Phase Coherence of Pairs of Cooper Pairs as Quasi-Long-Range Order of Half-Vortex Pairs in a Two-Dimensional Bilayer System

It is known that the loss of phase coherence of Cooper pairs in two-dimensional superconductivity corresponds to the unbinding of vortex-antivortex pairs with the quasi-long-range order in the order-parameter phase field, described by the Berezinskii-Kosterlizt-Thouless (BKT) transition of a 2D $XY$ model. Here we show that the second-order Josephson coupling can induce an exotic superconducting phase in a bilayer system. By using tensor-network methods, the partition function of the 2D classical model is expressed as a product of 1D quantum transfer operator, whose eigenequation can be solved by an algorithm of matrix product states rigorously. From the singularity shown by the entanglement entropy of the 1D quantum analog, various phase transitions can be accurately determined. Below the BKT phase transition, an interlayer Ising long-range order is established at $T_{\mathrm{Ising}}$, and the phase coherence of both intralayers and interlayers is locked together. For two identical layers, the Ising transition coincides with the BKT transition at a multicritical point. For two inequivalent layers, however, there emerges an intermediate quasi-long-range order phase ($T_{\mathrm{Ising}}<T<T_{\mathrm{BKT}}$), where the vortex-antivortex bindings occur in the layer with the larger intralayer coupling, but only half-vortex pairs with topological strings exist in the other layer, corresponding to the phase coherence of pairs of Cooper pairs. So our study provides a promising way to realize the charge-$4e$ superconductivity in a bilayer system.

Figure 1

(a) The finite-temperature phase diagram of the bilayer system. Here we choose $K=0.5J_1$. In the low-temperature phase, there emerges an interlayer Ising-like long-range order. The BKT and Ising transitions merge together at the point $P$. (b) The schematic picture of the quasi-LRO phase 2, while the quasi-LRO phase 1 is obtained by switching the upper and lower layers. (c) The schematic picture of the low-temperature ordered phase.

Figure 2

(a) Tensor-network representation of the partition function. (b) The construction of the local tensor $O$ in the partition function. (c) Eigenequation for the fixed-point UMPS $|\mathrm{\Psi }(A)⟩$ of the 1D quantum transfer operator $T$. (d) Two-point correlation function represented by contracting a sequence of channel operators.

Figure 3

(a),(b) The entanglement entropy as a function of temperature for $J_2/J_1=1.5$ and $J_1=J_2$, with $K=0.5J_1$. (c),(d) The specific heat and the local Ising order parameter along $J_2/J_1=1.5$ and $J_1=J_2$, respectively.

Figure 4

The properties of the quasi-LRO phase 2 when $J_2/J_1=1.5$, $K/J_1=0.5$, and $T/J_1=1.3$. (a) The correlation function of the $XY$ spins shows an exponential decay. The inset is an exponential fitting. (b) The correlation function of the nematic spins exhibits a power law decay. The inset is fitted by a power law.

Figure 5

The spin stiffness as a function of temperature for given values of $J_2/J_1$. The interlayer coupling is chosen as $K=0.5J_1$, and the bond dimension of the local tensor is $D=110$. The red straight line indicates the temperature of the Ising transition.