Lorenz Bartosch

Putting competing orders in their place near the Mott transition

We describe the localization transition of superfluids on two-dimensional lattices into commensurate Mott insulators with average particle density p/q (p, q relatively prime integers) per lattice site. For bosons on the square lattice, we argue that the superfluid has at least q degenerate species of vortices which transform under a projective representation of the square lattice space group (a PSG). The formation of a single vortex condensate produces the Mott insulator, which is required by the PSG to have density wave order at wavelengths of q/n lattice sites (n integer) along the principle axes; such a second-order transition is forbidden in the Landau-Ginzburg-Wilson framework. We also discuss the superfluid-insulator transition in the direct boson representation, and find that an interpretation of the quantum criticality in terms of deconfined fractionalized bosons is only permitted at special values of q for which a permutative representation of the PSG exists. We argue (and demonstrate in detail in a companion paper: L. Balents et al., cond-mat/0409470) that our results apply essentially unchanged to electronic systems with short-range pairing, with the PSG determined by the particle density of Cooper pairs. We also describe the effect of static impurities in the superfluid: the impurities locally break the degeneracy between the q vortex species, and this induces density wave order near each vortex. We suggest that such a theory offers an appealing rationale for the local density of states modulations observed by Hoffman et al. (cond-mat/0201348) in STM studies of the vortex lattice of BSCCO, and allows a unified description of the nucleation of density wave order in zero and finite magnetic fields. We note signatures of our theory that may be tested by future STM experiments.

GENERATION OF COLORED NOISE

In this work, we describe a simple Markovian algorithm to generate a typical sample path of colored noise described by an Ornstein–Uhlenbeck process. The algorithm works equally well to simulate a real or complex disorder potential with exponentially decaying covariance and higher correlation functions given by Wicks theorem. As an input, we only need independent Gaussian random numbers which can easily be generated by the well-known Box–Muller algorithm. Finally, we discuss an alternative method which can also be used to generate non-Gaussian colored noise.

Scaling Theory of the Mott Transition and Breakdown of the Grüneisen Scaling Near a Finite-Temperature Critical End Point

We discuss a scaling theory of the lattice response in the vicinity of a finite-temperature critical end point. The thermal expansivity is shown to be more singular than the specific heat such that the Grüneisen ratio diverges as the critical point is approached, except for its immediate vicinity. More generally, we express the thermal expansivity in terms of a scaling function which we explicitly evaluate for the two-dimensional Ising universality class. Recent thermal expansivity measurements on the layered organic conductor κ-(BEDT-TTF)2X close to the Mott transition are well described by our theory.

Renormalization of the BCS-BEC crossover by order-parameter fluctuations

We use the functional renormalization group approach with partial bosonization in the particle-particle channel to study the effect of order parameter fluctuations on the BCS-Bose-Einstein condensate (BEC) crossover of superfluid fermions in three dimensions. Our approach is based on a new truncation of the vertex expansion where the renormalization group flow of bosonic two-point functions is closed by means of Dyson-Schwinger equations and the superfluid order parameter is related to the single-particle gap via a Ward identity. We explicitly calculate the chemical potential, the single-particle gap, and the superfluid order parameter at the unitary point and compare our results with experiments and previous calculations.

Collective fields in the functional renormalization group for fermions, Ward identities, and the exact solution of the Tomonaga-Luttinger model

We develop a new formulation of the functional renormalization group (RG) for interacting fermions. Our approach unifies the purely fermionic formulation based on the Grassmannian functional integral, which has been used in recent years by many authors, with the traditional Wilsonian RG approach to quantum systems pioneered by Hertz [Phys. Rev. B 14, 1165 (1976)], which attempts to describe the infrared behavior of the system in terms of an effective bosonic theory associated with the soft modes of the underlying fermionic problem. In our approach, we decouple the interaction by means of a suitable Hubbard-Stratonovich transformation (following the Hertz-approach), but do not eliminate the fermions; instead, we derive an exact hierarchy of RG flow equations for the irreducible vertices of the resulting coupled field theory involving both fermionic and bosonic fields. The freedom of choosing a momentum transfer cutoff for the bosonic soft modes in addition to the usual band cutoff for the fermions opens the possibility of new RG schemes. In particular, we show how the exact solution of the Tomonaga-Luttinger model emerges from the functional RG if one works with a momentum transfer cutoff. Then the Ward identities associated with the local particle conservation at each Fermi point are valid at every stage of the RG flow and provide a solution of an infinite hierarchy of flow equations for the irreducible vertices. The RG flow equation for the irreducible single-particle self-energy can then be closed and can be reduced to a linear integro-differential equation, the solution of which yields the result familiar from bosonization. We suggest new truncation schemes of the exact hierarchy of flow equations, which might be useful even outside the weak coupling regime.

Breakdown of Hooke’s law of elasticity at the Mott critical endpoint in an organic conductor

The coupling of the critical electronic system to a compressible lattice is found to drastically alter the Mott criticality. The Mott metal-insulator transition, a paradigm of strong electron-electron correlations, has been considered as a source of intriguing phenomena. Despite its importance for a wide range of materials, fundamental aspects of the transition, such as its universal properties, are still under debate. We report detailed measurements of relative length changes ΔL/L as a function of continuously controlled helium-gas pressure P for the organic conductor κ-(BEDT-TTF)2Cu[N(CN)2]Cl across the pressure-induced Mott transition. We observe strongly nonlinear variations of ΔL/L with pressure around the Mott critical endpoint, highlighting a breakdown of Hooke’s law of elasticity. We assign these nonlinear strain-stress relations to an intimate, nonperturbative coupling of the critical electronic system to the lattice degrees of freedom. Our results are fully consistent with mean-field criticality, predicted for electrons in a compressible lattice with finite shear moduli. We argue that the Mott transition for all systems that are amenable to pressure tuning shows the universal properties of an isostructural solid-solid transition.

Mott metal-insulator transition on compressible lattices.

The critical properties of the finite temperature Mott end point are drastically altered by a coupling to crystal elasticity, i.e., whenever it is amenable to pressure tuning. Similar as for critical piezoelectric ferroelectrics, the Ising criticality of the electronic system is preempted by an isostructural instability, and long-range shear forces suppress microscopic fluctuations. As a result, the end point is governed by Landau criticality. Its hallmark is, thus, a breakdown of Hookes law of elasticity with a nonlinear strain-stress relation characterized by a mean-field exponent. Based on a quantitative estimate, we predict critical elasticity to dominate the temperature range ΔT*/T(c)≃8%, close to the Mott end point of κ-(BEDT-TTF)(2)X.

Putting competing orders in their place near the Mott transition. II. The doped quantum dimer model

We study the phases of doped spin S = 1/2 quantum antiferromagnets on the square lattice, as they evolve from paramagnetic Mott insulators with valence bond solid (VBS) order at doping δ = 0, to superconductors at moderate δ. The interplay between density wave/VBS order and superconductivity is efficiently described by the quantum dimer model, which acts as an effective theory for the total spin S = 0 sector. We extend the dimer model to include fermionic S = 1/2 excitations, and show that its mean-field, static gauge field saddle points have projective symmetries (PSGs) similar to those of ‘slave’ particle U(1) and SU(2) gauge theories. We account for the nonperturbative effects of gauge fluctuations by a duality mapping of the S = 0 dimer model. The dual theory of vortices has a PSG identical to that found in a previous paper (L. Balents et al., cond-mat/0408329) by a duality analysis of bosons on the square lattice. The previous theory therefore also describes fluctuations across superconducting, supersolid and Mott insulating phases of the present electronic model. Finally, with the aim of describing neutron scattering experiments, we present a phenomenological model for collective S = 1 excitations and their coupling to superflow and density wave fluctuations.

Exact numerical calculation of the density of states of the fluctuating gap model

We develop a powerful numerical algorithm for calculating the density of states rho(omega) of the fluctuating gap model, which describes the low-energy physics of disordered Peierls and spin-Peierls chains. We obtain rho(omega) with unprecedented accuracy from the solution of a simple initial value problem for a single Riccati equation. Generating Gaussian disorder with large correlation length xi by means of a simple Markov process, we present a quantitative study of the behavior of rho (omega) in the pseudogap regime. In particular, we show that in the commensurate case and in the absence of forward scattering the pseudogap is overshadowed by a Dyson singularity below a certain energy scale omega^{ast}, which we explicitly calculate as a function of xi.

Dynamic scaling in the vicinity of the Luttinger liquid fixed point

We calculate the single-particle spectral function A(k, ω) of a one-dimensional Luttinger liquid by means of a functional renormalization group (RG) approach. Given an infrared energy cut-off Λ = Λ0e−l, our approach yields the spectral function in the scaling form, AΛ(kF + p, ω) = τZl Al(pξ, ωτ), where kF is the Fermi momentum, Zl is the wavefunction renormalization factor, τ = 1/Λ is the timescale and ξ = vF /Λ is the length scale associated with Λ. At the Luttinger liquid fixed point (l → ∞) our RG result for A(k, ω) exhibits the correct anomalous scaling properties, and for k = ±kF agrees exactly with the well known bosonization result at weak coupling. Our calculation demonstrates that the field rescaling is essential for obtaining the crossover from Fermi liquid behaviour to Luttinger liquid behaviour from a truncation of the hierarchy of exact RG flow equations as the infrared cut-off Λ is reduced.