arXiv subjects

Walter Metzner

Publications and source records attributed to Walter Metzner.

At least 37 records · Page 2Linked to original sources

arXiv2018

Antiferromagnetic and d-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group

Using the dynamical mean-field theory (DMFT) as a `booster-rocket', the functional renormalization group (fRG) can be upgraded from a weak-coupling method to a powerful computation tool for strongly interacting fermion systems. The strong local correlations are treated non-perturbatively by the DMFT, while the fRG flow can be formulated such that it is driven exclusively by non-local correlations, which are more amenable to approximations. We show that the full frequency dependence of the two-particle vertex needs to be taken into account in this approach, and demonstrate that this is actually possible -- in spite of the singular frequency dependence of the vertex at strong coupling. We are thus able to present the first results obtained from the DMFT-boosted fRG for the two-dimensional Hubbard model in the strongly interacting regime. We find strong antiferromagnetic correlations from half-filling to 18 percent hole-doping, and, at the lowest temperature we can access, a sizable $d$-wave pairing interaction driven by magnetic correlations at the edge of the antiferromagnetic regime.

cond-mat.str-el↗

arXiv2018

Dynamically enhanced magnetic incommensurability: Effects of local dynamics on non-local spin-correlations in a strongly correlated metal

We compute the spin susceptibility of the two-dimensional Hubbard model away from half-filling, and analyze the impact of frequency dependent vertex corrections as obtained from the dynamical mean field theory (DMFT). We find that the local dynamics captured by the DMFT vertex strongly affects non-local spin correlations, and thus the momentum dependence of the spin susceptibility. While the widely used random phase approximation yields commensurate Néel-type antiferromagnetism as the dominant instability over a wide doping range, the vertex corrections favor incommensurate ordering wave vectors away from $(π,π)$. Our results indicate that the connection between the magnetic ordering wave vector and the Fermi surface geometry, familiar for weakly interacting systems, can hold in a strongly correlated metal, too.

cond-mat.str-el↗

arXiv2018

Non-separable frequency dependence of two-particle vertex in interacting fermion systems

We derive functional flow equations for the two-particle vertex and the self-energy in interacting fermion systems which capture the full frequency dependence of both quantities. The equations are applied to the hole-doped two-dimensional Hubbard model as a prototype system with entangled magnetic, charge and pairing fluctuations. Each fluctuation channel acquires substantial dependencies on all three Matsubara frequencies, such that the frequency dependence of the vertex cannot be accurately represented by a channel sum with only one frequency variable in each term. At the temperatures we are able to access, the leading instabilities are mostly antiferromagnetic, with an incommensurate wave vector. However, at large doping, a divergence in the charge channel occurs at a finite frequency transfer, if the vertex flow is computed without self-energy feedback. This enigmatic instability was already observed in a calculation by Husemann et al. [Phys. Rev. B 85, 075121 (2012)], who used an approximate separable ansatz for the frequency dependence of the vertex. We identify a simple mechanism for this instability in terms of a random phase approximation for the charge channel with a frequency dependent effective magnetic interaction as input. In spite of the strong momentum and frequency dependence of the vertex, the self-energy has a Fermi liquid form. At the moderate interaction strength where our approach is applicable, we obtain a moderate reduction of the quasi-particle weight and a sizable decay rate with a pronounced momentum dependence. Nevertheless, the self-energy feedback into the vertex flow turns out to be crucial, as it suppresses the unphysical finite frequency charge instability.

cond-mat.str-el↗

arXiv2017

Longitudinal fluctuations in the Berezinskii-Kosterlitz-Thouless phase

We analyze the interplay of longitudinal and transverse fluctuations in a $U(1)$ symmetric two-dimensional $ϕ^4$-theory. To this end, we derive coupled renormalization group equations for both types of fluctuations obtained from a linear (cartesian) decomposition of the order parameter field. Discarding the longitudinal fluctuations, the expected Berezinskii-Kosterlitz-Thouless (BKT) phase characterized by a finite stiffness and an algebraic decay of order parameter correlations is recovered. Renormalized by transverse fluctuations, the longitudinal mass scales to zero, so that longitudinal fluctuations become increasingly important for small momenta. Within our expansion of the effective action, they generate a logarithmic decrease of the stiffness, in agreement with previous functional renormalization group calculations. The logarithmic terms imply a deviation from the vanishing beta-function for the stiffness in the non-linear sigma model describing the phase fluctuations at three-loop order. To gain further insight, we also compute the flow of the parameters characterizing longitudinal and transverse fluctuations from a density-phase representation of the order parameter field, with a cutoff on phase fluctuations. The power-law flow of the longitudinal mass and other quantities is thereby confirmed, but the stiffness remains finite in this approach. We conclude that the marginal flow of the stiffness obtained in the cartesian representation is an artifact of the truncated expansion of momentum dependences.

cond-mat.stat-mech↗

arXiv2016

Fermi surface reconstruction and drop of Hall number due to spiral antiferromagnetism in high-$T_c$ cuprates

We show that a Fermi surface reconstruction due to spiral antiferromagnetic order may explain the rapid change in the Hall number as recently observed near optimal doping in cuprate superconductors [Badoux~\textit{et. al.}, Nature \textbf{531}, 210 (2016)]. The single-particle spectral function in the spiral state exhibits hole pockets which look like Fermi arcs due to a strong momentum dependence of the spectral weight. Adding charge-density wave order further reduces the Fermi surface to a single electron pocket. We propose quantum oscillation measurements to distinguish between commensurate and spiral antiferromagnetic order. Similar results apply to certain metals in which topological order replaces antiferromagnetic order.

cond-mat.str-el↗

arXiv2016

Coexistence of incommensurate magnetism and superconductivity in the two-dimensional Hubbard model

We analyze the competition of magnetism and superconductivity in the two-dimensional Hubbard model with a moderate interaction strength, including the possibility of incommensurate spiral magnetic order. Using an unbiased renormalization group approach, we compute magnetic and superconducting order parameters in the ground state. In addition to previously established regions of Neel order coexisting with d-wave superconductivity, the calculations reveal further coexistence regions where superconductivity is accompanied by incommensurate magnetic order.

cond-mat.str-el↗

arXiv2016

Fermion loops and improved power-counting in two-dimensional critical metals with singular forward scattering

We analyze general properties of the perturbation expansion for two-dimensional quantum critical metals with singular forward scattering, such as metals at an Ising nematic quantum critical point and metals coupled to a U(1) gauge field. We derive asymptotic properties of fermion loops appearing as subdiagrams of the contributing Feynman diagrams -- for large and small momenta. Substantial cancellations are found in important scaling limits, which reduce the degree of divergence of Feynman diagrams with boson legs. Implementing these cancellations we obtain improved power-counting estimates that yield the true degree of divergence. In particular, we find that perturbative contributions to the boson self-energy are generally ultraviolet convergent for a dynamical critical exponent $z<3$, and divergent beyond three-loop order for $z \geq 3$.

cond-mat.str-el↗

arXiv2015

Anomalous dynamical scaling from nematic and U(1)-gauge field fluctuations in two dimensional metals

We analyze the scaling theory of two-dimensional metallic electron systems in the presence of critical bosonic fluctuations with small wave vectors, which are either due to a U(1) gauge field, or generated by an Ising nematic quantum critical point. The one-loop dynamical exponent z=3 of these critical systems was shown previously to be robust up to three-loop order. We show that the cancellations preventing anomalous contributions to z at three-loop order have special reasons, such that anomalous dynamical scaling emerges at four-loop order.

cond-mat.str-el↗

arXiv2014

Non-Fermi liquid behavior at the onset of incommensurate 2k_F charge or spin density wave order in two dimensions

We analyze the influence of quantum critical fluctuations on single-particle excitations at the onset of incommensurate $2k_F$ charge or spin density wave order in two-dimensional metals. The case of a single pair of hot spots at high symmetry positions on the Fermi surface needs to be distinguished from the case of two hot spot pairs. We compute the fluctuation propagator and the electronic self-energy perturbatively in leading order. The energy dependence of the single-particle decay rate at the hot spots obeys non-Fermi liquid power laws, with an exponent 2/3 in the case of a single hot spot pair, and exponent one for two hot spot pairs. The prefactors of the linear behavior obtained in the latter case are not particle-hole symmetric.

cond-mat.str-el↗

arXiv2014

Competing order in correlated electron systems made simple: Consistent fusion of functional renormalization and mean-field theory

We derive an efficient and unbiased method for computing order parameters in correlated electron systems with competing instabilities. Charge, magnetic and pairing fluctuations above the energy scale of spontaneous symmetry breaking are taken into account by a functional renormalization group flow, while the formation of order below that scale is treated in mean-field theory. The method captures fluctuation driven instabilities such as $d$-wave superconductivity. As a first application we study the competition between antiferromagnetism and superconductivity in the ground state of the two-dimensional Hubbard model.

cond-mat.str-el↗

arXiv2014

Low energy singularities in the ground state of fermionic superfluids

We analyze the effects of order parameter fluctuations on the ground state of fully gapped charge-neutral fermionic superfluids. The Goldstone mode associated with the spontaneously broken symmetry leads to a problem of coupled singularities in $d \leq 3$ dimensions. We derive a minimal set of one-loop renormalization group equations which fully captures the interplay of the singularities. The flow equations are based on a symmetry conserving truncation of a scale dependent effective action. We compute the low energy behavior of longitudinal, transverse and mixed order parameter correlations, and their impact on the fermionic gap. We demonstrate analytically that cancellations protecting the Goldstone mode are respected by the flow, and we present a numerical solution of the flow equations for the two-dimensional attractive Hubbard model.

cond-mat.str-el↗

arXiv2014

Superconductivity in the two-dimensional $t$-$t'$-Hubbard model

Using a recently developed renormalization group method for fermionic superfluids, we determine conditions for d-wave superconductivity in the two-dimensional Hubbard model at moderate interaction strength, and we compute the pairing gap in the superconducting regime. A pairing instability signaled by a divergent flow in the Cooper channel leads to a superconducting state in all studied cases. The next-to-nearest neighbor hopping $t'$ plays a crucial role in the competition between antiferromagnetism and superconductivity. A sizable $t'$ is necessary to obtain a sizable pairing gap.

cond-mat.str-el↗

arXiv2013

Effective interactions and fluctuation effects in spin-singlet superfluids

We derive and evaluate one-loop functional flow equations for the effective interactions, self-energy and gap function in spin-singlet superfluids. The flow is generated by a fermionic frequency cutoff, which is supplemented by an external pairing field to treat divergencies associated with the Goldstone boson. To parametrize the singular momentum and frequency dependences of the effective interactions, the Nambu interaction vertex is decomposed in charge, magnetic, and normal and anomalous pairing channels. The one-loop flow solves reduced (mean-field) models for superfluidity exactly, and captures also important fluctuation effects. The Ward identity from charge conservation is generally violated, but can be enforced by projecting the flow. Applying the general formalism to the two-dimensional attractive Hubbard model, we obtain detailed results on the momentum and frequency dependences of the effective interactions for weak and moderate bare interactions. The gap is reduced by fluctuations, with a stronger reduction at weaker interactions, as expected.

cond-mat.supr-con↗

arXiv2012

Incommensurate nematic fluctuations in the two-dimensional Hubbard model

We analyze effective d-wave interactions in the two-dimensional extended Hubbard model at weak coupling and small to moderate doping. The interactions are computed from a renormalization group flow. Attractive d-wave interactions are generated via antiferromagnetic spin fluctuations in the pairing and charge channels. Above Van Hove filling, the d-wave charge interaction is maximal at incommensurate diagonal wave vectors, corresponding to nematic fluctuations with a diagonal modulation. Below Van Hove filling a modulation along the crystal axes can be favored. The nematic fluctuations are enhanced by the nearest-neighbor interaction in the extended Hubbard model, but they always remain smaller than the dominant antiferromagnetic, pairing, or charge density wave fluctuations.

cond-mat.str-el↗

arXiv2012

Functional renormalization group approach to correlated fermion systems

Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems, (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals, (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and non-equilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. We present a self-contained derivation of the exact flow equations and describe frequently used truncation schemes. Reviewing selected applications we then show how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems.

cond-mat.str-el↗

arXiv2012

Fermi surface truncation from thermal nematic fluctuations

We analyze how thermal fluctuations near a finite temperature nematic phase transition affect the spectral function $A({\bf k},ω)$ for single-electron excitations in a two-dimensional metal. Perturbation theory yields a splitting of the quasi-particle peak with a d-wave form factor, reminiscent of a pseudogap. We present a resummation of contributions to all orders in the Gaussian fluctuation regime. Instead of a splitting, the resulting spectral function exhibits a pronounced broadening of the quasi-particle peak, which varies strongly around the Fermi surface and vanishes upon approaching the Brillouin zone diagonal. The Fermi surface obtained from a Brillouin zone plot of $A({\bf k},0)$ seems truncated to Fermi arcs.

cond-mat.str-el↗

arXiv2012

Incommensurate nematic fluctuations in two dimensional metals

To assess the strength of nematic fluctuations with a finite wave vector in a two-dimensional metal, we compute the static d-wave polarization function for tight-binding electrons on a square lattice. At Van Hove filling and zero temperature the function diverges logarithmically at q=0. Away from Van Hove filling the ground state polarization function exhibits finite peaks at finite wave vectors. A nematic instability driven by a sufficiently strong attraction in the d-wave charge channel thus leads naturally to a spatially modulated nematic state, with a modulation vector that increases in length with the distance from Van Hove filling. Above Van Hove filling, for a Fermi surface crossing the magnetic Brillouin zone boundary, the modulation vector connects antiferromagnetic hot spots with collinear Fermi velocities.

cond-mat.str-el↗

arXiv2012

Singular order parameter interaction at nematic quantum critical point in two dimensional electron systems

We analyze the infrared behavior of effective N-point interactions between order parameter fluctuations for nematic and other quantum critical electron systems with a scalar order parameter in two dimensions. The interactions exhibit a singular momentum and energy dependence and thus cannot be represented by local vertices. They diverge for all N greater or equal 4 in a collinear static limit, where energy variables scale to zero faster than momenta, and momenta become increasingly collinear. The degree of divergence is not reduced by any cancellations and renders all N-point interactions marginal. A truncation of the order parameter action at quartic or any other finite order is therefore not justified. The same conclusion can be drawn for the effective action describing fermions coupled to a U(1) gauge field in two dimensions.

cond-mat.str-el↗