A system of two-species, one-dimensional fermions, with an attractive two-body interaction of the derivative-delta type, features a scale anomaly. In contrast to the well-known two-dimensional case with contact interactions, and its one-dimensional cousin with three-body interactions (studied recently by some of us and others), the present case displays dimensional transmutation featuring a power-law rather than a logarithmic behavior. We use both the Schrödinger equation and quantum field theory to study bound and scattering states, showing consistency between both approaches. We show that the expressions for the reflection $\left(R\right)$ and the transmission $\left(T\right)$ coefficients of the renormalized, anomalous derivative-delta potential are identical to those of the regular delta potential. The second-order virial coefficient is calculated analytically using the Beth–Uhlenbeck formula, and we make comments about the proper ${ϵ}_B\to 0$ (where ${ϵ}_B$ is the bound-state energy) limit. We show the impact of the quantum anomaly (which appears as the binding energy of the two-body problem, or equivalently as Tan’s contact) on the equation of state and on other universal relations. Our emphasis throughout is on the conceptual and structural aspects of this problem.

具有导数-德尔塔型有吸引力的两体相互作用的两物种一维费米子系统具有尺度异常。与众所周知的具有接触相互作用的二维情况及其具有三体相互作用的一维表亲（我们中的一些人和其他人最近研究）不同，本案例显示的是具有幂律而不是幂律的二维trans变。对数行为。我们同时使用Schrödinger方程和量子场论来研究束缚态和散射态，显示了两种方法之间的一致性。我们展示了反射的表达式$（\mathrm{[R}）$ 和传输 $（Ť）$重新归一化的异常导数-δ势的系数与规则δ势的系数相同。使用Beth–Uhlenbeck公式解析计算二阶维里系数，并对适当的${ϵ}_{乙}\to 0$ （在哪里 ${ϵ}_{乙}$是束缚态能量极限。我们展示了量子异常（以两体问题的结合能或等效地以Tan接触的形式出现）对状态方程和其他普遍关系的影响。我们在整个过程中都重点关注此问题的概念和结构方面。