Euclidean operator growth and quantum chaos
Euclidean operator growth and quantum chaos
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DOI：
10.1103/physrevresearch.2.043234
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发表时间：
2019-11
期刊：
影响因子：
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通讯作者：
Alexander Avdoshkin;A. Dymarsky
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文献类型：
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作者：
Alexander Avdoshkin;A. Dymarsky
考虑具有局部相互作用的格系在欧几里得时间演化下的局部算子的增长。我们推导了算子范数增长的严格界，然后建立了空间增长的Lieb-Robinson界的模拟。与闵可夫斯基情况相反，当算子的弹道扩散是全称的，在欧几里得情况下，空间增长是系统相关的，并表明系统是可积的还是混沌的。在可积情况下，欧几里得空间增长至多是多项式。在混沌情况下，它是最快的：一维指数，而在高维和贝特格上，局部算子可以在有限的欧几里德时间内达到空间无穷大。我们利用欧几里得增长的边界来建立对单个矩阵元素和算子功率谱的约束。我们证明了一维系统是特殊的，它的功率谱在大频率下总是被超指数抑制。最后，我们将欧几里得增长的界与Lanczos系数增长的界联系起来。为此，我们提出了加权Dyck路径的路径积分形式，并利用鞍点近似对其进行了计算。利用Lanczos系数的增长和控制OTOCs增长的Lyapunov指数之间的推测联系，我们提出了在所有温度下有效的改进的混沌界。
We consider growth of local operators under Euclidean time evolution in lattice systems with local interactions. We derive rigorous bounds on the operator norm growth and then proceed to establish an analog of the Lieb-Robinson bound for the spatial growth. In contrast to the Minkowski case when ballistic spreading of operators is universal, in the Euclidean case spatial growth is system-dependent and indicates if the system is integrable or chaotic. In the integrable case, the Euclidean spatial growth is at most polynomial. In the chaotic case, it is the fastest possible: exponential in 1D, while in higher dimensions and on Bethe lattices local operators can reach spatial infinity in finite Euclidean time. We use bounds on the Euclidean growth to establish constraints on individual matrix elements and operator power spectrum. We show that one-dimensional systems are special with the power spectrum always being superexponentially suppressed at large frequencies. Finally, we relate the bound on the Euclidean growth to the bound on the growth of Lanczos coefficients. To that end, we develop a path integral formalism for the weighted Dyck paths and evaluate it using saddle point approximation. Using a conjectural connection between the growth of the Lanczos coefficients and the Lyapunov exponent controlling the growth of OTOCs, we propose an improved bound on chaos valid at all temperatures.