Quantum criticality in dissipative quantum two-dimensional $XY$ and Ashkin-Teller models: Application to the cuprates

In a recent paper [V. Aji and C. M. Varma, Phys. Rev. Lett. 99, 067003 (2007)] we have shown that the dissipation driven quantum phase transition of the two-dimensional $XY$ model represents a universality class where the correlations at criticality are local in space and power law in time. Here we provide a detailed analysis of the model. The local criticality is brought about by the decoupling of infrared singularities in space and time. The former leads to a Kosterlitz-Thouless transition whereby the excitations of the transverse component of the velocity field (vortices) unbind in space. The latter, on the other hand, leads to a transition among excitations (warps) in the longitudinal component of the velocity field, which unbind in time. The quantum Ashkin-Teller model, with which the observed loop order in the cuprates is described, maps in the critical regime to the quantum $XY$ model. We also discuss other models which are expected to have similar properties.

Figure 1

(Color online) A warp is an event in time where the divergence of $\mathbf{m}$ field changes.

Figure 2

Dissipative driven proliferation of warps occurs for $1/\alpha >1$. The power-law scaling at criticality is cutoff at an energy scale $T_x$ as discussed in the text.

Figure 3

Path between $i\mu$ and $j\nu$ used to compute the correlation function.

Figure 4

(Color online) The zero-temperature phase diagram in the $1/\alpha \text{-JC}$ plane for the dissipative 2D $XY$ model. For $\alpha =0$, the system undergoes a superconductor to insulator phase transition with a 3D $XY$ critical point. For infinite $\alpha$, the dynamics is quenched and the we recover the 2D $XY$ behavior. There are two phases for finite dissipation. The disordered phase is characterized with a diverging fugacity of warps and vortices. The algebraically ordered phase is one in which both the vortices and warps are bound, and the correlations are power laws separately in space and time with continuously varying exponents.

Figure 5

The four domains of the circulating current phase are shown. They can be parameterized by two Ising variables which represent the currents in the $x$ and $y$ directions.

Figure 6

Equations for the full Green’s function. The thick lines represent the full propagator $⟨p\left(\mathbf{k},\omega \right)p\left(-\mathbf{k},-\omega \right)⟩$ and the thick dashed lines represent $⟨{\rho }_w\left(\mathbf{k},\omega \right){\rho }_w\left(-\mathbf{k},-\omega \right)⟩$, while the thin full and dashed lines represent the same propagators computed in the absence of the linear coupling with the warps.