Nonzero-temperature transport near quantum critical points

Abstract

We describe the nature of charge transport at non-zero temperatures (T ) above the two-dimensional (d) superfluid-insulator quantum critical point. We argue that the transport is characterized by inelastic collisions among thermally excited carriers at a rate of order k B T /h. This implies that the transport at frequencies ω ≪ k B T /h is in the hydrodynamic, collision-dominated (or 'incoherent') regime, while ω ≫ k B T /h is the collisionless (or 'phasecoherent') regime. The conductivity is argued to be e 2 /h times a non-trivial universal scaling function ofhω/k B T , and not independent ofhω/k B T , as has been previously claimed, or implicitly assumed. The experimentally measured d.c. conductivity is the hydrodynamichω/k B T → 0 limit of this function, and is a universal number times e 2 /h, even though the transport is incoherent. Previous work determined the conductivity by incorrectly assuming it was also equal to the collisionlesshω/k B T → ∞ limit of the scaling function, which actually describes phase-coherent transport with a conductivity given by a different universal number times e 2 /h. We provide the first computation of the universal d.c. conductivity in a disorder-free boson model, along with explicit crossover functions, using a quantum Boltzmann equation and an expansion in ǫ = 3 − d. The case of spin transport near quantum critical points in antiferromagnets is also discussed. Similar ideas should apply to the transitions in quantum Hall systems and to metal-insulator transitions. We suggest experimental tests of our picture and speculate on a new route to self-duality at two-dimensional quantum critical points.

Quantum phases and phase transitions of Mott insulators

“…As we noted earlier, the paramagnetic phase has well-defined 'triplon' or 'spin exciton' excitations t α , and these have an infinite lifetime at T = 0. At T > 0, thermally excited t α quasiparticles will collide with each other via their scattering amplitude, u, and this will lead to a finite lifetime [37,38]. Now approach λ = λ c .…”

Dynamics and Transport Near Quantum-Critical Points

“…Indeed, it was conjectured in Ref [63] that 2πΣ(0) = 1 exactly for N = 2 in d = 2. Definitively establishing this self-duality however requires techniques other than expansion in ǫ = 3− d, or 1/N , as it is only possible precisely at d = 2, N = 2.…”

“…We will not display the explicit form of this equation, or discuss its solution here: the reader should consult Ref [63]. That analysis leads to an explicit solution for Σ which has the qualitative form of Fig 14. We quote quantitative results for the two limiting values [63,78]: …”

“…We will outline the recent computation of Σ in an expansion in ǫ = 3 − d [63]. For small ǫ the quartic interaction, u 0 , in S reaches a universal (to leading order in ǫ) fixed point value [1] u 0 = 24π 2 5 ǫ…”

“…There we determined the spin diffusion constant in the quantum-disordered regime (T ≪ ∆) of the N = 3 model in d = 1. Here, following Ref [63], we shall describe transport for N ≥ 2 in d = 2 in the "High T" region (Fig 1 and 3), where the dynamics is quantum-relaxational (Fig 8) . The methods discussed here can also be applied to the quantum-disordered region of the N ≥ 2, d = 2 models, but we shall not present that extension here.…”