Half猜想的格点规范理论与中心支Wilson费米子
Lattice gauge theory for Haldane conjecture and central-branch Wilson fermion
摘要：
我们发展了$(1+1)$ d格点$U(1)$规范理论，定义了$2$味的无质量Schwinger模型，并讨论了它与Haldinger猜想的联系.我们建议使用中心分支威尔逊费米子，它是通过将质量$m$和威尔逊参数$r$关联起来定义为$m+2r=0$的。这种设置在连续极限下给出了两个无质量的狄拉克费米子，结果证明不需要对$m$进行微调，因为中心分支$U(1)_{\bar{V}}$处的额外$U(1)$对称性禁止了附加质量重整化。此外，我们还证明了Dirac行列式是半正定的，并且该公式不存在符号问题，从而使路径积分的蒙特卡罗模拟成为可能.通过对低能对称性的识别，我们证明了该晶格模型具有介于$U(1)_{\bar{V}}$、晶格平移和晶格旋转之间的混合t Hooft反常。我们讨论了它与半整数反铁磁自旋链反常的关系，从而使我们的格点规范理论适合于Halfton猜想的数值模拟。进而对$2$ d Wilson费米子的宇称破缺相（青木相）给出了新的严格的理解。
英文摘要：
We develop the $(1+1)$d lattice $U(1)$ gauge theory in order to define $2$-flavor massless Schwinger model, and discuss its connection with Haldane conjecture. We propose to use the central-branch Wilson fermion, which is defined by relating the mass, $m$, and the Wilson parameter, $r$, as $m+2r=0$. This setup gives two massless Dirac fermions in the continuum limit, and it turns out that no fine-tuning of $m$ is required because the extra $U(1)$ symmetry at the central branch, $U(1)_{\bar{V}}$, prohibits the additive mass renormalization. Moreover, we show that Dirac determinant is positive semi-definite and this formulation is free from the sign problem, so the Monte Carlo simulation of the path integral is possible. By identifying the symmetry at low energy, we show that this lattice model has the mixed 't Hooft anomaly between $U(1)_{\bar{V}}$, lattice translation, and lattice rotation. We discuss its relation to the anomaly of half-integer anti-ferromagnetic spin chains, so our lattice gauge theory is suitable for numerical simulation of Haldane conjecture. Furthermore, it gives new and strict understanding on parity-broken phase (Aoki phase) of $2$d Wilson fermion.
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