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arXiv:1009.2551 · Physical Review Letters

Complex Critical Exponents for Percolation Transitions in Josephson-Junction Arrays, Antiferromagnets, and Interacting Bosons

Rafael M. Fernandes, Jörg Schmalian

Published Feb 11, 2011• 9 citations
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Abstract

We show that the critical behavior of the XY quantum-rotor model undergoing a percolation transition is dramatically affected by its topological Berry phase 2π ρ. In particular, for irrational ρ, its low-energy excitations emerge as spinless fermions with fractal spectrum. As a result, critical properties not captured by the usual Ginzburg-Landau-Wilson description of phase transitions arise, such as complex critical exponents, log-periodic oscillations and dynamically broken scale invariance.

DOI
10.1103/physrevlett.106.067004

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We show that the critical behavior of the XY quantum-rotor model undergoing a percolation transition is dramatically affected by its topological Berry phase 2π ρ. In particular, for irrational ρ, its low-energy excitations emerge as spinless fermions with fractal spectrum. As a result, critical properties not captured by the usual Ginzburg-Landau-Wilson description of phase transitions arise, such as complex critical exponents, log-periodic oscillations and dynamically broken scale invariance.
We show that the critical behavior of quantum systems undergoing a percolation transition is dramatically affected by their topological Berry phase 2πρ2\pi\rho2πρ. For irrational ρ\rhoρ, we demonstrate that the low-energy excitations of diluted Josephson-junctions arrays, quantum antiferromagnets, and interacting bosons are spinless fermions with fractal spectrum. As a result, critical properties not captured by the usual Ginzburg-Landau-Wilson description of phase transitions emerge, such as complex critical exponents, log-periodic oscillations and dynamically broken scale-invariance.
The complex conductance $G(\ensuremath{\omega})$ of site-diluted Josephson junction arrays close to the percolation threshold was measured over a wide range of frequencies \ensuremath{\omega}. Well below Tc{T}_{c}Tc​ both the superfluid $[\ensuremath{\omega}\mathrm{Im}G]$ and dissipative [ReG][\mathrm{Re}G][ReG] components are independent of \ensuremath{\omega} below a critical frequency ${\ensuremath{\omega}}_{c}$, whereas $G(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}u}$ with $u\ensuremath{\approx}\frac{1}{2}$ for $\ensuremath{\omega}>{\ensuremath{\omega}}_{c}$. This is shown to reflect the crossover from a Euclidean regime $(\ensuremath{\omega}<{\ensuremath{\omega}}_{c})$ dominated by phononlike modes of the phase system to a fractal regime $(\ensuremath{\omega}>{\ensuremath{\omega}}_{c})$, where the relevant excitations are localized fractons. Percolative critical exponents extracted from the data are consistent with theoretical predictions.
Measurements of the IV characteristics of site-diluted Josephson-junction arrays have revealed intriguing effects of percolative disorder on the phase transition and the vortex dynamics in a two-dimensional XY system. Different from other types of phase transitions, the Kosterlitz-Thouless transition was eliminated with the introduction of percolative disorder far below the percolation threshold. Even after the Kosterlitz-Thouless order had been removed, the system remained superconducting at low temperatures by establishing a different type of order. Near the percolation threshold, evidence was found that, as a consequence of the underlying fractal structure, the critical dynamics of the phase degrees of freedom persisted down to zero temperature.
We consider the effect of the coupling between two-dimensional (2D) quantum rotors near an XY ferromagnetic quantum critical point and spins of itinerant fermions. We analyze how this coupling affects the dynamics of rotors and the self-energy of fermions. A common belief is that near a q=0q=0q=0 ferromagnetic transition, fermions induce an $\mathrm{\ensuremath{\Omega}}/q$ Landau damping of rotors (i.e., the dynamical critical exponent is z=3z=3z=3) and Landau overdamped rotors give rise to non-Fermi liquid fermionic self-energy $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}$. This behavior has been confirmed in previous quantum Monte Carlo (QMC) studies. Here we show that for the XY case the behavior is different. We report the results of large-scale quantum Monte Carlo simulations, which show that at small frequencies z=2z=2z=2 and $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{1/2}$. We argue that the new behavior is associated with the fact that a fermionic spin is by itself not a conserved quantity due to spin-spin coupling to rotors, and a combination of self-energy and vertex corrections replaces 1/q1/q1/q in the Landau damping by a constant. We discuss the implication of these results to experiments.
We present the critical theory of a number of zero-temperature phase transitions of quantum antiferromagnets and interacting boson systems in two dimensions. The most important example is the transition of the S=1∕2S=1∕2S=1∕2 square lattice antiferromagnet between the N\'eel state (which breaks spin rotation invariance) and the paramagnetic valence bond solid (which preserves spin rotation invariance but breaks lattice symmetries). We show that these two states are separated by a second-order quantum phase transition. This conflicts with Landau-Ginzburg-Wilson theory, which predicts that such states with distinct broken symmetries are generically separated either by a first-order transition, or by a phase with co-existing orders. The critical theory of the second-order transition is not expressed in terms of the order parameters characterizing either state, but involves fractionalized degrees of freedom and an emergent, topological, global conservation law. A closely related theory describes the superfluid-insulator transition of bosons at half filling on a square lattice, in which the insulator has a bond density wave order. Similar considerations are shown to apply to transitions of antiferromagnets between the valence bond solid and the Z2{Z}_{2}Z2​ spin liquid: the critical theory has deconfined excitations interacting with an emergent U(1)\mathrm{U}(1)U(1) gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.
We investigate the zero-temperature superfluid to insulator transitions in a diluted two-dimensional quantum rotor model with particle-hole symmetry. We map the Hamiltonian onto a classical (2+1)(2+1)(2+1)-dimensional XYXYXY model with columnar disorder which we analyze by means of large-scale Monte Carlo simulations. For dilutions below the lattice percolation threshold, the system undergoes a generic superfluid-Mott glass transition. In contrast to other quantum phase transitions in disordered systems, its critical behavior is of conventional power-law type with universal (dilution-independent) critical exponents $z=1.52(3), \ensuremath{\nu}=1.16(5), \ensuremath{\beta}/\ensuremath{\nu}=0.48(2), \ensuremath{\gamma}/\ensuremath{\nu}=2.52(4)$, and $\ensuremath{\eta}=\ensuremath{-}0.52(4)$. These values agree with and improve upon earlier Monte Carlo results [Phys. Rev. Lett. 92, 015703 (2004)] while (partially) excluding other findings in the literature. As a further test of universality, we also consider a soft-spin version of the classical Hamiltonian. In addition, we study the percolation quantum phase transition across the lattice percolation threshold; its critical behavior is governed by the lattice percolation exponents in agreement with recent theoretical predictions. We relate our results to a general classification of phase transitions in disordered systems, and we briefly discuss experiments.