利用置换对称性求解耗散相变的平均场解
Resolving mean-field solutions of dissipative phase transitions using permutational symmetry
AI总结：
本文利用无穷维下自旋全连接带来的置换对称性将耗散横场伊辛模型的计算复杂度从指数级降至$\mathcal{O}(N^3)$，通过数值求解Lindblad方程得到全部临界指数，并据此判定$d=3$处的不连续相变并非平均场解。
摘要：
耗散量子系统中的相变已通过多种解析方法进行研究，特别是在平均场（MF）极限下。然而，解析结果往往依赖于具体的方法论。例如，Keldysh形式体系表明耗散横场伊辛（DTI）模型在上临界维度$d_c=3$处表现出不连续相变，而无涨落的平均场方法则预言在无穷维（$d_\infty$）下发生连续相变。由于$d_c$以上的平均场解应当相同，这两种解无法调和，因此有必要进行数值验证。然而，由于计算复杂度随系统尺寸$N$呈$\mathcal{O}(2^{2N})$指数增长，对大系统的数值研究可能不可行。本文注意到，由于自旋在$d_\infty$下可被视为全连接，自旋指标可以具有置换不变性，量子态数目可被大幅压缩，计算复杂度降为$\mathcal{O}(N^3)$。Lindblad方程被转化为基于压缩态的动力学方程。将Runge--Kutta算法应用于该动力学方程，我们得到了所有临界指数，包括动力学指数$z\approx 0.5$。此外，由于DTI模型具有$\mathbb{Z}_2$对称性，超标度关系具有$2β+γ=ν(d+z)$的形式，我们得到平均场极限下的关系$d_c+z=4$。因此$d_c\approx 3.5$，从而$d=3$处的不连续相变不能被视为平均场解。我们得出结论：$d_\infty$下的置换不变性可以有效地用于检验量子相变中解析平均场解的有效性。
英文摘要：
Phase transitions in dissipative quantum systems have been investigated using various analytical approaches, particularly in the mean-field (MF) limit. However, analytical results often depend on specific methodologies. For instance, Keldysh formalism shows that the dissipative transverse Ising (DTI) model exhibits a discontinuous transition at the upper critical dimension, $d_c= 3$, whereas the fluctuationless MF approach predicts a continuous transition in infinite dimensions ($d_\infty$). These two solutions cannot be reconciled because the MF solutions above $d_c$ should be identical. This necessitates a numerical verification. However, numerical studies on large systems may not be feasible because of the exponential increase in computational complexity as $\mathcal{O}(2^{2N})$ with system size $N$. Here, we note that because spins can be regarded as being fully connected at $d_\infty$, the spin indices can be permutation invariant, and the number of quantum states can be considerably contracted with the computational complexity $\mathcal{O}(N^3)$. The Lindblad equation is transformed into a dynamic equation based on the contracted states. Applying the Runge--Kutta algorithm to the dynamic equation, we obtain all the critical exponents, including the dynamic exponent $z\approx 0.5$. Moreover, since the DTI model has $\mathbb{Z}_2$ symmetry, the hyperscaling relation has the form $2β+γ=ν(d+z)$, we obtain the relation $d_c+z=4$ in the MF limit. Hence, $d_c\approx 3.5$; thus, the discontinuous transition at $d=3$ cannot be treated as an MF solution. We conclude that the permutation invariance at $d_\infty$ can be used effectively to check the validity of an analytic MF solution in quantum phase transitions.
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