We extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington–Kirkpatrick spin-glass model without external magnetic field to the quantum case with a “transverse field” of strength \(\mathsf {b}\). More precisely, if the Gaussian disorder is weak in the sense that its standard deviation \(\mathsf {v}>0\) is smaller than the temperature \(1/\beta \), then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any \(\mathsf {b}/\mathsf {v}\ge 0\). The macroscopic annealed free energy turns out to be non-trivial and given, for any \(\beta \mathsf {v}>0\), by the global minimum of a certain functional of square-integrable functions on the unit square according to a Varadhan large-deviation principle. For \(\beta \mathsf {v}<1\) we determine this minimum up to the order \((\beta \mathsf {v})^{4}\) with the Taylor coefficients explicitly given as functions of \(\beta \mathsf {b}\) and with a remainder not exceeding \((\beta \mathsf {v})^{6}/16\). As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong \(\beta \mathsf {b}\)-dependence even to lowest order. Our main tool for dealing with the non-commutativity of the spin-operator components is a probabilistic representation of the Boltzmann–Gibbs operator by a Feynman–Kac (path-integral) formula based on an independent collection of Poisson processes in the positive half-line with common rate \(\beta \mathsf {b}\). Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman.

我们将Aizenman，Lebowitz和Ruelle在1987年关于Sherrington - Kirkpatrick自旋玻璃模型的开创性论文中将两个严格的结果扩展到没有外磁场的量子情况下，该情况具有“ \” \ mathsf {b} \）。更准确地说，如果高斯无序在其标准偏差\（\ mathsf {v}> 0 \）小于温度\（1 / \ beta \）的意义上是弱的，则（随机）自由能几乎肯定等于宏观极限中的退火自由能，并且对于任何\（\ mathsf {b} / \ mathsf {v} \ ge 0 \）没有自旋玻璃相。对于任何\（\ beta \ mathsf {v}> 0 \），宏观退火后的自由能证明是不平凡的，并且根据一个Varadhan大偏差原理。对于\（\测试\ mathsf {V} <1 \）我们确定该最小可达顺序\（（\测试\ mathsf {V}）^ {4} \）与泰勒明确给定为的函数的系数\（ \ beta \ mathsf {b} \），并且不超过\（（\ beta \ mathsf {v}）^ {6} / 16 \）。作为副产品，我们证明对最小化问题的所谓静态近似会产生错误的\（\ beta \ mathsf {b} \）-甚至最低顺序的依赖。我们的用于处理自旋操作部件的非交换性主要工具是的概率表示玻尔兹曼-吉布斯由操作者费曼-卡茨基于独立集合（路径积分）式泊松过程中的正半符合共同利率\（\ beta \ mathsf {b} \）。它的本质可以追溯到1956年的Kac，但该公式仅由Gaveau和Schulman于1989年发布。