Instanton confinement-deconfinement transitions: Stability of pseudogap phases and topological order

We explore the stability of certain many-body quantum states which may exist at zero or finite temperatures, may lack long-range order and even topological order, and still are thermodynamically distinct from uncorrelated disordered phases. We sharply characterize such states by the conservation of topological charge, or equivalently confinement of instantons, using a generalization of the Wilson loop and the correlation length of an emergent gauge field. Our main conclusions are (i) topological orders can exist at finite temperatures, (ii) relativistic liquids of topological defects can also exist as stable phases at finite temperatures, and (iii) there are two universality classes of instanton suppression. We also relate the instanton dynamics to the problem of the pseudogap state in underdoped cuprates. A universal experimental signature of the instanton deconfinement transition is a change of the quantum noise spectrum, which can perhaps be measured in some situations, for example, via a quantum anomaly, or indirectly detected with a specific heat jump. The method of analysis is a functional renormalization group that generalizes the Coulomb gas treatment of Kosterlitz and Thouless to arbitrary interactions and dimensions. In particular, we construct an exact nonperturbative technique for confining interactions between instantons that introduce irreparable infrared divergences in the standard perturbative approaches.

Figure 1

An illustration of instantons for vortices in $D=2+1$ space-time dimensions. (a) Worldlines of a vortex and antivortex (blue) are terminated by a pair of opposite-charge instantons. In this example, the two instantons occur at the same time $t=t_0$, but this is not generally required. The phase gradients $\mathbf{\nabla }\theta$ of $\psi \sim e^{i\theta }$, which build the action cost, are indicated with colored clouds. Far away from the instantons, the phase gradients are minimal (green), and typically produce a Coulomb-like interaction between the topological defects (screened to a finite range in the presence of gauge fields). But, the gradients must smoothly deform near the instantons in order to minimize the impact of a sudden change across $t=t_0$. In the best case scenario, there is a large action cost $V\left(r\right)$ on the temporal space-time lattice links (due to ${\partial }_t\theta$) at the time $t=t_0$ and within a region of space which grows in proportion to the distance $r$ between the instantons. The resulting $V\left(r\right)\propto r$ confines the instantons and suppresses the shown vortex creation/annihilation events. However, the underlying assumption was that discontinuous changes of $\theta$ are very costly. Instead, phase fluctuations on a lattice can be abundant enough to uncorrelate $\theta$ even across distances comparable to the lattice constant. This situation characterizes Mott insulators and removes the linear confinement of instantons [i.e., $V\left(r\right)$ becomes short-ranged]. (b) The case of confined instantons: vortex flux lines cannot terminate. In this example, a vortex-antivortex pair annihilates, but the horizontal connecting flux line $\mathbf{\nabla }\times \mathbf{A}\ne 0$ corresponds to the electric field $\mathbf{E}={\partial }_t\mathbf{A}$ which is generated via Faraday's law. The Maxwell action cost of this electric field suppresses vortex creation/annihilation process across large distances.

Figure 2

(a) Stage 1 of the renormalization group: the temporal (vertical) extent of the world $\beta =1/T$ flows toward zero. While $\beta$ is large enough, the configurations of topological defects and instantons can comfortably fit into the world view. In particular, the field configuration (e.g. the shown phase gradient $\mathbf{\nabla }\theta$) can maintain a confining action potential $V\left(r\right)\propto r$ between the instantons. Such confined instantons are macroscopically irrelevant if their confinement length $\lambda$ scales down to zero during this stage. (b) Stage 2 of the renormalization group: $\beta$ has scaled down to the cutoff length, and the world has lost its imaginary time dimension. If the instantons have not been confined under $\lambda \to 0$ yet, they cannot be properly defined any more. Any surviving spatial gradients, which build topological defects, can take their minimum-action form. This typically produces an algebraic $U\left(r\right)\propto r^{-\alpha }$ potential energy between the defects ($\alpha >0$). One could interpret $U\left(r\right)$ as the interaction potential between instanton remnants, but this is not enough for confinement. An exception is $U\left(r\right)\propto ln\left(r\right)$ in $d=2$ spatial dimensions, which permits a Kosterlitz-Thouless transition.