Interacting many-body systems with explicitly accessible spatiotemporal correlation functions are extremely rare, especially in the absence of Bethe-ansatz or Yang-Baxter integrability. Recently, we identified a remarkable class of such systems and termed them dual-unitary quantum circuits. These are brickwork-type local quantum circuits whose dynamics are unitary in both time and space. The spatiotemporal correlation functions of these systems turn out to be nontrivial only at the edge of the causal light cone and can be computed in terms of one-dimensional transfer matrices. Dual unitarity, however, requires fine-tuning, and the degree of generality of the observed dynamical features remains unclear. Here, we address this question by studying perturbed dual-unitary quantum circuits. Considering arbitrary perturbations of the local gates, we prove that for a particular class of unperturbed elementary dual-unitary gates the correlation functions are still expressed in terms of one-dimensional transfer matrices. These matrices, however, are now contracted over generic paths connecting the origin to a fixed end point inside the causal light cone. The correlation function is given as a sum over all such paths. Our statement is rigorous in the “dilute limit,” where only a small fraction of the gates is perturbed, and in the presence of random longitudinal fields, but we provide theoretical arguments and stringent numerical checks supporting its validity even in the clean case (no randomness) and when all gates are perturbed. As a by-product of our analysis, in the case of random longitudinal fields—which turns out to be equivalent to certain classical Markov chains—we find four types of non-dual-unitary (and nonintegrable) interacting many-body systems where the correlation functions are exactly solvable and given—without approximations—by the path-sum formula.

中文翻译：

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扰动双-电路中的相关性：有效的路径积分公式

具有明确可访问的时空相关函数的交互多体系统极为罕见，尤其是在没有Bethe-ansatz或Yang-Baxter可积性的情况下。最近，我们发现了这类系统的杰出代表，并将其称为双unit量子电路。这些是砖砌型局部量子电路，其动力学在时间和空间上都是统一的。这些系统的时空相关函数仅在因果光锥的边缘处是不平凡的，并且可以根据一维传递矩阵进行计算。但是，双重统一性需要进行微调，并且所观察到的动力学特征的普遍程度仍然不清楚。在这里，我们通过研究扰动的双-量子电路来解决这个问题。考虑到本地门的任意扰动，我们证明，对于一类不受干扰的基本双gate门，相关函数仍以一维传递矩阵表示。但是，这些矩阵现在在将原点连接到因果光锥内的固定端点的通用路径上收缩了。相关函数作为所有此类路径的总和给出。我们的陈述在“稀释极限”中非常严格，在这种情况下，只有一小部分门受到干扰，并且存在随机纵向场，但是即使在干净的情况下，我们也提供了理论依据和严格的数值检验来支持其有效性（无随机性）以及所有门都受到干扰时。作为我们分析的副产品，