The Superfluid–Mott Quantum Phase Transition

The Bose–Hubbard transition has two generic critical theories. At the tip of an integer-filling Mott lobe, emergent particle–hole symmetry removes the term first order in imaginary time and gives a $(d+1)$-dimensional XY theory with $z=1$. At a generic density-driven edge that term is present, giving the dilute-Bose-gas universality class with $z=2$.

Required background. Use the controlled Bose–Hubbard limits and universality and scaling functions.

Continuum actions at a Mott-lobe tip and edge

Let $\Psi$ create the incipient superfluid order. The most general local Euclidean action through two time derivatives is

$$S=\int\mathrm d\tau\,\mathrm d^dx\left[ K_1\Psi^*\partial_\tau\Psi+K_2|\partial_\tau\Psi|^2 +c^2|\nabla\Psi|^2+r|\Psi|^2+u|\Psi|^4+\cdots \right].$$

At a generic lobe edge, changing $\mu$ adds particles or holes and $K_1\ne0$. Balancing $\omega$ with $k^2$ gives $z=2$; the upper critical spatial dimension is $d_c=2$. The excess particle or hole density is the tuning response and becomes nonzero continuously on the compressible side.

At a lobe tip, the path through parameter space holds integer density while particle and hole gaps close together. Emergent particle–hole symmetry makes $K_1=0$, so $\omega^2$ balances $k^2$ and $z=1$. The action is the classical XY model in $D=d+1$ dimensions. In $d=2$, for example, this is the three-dimensional XY universality class, not the dilute two-dimensional Bose fixed point. This distinction is part of the original scaling theory Fisher et al. 1989, §§ IV–V, pp. 555–562.

Diagnosing the path

One must specify both the location on the lobe and the direction of approach. Useful observables are the particle and hole gaps, compressibility, stiffness, and finite-size winding or correlation ratios. At a tip, simultaneous gap closure and approximate particle–hole symmetry should improve with size. At an edge, density changes immediately across the transition.

Finite-size scaling uses $L_\tau\propto L^z$; choosing the wrong $z$ can manufacture a crossing. A trap sweeps through local chemical potentials and rounds the singularity. Disorder can change the universality class, and above an upper critical dimension dangerously irrelevant $u$ modifies naive hyperscaling Sachdev 2011, ch. 10. Generic RG machinery and error certification remain in their dedicated treatments elsewhere on the site.

Exercises

Power-count the quartic coupling in the two actions and identify their upper critical dimensions.

Solution

For $z=1$, the effective dimension is $D=d+1$ and $|\Psi|^4$ is marginal at $D=4$, so $d_c=3$. For $z=2$ with the first-order time derivative, $[u]=2-d$, so $d_c=2$. These statements concern Gaussian power counting; below the upper critical dimension the interacting fixed point determines exponents.

References

• Michael P. A. Fisher, Peter B. Weichman, Geoffrey Grinstein, and Daniel S. Fisher, “Boson Localization and the Superfluid–Insulator Transition,” Physical Review B 40 (1989) 546–570, §§ IV–V, doi:10.1103/PhysRevB.40.546.
• Subir Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011), ch. 10, doi:10.1017/CBO9780511973765.