors ($w=q^{\pm r}$).
• Figure 5: As an example of the incommensurate case, we present the sequences of sectors connected by intertwiners $F^{\pm}$ for $N=7$, $q=e^{\frac{2}{3}\pi i}$, $\ell=3$. The red sectors highlight the hidden sectors with non-zero twists and $\ell\mid d$ in the sequences, which explain the multiplicity of the zero twist sectors through the exact sequence construction.

Theorems & Definitions (12)

• Theorem 1: Lower bound for commensurate case
• Corollary 2: Lower bound for incommensurate case
• Remark 3: Special cases: $\Delta = 0, \pm 1$
• Theorem 4: Pinet, Saint-Aubin Pinet_2022
• Lemma 5: Transport of generalized eigenvalues
• Lemma 6: Odd $N$ and $q^{\ell}=-1$
• proof
• Lemma 7: Restriction of exact sequences
• Corollary 8: Dimension of restricted exact sequences
• proof : Proof of Lemma \ref{['lem:transport']}