发表机构
Hanyang University（汉阳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立O(2N)分数阶拉普拉斯模型的吉布斯熵与有限温度O(N)模型自由能间的对应，通过β=2t映射将温度依赖转化为薛定谔型动力学，并引入l-形变处理量子修正，在一圈水平验证对应关系。
AI 中文摘要
我们研究了在d维空间中具有分数阶拉普拉斯算子$\sqrt{-\nabla^2}$的O(2N)矢量模型，及其由薛定谔型方程描述的哈密顿动力学。该方程是一种流守恒方程，其中可以定义概率$P(\phi^a)$的流$j(\phi^a)$，这里$\phi^a$是O(2N)矢量场。自然地，可以考虑吉布斯熵$S=-\int [D\phi^a] P(\phi^a)\log P(\phi^a)$来探索该系统。我们意识到，具有分数阶拉普拉斯算子的O(2N)矢量模型的吉布斯熵与d维中具有形变参数$\mu$的有限温度（$1/\beta$）O(N)矢量模型的自由能相匹配。随机虚时间$t$与逆温度$\beta$之间的精确映射为$\beta=2t$。因此，热O(N)矢量模型的温度依赖性可以实现为满足薛定谔型方程的时间依赖解的动力学。该自由能通过将O(N)矢量模型置于$S^1\times \mathbb R_d$中获得，其中$S^1$是周期为$\beta$的热圆。为了得到d维理论，我们求和圆上所有可能的频率（即所谓的Matsubara频率求和），从而得到d维热配分函数。我们注意到，非平凡的t依赖性出现在经典极限之外。为了考虑量子效应，我们通过保留$\hbar$修正来求解哈密顿动力学。谱形变由参数$l$介导，使得$\mu=\beta^{-1}\log l$，因此我们称之为l-形变。这与薛定谔方程的初始边界条件有关。我们还注意到，这两个理论并不等价，我们仅在一圈行列式（即零点点函数）的水平上检查它们的对应关系。
英文摘要
We explore O(2N) vector model with fractional Laplacian, $\sqrt{-\nabla^2}$ in $d$-dimension and its Hamiltonain dynamics which is described by a Schrodinger type equation. This equation is a kind of current conservation equation, where one can define a current $j(ϕ^a)$ of a probability $P(ϕ^a)$, where $ϕ^a$ is the O(2N) vector field. Naturally, Gibbs entropy $S=-\int [Dϕ^a] P(ϕ^a)\log P(ϕ^a)$ can be considered to explore the system. We realize that this Gibbs entropy of the O(2N) vector model with fractional Laplacian is matched with free energy of O(N) vector model in finite temperature, $1/β$ with a deformation, $μ$ in $d$-dimension. The precise map between the stochastic fictitious time $t$ and the inverse temperature $β$ is $β=2t$. Therefore, the temperature dependence of the thermal O(N) vector model can be realized as a dynamics of time dependent solution satisfying Schrodinger type equation. This free energy is obtained by putting O(N) vector model in $S^1\times \mathbb R_d$, where $S^1$ is thermal circle with its periodicity $β$. To get $d$-dimensional theory, we sum up all possible frequencies on the circle(so called Matsubara frequency summation) which gives $d$-dimensional thermal partition function. We note that the nontrivial $t$-dependence appears beyond classical limit. To take into account quantum effects, we solve the Hamiltonian dynamics by keeping $\hbar$ corrections. The spectral deformation is mediated by a parameter $l$ such that $μ=β^{-1}\log l$ and so we call this $l$-deformation. This is related to the initial boundary condition of the Schrodinger equation. We also note that the two theoreis are not equivalent each other and we just check their correspondence in the level of one-loop determinant, i.e. zero point function in the note.
Comments47+1 pages, 9 figures
