纯化吉布斯态的模零化子母哈密顿量：谱设计与可控近似
Modular-Annihilator Parent Hamiltonians for Purified Gibbs States: Spectral Design and Controlled Approximation
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中文总结 AI 辅助
本研究针对纯化吉布斯态，提出基于模零化子的无挫平方和母哈密顿量精确构造方法，在自由费米子体系实现谱优化与快速混合，针对相互作用体系引入Krylov-Lanczos近似方案并给出误差界，推进有限温物理与基态方法的关联研究。
中文摘要 AI 辅助
纯化吉布斯态为有限温物理、耗散动力学与基态方法之间搭建了桥梁。本文研究了基于模零化子的纯化吉布斯态母哈密顿量及相应林德布拉德超算符的精确有限平方和（SoS）构造方法。给定一组有限的厄米生成元，对应的模零化子可给出无挫的平方和表示，无需连续时间积分，也无需显式分解为玻尔频率扇区。纯化吉布斯态始终是共同零模，而选择与组合生成元的自由度可用于优化母哈密顿量的谱性质。对于自由费米子哈密顿量，模变换对马约拉纳算符呈线性作用，由此得到一族可解析求解的母哈密顿量，其由实对称系数矩阵\boldsymbol{S}参数化。对于标量泛函子类$S=f(h)$，我们证明在算符范数固定时，选取$S_{\mathrm{opt}}\propto 1/\sqrt{\cosh(2βh)}$对任意$β$均有混合时间上界$2\log(2N/ε)$，该上界表现出快速混合特性，且与逆温度$β$无关。对于模修饰生成元无闭合形式的相互作用体系，我们提出了一种Krylov-Lanczos近似方案，并根据模近似误差和母哈密顿量能隙给出了所得基态误差的界。数值结果验证了自由费米子体系的谱优势，并展示了相互作用体系构造的精度如何依赖于温度、相互作用强度以及Krylov子空间维数。
英文摘要
Purified Gibbs states provide a bridge between finite-temperature physics, dissipative dynamics, and ground-state methods. In this work, we study the exact finite sum-of-squares (SoS) construction of their parent Hamiltonians and the associated Lindbladian based on modular annihilators. Given a finite set of Hermitian generators, the corresponding modular annihilators yield a frustration-free SoS representation without continuous time integrals or an explicit decomposition into Bohr-frequency sectors. The purified Gibbs state remains a common zero mode while the freedom to choose and combine the generators can be used to optimize the spectral properties of the parent Hamiltonian. For free-fermion Hamiltonians, modular transformations act linearly on Majorana operators, leading to an analytically solvable family of parent Hamiltonians parameterized by a real symmetric coefficient matrix \(S\). For the scalar-functional subclass $S=f(h)$, we show that, at fixed operator norm, the choice $S_{\mathrm{opt}}\propto 1/\sqrt{\cosh(2βh)}$ has mixing time upper bound $2\log(2N/ε)$ for any $β$, which exhibits rapid mixing and is irrelevant to the inverse temperature $β$. For interacting systems, where the modularly dressed generators are not available in closed form, we introduce a Krylov--Lanczos approximation scheme and bound the resulting ground-state error in terms of the modular-approximation error and the parent-Hamiltonian gap. Numerical results illustrate the free-fermion spectral advantage and show how the accuracy of the interacting construction depends on temperature, interaction strength, and Krylov dimension.
发表机构
- Shanghai University（上海大学）
- University of Tokyo（东京大学）
- Université de Sherbrooke（舍布鲁克大学）
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