Bootstrapping Symmetries in Quantum Many-Body Systems from the Cross Spectral Form Factor

Abstract

Symmetries play a central role in quantum many-body physics, yet uncovering them systematically remains challenging. We introduce a bootstrap framework designed to reconstruct the representation theory of hidden finite group symmetries of quantum many-body lattice Hamiltonians, using only a known symmetry subgroup $N$ and spectral correlations between its symmetry sectors. We introduce a novel variant of the spectral form factor, the cross spectral form factor (xSFF), which we compute via exact diagonalization to seed the bootstrap algorithm. By applying the constraints derived from these data alongside the algebraic conditions of the fusion rules, our bootstrap procedure sharply restricts the set of candidate groups $G$. Remarkably, without any prior assumptions regarding the full symmetry group $G$, our method can systematically recover its representation-theoretic data, including the number and dimensions of the irreducible representations, their branching rules with respect to $N$, the fusion algebra, and the full character table. This framework applies equally well to chaotic and integrable many-body systems and accommodates both unitary and anti-unitary symmetries. Through various examples, we demonstrate that the underlying group $G$ can be uniquely identified. In particular, our bootstrap independently recovers the $\mathbb{Z}_4$ symmetry at the self-dual point of the three-state quantum torus chain, detects signatures of projective representations in the effective Hamiltonian of the driven Bose-Hubbard model, and rediscovers the $η$-pairing $\mathrm{SO}(4)$ symmetry of the one-dimensional Fermi-Hubbard model. Our framework thus establishes a practical route to identify symmetries directly from dynamical spectral observables.

Figures (9)

• Figure 1: Computational workflow of the bootstrap algorithm. At each trial rank $r$, the algorithm deduces allowable branching vector types (Step 1), assembles candidate branching matrices (Step 2), enumerates fusion decompositions (Step 3), and backtracks over the full fusion set enforcing associativity and rigidity (Step 4). If no consistent solution is found, $r$ is incremented. Otherwise, the surviving solutions at the termination rank $r=r_*$ are classified by their dimensions and character tables (Step 5) and split into linear representation or corepresentation branches (Step 6).
• Figure 3: The xSFF for the $S_3$-invariant O'Brien-Fendley model with $L=8$ sites, projected onto the $\mathbb{Z}_3$ irreps. The data are averaged over 200 disorder realizations, where the bond couplings $J_j\sim\mathcal{N}(1,\,0.25^2)$ and the field couplings $h_j\sim\mathcal{N}(1,\,0.25^2)$ are independent Gaussian random variables. To clearly resolve the plateau heights, the xSFF data throughout this work are processed using auto-scale Gaussian smoothing. Solid colored lines denote the smoothed xSFF, while the underlying shaded curves represent the raw data. Left: The diagonal elements, whose late-time plateaus align perfectly with the corresponding benchmark lines $R_{\lambda}=R_{\text{all}}$ for all $\lambda\in\hat{\mathbb{Z}}_3$. Right: The off-diagonal elements of the xSFF. In particular, $K_{1,1}^{(\mathbb{Z}_3)} = K_{1,2}^{(\mathbb{Z}_3)} = K_{2,2}^{(\mathbb{Z}_3)}$, implying that the charge sectors $\mathcal{H}_1$ and $\mathcal{H}_2$ are degenerate.
• Figure 4: xSFF for the Kennedy-Tasaki transformed spin-1 model with $L=8$ sites, projected onto the $\mathbb{Z}_2 \times \mathbb{Z}_2$ irreps, averaged over 200 disorder realizations. The couplings are drawn independently as $J_j\sim\mathcal{N}(1,\,0.25^2)$, $\Delta_j\sim\mathcal{N}(0.7,\,0.25^2)$, $g_j\sim\mathcal{N}(0.2,\,0.25^2)$, and $D_j\sim\mathcal{N}(0.3,\,0.25^2)$. Left: The diagonal elements, whose late-time plateaus align perfectly with the corresponding benchmark lines $R_{\lambda}$. Right: The off-diagonal elements of the xSFF. In particular, $K_{11,11}^{(V_4)}=K_{01,01}^{(V_4)} = K_{11,01}^{(V_4)}$, implying that the charge sectors $\mathcal{H}_{01}$ and $\mathcal{H}_{11}$ are degenerate.
• Figure 5: xSFF for the extended Ashkin-Teller chain at the Potts point with $L = 7$ sites, projected onto $N = \mathbb{Z}_2 \times \mathbb{Z}_2$ irreps, averaged over $200$ disorder realizations. The couplings are drawn independently from uniform distributions: nearest-neighbor $J_j\sim\mathrm{Unif}(0.5,\,1.5)$, next-nearest-neighbor $J_j\sim\mathrm{Unif}(0.5,\,1.5)$, and transverse field $h_j\sim\mathrm{Unif}(0.5,\,1.5)$. Left: The diagonal elements exhibit late-time plateaus that align perfectly with the corresponding benchmark lines $R_{\lambda}=R_{\text{all}}$ for all $\lambda\in\hat{V}_4$, with the exception that $K_{00,00}^{(V_4)}/R_{\text{all}}>1$. Right: Off-diagonal elements. The three non-trivial sectors are degenerate, with $K_{10,01}^{(V_4)} = K_{10,11}^{(V_4)} = K_{01,11}^{(V_4)} = K_{10,10}^{(V_4)}$, while the cross-correlations with the trivial sector vanish.
• Figure 6: xSFF for the quantum torus chain at the non-self-dual point $\theta=\pi/6$, with $L = 9$ sites, projected onto $N \cong \mathbb{Z}_3^2 \rtimes \mathbb{Z}_2$ irreps, averaged over $200$ disorder realizations. The bond couplings are drawn as $J_j\sim\mathcal{N}(1,\,0.25^2)$, with $J_j^Z = J_j\cos\theta$ and $J_j^X = J_j\sin\theta$. Left: The diagonal elements exhibit late-time plateaus that align perfectly with the corresponding benchmark lines $R_{\lambda}$. Right: The off-diagonal elements. The paired sectors $[1,1] \leftrightarrow [1,2]$ exhibit non-vanishing xSFF values, while all other cross-block correlations vanish.