[challenge]: Certified bulk spectral-gap bounds for frustrated spin-1/2 models

Released by

许湘灵（Xiangling Xu）, Inria Saclay; 王杰（Jie Wang）, AMSS-CAS

Contact email

xu.xiangling@inria.fr; wangjie212@amss.ac.cn

Method

Semidefinite programming / Noncommutative polynomial optimization

Background

Spectral gaps determine low-energy excitations and are closely related to stability and quantum phases, but rigorous thermodynamic-limit gap bounds are difficult to obtain. Exact diagonalization, DMRG, and tensor-network methods can give estimates, but they usually rely on finite geometries, extrapolations, or variational ansätze, and thus not certified values for the states in the thermodynamic-limit.

The recent work The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds introduced a hierarchy of semidefinite programs (SDPs) that produces certified upper bounds on the thermodynamic-limit bulk spectral gap $\Delta_{\mathrm{bulk}}$. The accompanying spin-$1/2$ code is available at wangjie212/SpectralGap. The hierarchy is based the state polynomial optimization (see Ref. 2), a novel technique that allows SDPs to deal with nonlinear constraints on the states.

At any relaxation level $(L,d)$, if the SDP is infeasible at $\gamma \geq 0$, then no ground state has bulk spectral gap at least $\gamma$. (The case $\gamma=0$ coincides with the set of ground states.) Consequently, at relaxation level $(L,d)$, maximizing the gap parameter $\gamma$ for which the SDP relaxation is feasible gives a monotonic convergent sequence

$$
\Gamma_{L,d} \searrow \Delta_{\mathrm{bulk}}.
$$

At a fixed assumed gap $\gamma$, the same SDP can give certified bounds

$$
O_{\min}(L,d;\gamma) \leq \omega(O) \leq O_{\max}(L,d;\gamma)
$$

for a chosen observable $O$ that are also monotonically convergent. That is, any ground state $\omega$ with bulk spectral gap of at least $\gamma$ must has its expectation value on $O$ bounded between $O_{\min}(L,d;\gamma)$ and $O_{\max}(L,d;\gamma)$. One can further incorporate symmetries for certified statements on symmetric ground states, which reduces the size of moment matrices. See also Ref. 3 for more techniques to improve the numerical values in practice.

The goal is to extend the existing code to new two-dimensional lattice geometries and study the following models in order:

1. square-lattice $J_1-J_2$ Heisenberg model;
2. Shastry-Sutherland model;
3. triangular-lattice $J_1-J_2$ Heisenberg model.

Consider the spin vector $S_i^a=\sigma_i^a/2$, where $\sigma^a_i$ is the Pauli operator at site $I$, and that

$$
\boldsymbol{S}_i \cdot \boldsymbol{S}_j = \frac{1}{4}(X_iX_j+Y_iY_j+Z_iZ_j).
$$

Target 1: square-lattice $J_1-J_2$ model

The Hamiltonian is

$$
H_{\square}(g) = \sum_{\langle i,j\rangle}\boldsymbol S_i\cdot\boldsymbol S_j + g\sum_{\langle \langle i,j\rangle \rangle} \boldsymbol S_i\cdot\boldsymbol S_j, \qquad g=J_2/J_1,
$$

the first sum ${\langle i,j\rangle}$ is the nearest-neighbor interaction, and the second sum ${\langle \langle i,j\rangle \rangle}$ is the next-nearest-neightbor interaction.

Study

$$
g = 0, 0.50, 0.535.
$$

The point $g=0$ is the unfrustrated square-lattice antiferromagnet. Around $g=0.50$, weak Néel order and a nonmagnetic phase have been debated. The value $g=0.535$ is a recent estimate for the Néel-to-valence-bond-solid transition.

For a square $2R \times 2R$-sublattice $W_R$, define the squared Néel order parameter

$$
M_{\mathrm N}^2(W_R) = \frac{1}{|W_R|^2} \sum_{i,j\in W_R} (-1)^{x_i+y_i+x_j+y_j} \omega(\boldsymbol S_i\cdot\boldsymbol S_j),
$$

where $x_i, y_i$ is the coordinate of site $i$.

Calculate:

• certified gap upper bounds $\Gamma_{L,d}(g)$ for several accessible $L,d$;
• certified maximum/minimum for $M_{\mathrm N}^2(W_R)$ at $g=0.50$ and $g=0.535$ for

$$
\gamma = 0, 0.05, 0.10.
$$

These are just suggested values since no values of gaps are known. It shows that, for all ground states with bulk gap of at least $\gamma$, the value of $M_{\mathrm N}^2(W_R)$ must be bounded by optimal values from the SDP relaxation. Bounds on $M_{\mathrm N}^2(W_R)$ should be compared for increasing $R$: persistent positive lower bounds is an evidence for Néel-ordered phase, while upper bounds decreasing with $R$ constrain Néel order.

Target 2: Shastry-Sutherland model

The Shastry-Sutherland lattice is a square lattice with alternating orthogonal diagonal bonds $D$, arranged so that every site belongs to exactly one diagonal bond (such as designated pairs of spins are called dimer). With the diagonal coupling set to $J=1$, define

$$ H_{\mathrm{SS}}(g) = \sum_{{i,j}\in D}\boldsymbol S_i\cdot\boldsymbol S_j + g\sum_{\langle i,j\rangle_{\square}} \boldsymbol S_i\cdot\boldsymbol S_j, \qquad g=J'/J.
$$

Study

$$
g = 0, 0.80.
$$

At $g=0$, the ground state is exactly a product of singlets on the diagonal dimers and

$$
\Delta_{\mathrm{bulk}}=1.
$$

For a diagonal dimer ${i,j}\in D$, the singlet projector is

$$
P^{(s)}_{ij} = \frac14I-\boldsymbol S_i\cdot\boldsymbol S_j,
$$

and the exact benchmark at $g=0$ is $\omega(P^{(s)}_{ij})=1$.

Near $g=0.80$, numerical studies disagree between a narrow gapless spin-liquid region and a direct transition from plaquette valence-bond order to Néel antiferromagnetism. Use two observables:

1. the squared Néel order parameter $M_{\mathrm N}^2(W_R)$ defined above;
2. a squared plaquette-order parameter

$$
P_R^2 = \omega \left[ \left( \frac{1}{|\mathcal P_R|} \sum_{p\in\mathcal P_R}\tau_p\widehat B_p \right)^2 \right], \qquad \widehat B_p = \frac14\sum_{{i,j}\in\partial p} \boldsymbol S_i\cdot\boldsymbol S_j,
$$

where $\tau_p=\pm1$ distinguishes the two symmetry-related classes of empty plaquettes. The sum runs over empty plaquettes, see also, e.g., Fig. 1 of Ref. 7 where "A" and "B" denotes the two classes.

Calculate:

• $\Gamma_{L,d}(0)$ and verify the exact dimer benchmark;
• $\Gamma_{L,d}(0.80)$ for several accessible $L,d$;
• certified intervals for $P_R^2$ and $M_{\mathrm N}^2(W_R)$ at $g=0.80$ for

$$
\gamma = 0, 0.05, 0.10,
$$

or other appropriate values suggested by $\Gamma_{L,d}(0.80)$. Finite-cylinder studies near $g=0.80$ find low-energy gaps of order $0.1J$ on accessible sizes and extrapolate them toward zero. Thus $0.05J$ and $0.10J$ are relevant conditional scales. Infeasibility excludes a symmetric gap of the tested size. If the SDP remains feasible, the observable intervals test whether every such gapped state must have plaquette or Néel correlations.

Target 3: triangular-lattice $J_1-J_2$ model

Let $E_1$ contain nearest-neighbor bonds of the triangular lattice and $E_2$ its next-nearest-neighbor bonds. With $J_1=1$, define

$$
H_{\triangle}(g) = \sum_{\langle i,j\rangle}\boldsymbol S_i\cdot\boldsymbol S_j + g\sum_{\langle \langle i,j\rangle \rangle}\boldsymbol S_i\cdot\boldsymbol S_j, \qquad g=J_2/J_1.
$$

Study

$$
g = 0.10, 0.12 .
$$

These points lie in the intermediate region where the literature contains competing interpretations as a gapped $\mathbb Z_2$ spin liquid, a gapless Dirac spin liquid, or a weakly ordered state.

For a finite triangular sublattice $W_R$, define the $120^\circ$ structure factor

$$
S_{120,R} = \frac{1}{|W_R|^2} \sum_{i,j\in W_R} \cos\left(K\cdot(r_i-r_j)\right) \omega(\boldsymbol S_i\cdot\boldsymbol S_j), \qquad K=(4\pi/3,0).
$$

Calculate:

• $\Gamma_{L,d}(g)$ for several accessible $L,d$;
• certified intervals for $S_{120,R}$ for

$$
\gamma = 0, 0.10, 0.20.
$$

Early DMRG work at $g=0.10$ reported finite-cylinder singlet and triplet gaps around $0.3$, while later work supports gapless interpretations. A certified full bulk-gap upper bound below $0.20J_1$ would exclude a robust many-body gap at that scale; a bound below $0.10$ would be stronger. Bounds on $S_{120,R}$ test whether an assumed gapped state is compatible with weak or substantial $120^\circ$ magnetic correlations.

Results to report

For every calculation, report the model parameters, imposed symmetries (if any), $L$ and $d$, moment-matrix size, solver status, runtime, certified gap upper bound, and certified observable optima.

The discussion should state precisely:

• what gap values are excluded by the certificate;
• whether the result is unrestricted or symmetry-restricted;
• how the observable bounds compare with the phases proposed in the literature;
• how the gap and observable bounds change as $L$, $d$, and the observable bounds relation to the size $R$.

References

1. X. Xu et al., The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds.
2. I. Klep et al., State polynomials: positivity, optimization and nonlinear Bell inequalities
3. J. Wang et al., Scalable ground-state certification of quantum spin systems via structured noncommutative polynomial optimization
4. X. Qian and M. Qin, Absence of spin liquid phase in the $J_1-J_2$ Heisenberg model on the square lattice.
5. J. Yang, A. W. Sandvik, and L. Wang, Quantum criticality and spin liquid phase in the Shastry-Sutherland model.
6. P. Corboz et al., Quantum spin liquid phase in the Shastry-Sutherland model revealed by high-precision infinite projected entangled-pair states.
7. M. Mezera et al., Neural Network Quantum States Analysis of the Shastry-Sutherland Model
8. Z. Zhu and S. R. White, Spin liquid phase of the $S=1/2$ $J_1-J_2$ Heisenberg model on the triangular lattice.
9. S. Jiang et al., Competing states in the $S=1/2$ triangular-lattice $J_1$-$J_2$ Heisenberg model.

Start by reading and running the existing spin-1/2 implementation in the linked wangjie212/SpectralGap repository, then trace how lattice geometries, Hamiltonians, observables, and SDP parameters are represented. Extend those entry points for the three requested models and report the specified solver results, matrix sizes, runtimes, gap bounds, and observable intervals for each parameter set.