Unusual superconducting isotope effect in the presence of a quantum criticality

The isotope effect in superconductivity (SC) is used to make a concrete connection to a quantum critical point (QCP) that is tunable by isotopic mass substitution. We find a distinct contribution to the isotope exponent in SC and derive an explicit relation to the critical exponent of a QCP. The relation between the two exponents is general and can be used as an experimental signature for the connection between SC and a QCP. We demonstrate it in a scenario where the SC pairing is due to modes related to a structural instability. Within this model the isotope exponent is derived in terms of microscopic parameters.

Figure 1

An illustration of the dependency of $T_c$ on the isotopic mass $M$, in the vicinity of a QCP. There is a structural phase transition to an ordered, or symmetry breaking, phase (shown as a shaded area in the bottom right corner). The transition temperature to this phase depends on the isotopic mass $T\left(M\right)$ and defines the edge of the shaded area. At the QCP, where $M=M_c$ this temperature vanishes, $T\left(M_c\right)=0$, so the system cannot reach the ordered phase for lower values of $M$. The dashed lines represent a crossover to the quantum critical region in which $\hslash {\omega }_0<T$, where ${\omega }_0$ is a frequency of a structural mode related to the phase transition and it vanishes at the QCP as shown in (5). Assuming this mode is responsible for superconductivity, we also plot the superconducting transition temperature in the form $T_c\sim e^{-{\omega }_0/a}$ (solid line). Since the crossover line is given by $T=\hslash {\omega }_0$, it is a mirror image of $T_c\sim 1-{\omega }_0$. At the QCP, where ${\omega }_0\sim {|M-M_C|}^{z\nu }$, these lines can have a cusp, with diverging derivative, or be smooth, with vanishing derivative, depending whether $z\nu >1$ or $z\nu <1$, as illustrated in the inset.

Figure 2

An illustration of a system described by (9), with a potential of the form $V\left(A\right)=V_0[{\left(\frac{A}{L}\right)}^4-2{\left(\frac{A}{L}\right)}^2]$. The double-well potential is shown in scale with the lowest three energy levels. For the chosen parameter regime, $M=1,L=1,V_0=4$, and $\hslash =1$, the first two levels are much closer so higher levels can be ignored. The wave functions of these levels are also shown, where the (anti) symmetric one is the (first excited) ground state.

Figure 3

The ratio between the energy differences of the first excited state and the ground state to the second excited state and the first $\frac{E_1-E_0}{E_2-E_1}$. When the ratio is much smaller than 1 the second excited state and higher states can be neglected, so the two-level approximation is valid. One can see that this occurs roughly when $M{V}_0{L}^2>\hslash$.