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We study the hybridization between plasmons, phonons, and electronic sound in ionic crystals using the Debye model, where the ionic background is modeled as a homogeneous, isotropic, elastic medium. We explicitly obtain the energies and the damping of the hybrid plasmon-sound modes in the hydrodynamic regime and calculate the corresponding dynamic structure factor. We show that the direct Coulomb interaction between the ions is essential to obtain a collective sound mode with linear dispersion., Comment: 10 pages, 5 figures

We show that the high-temperature expansion of the free energy and arbitrary imaginary-time-ordered connected correlation functions of quantum spin systems can be recursively obtained from the exact renormalization group flow equation for the generating functional of connected spin correlation functions derived by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)]. Our recursive algorithm can be explicitly written down in closed form including all combinatorial factors. We use our method to estimate critical temperatures of Heisenberg magnets from low-order truncations of the inverse spin susceptibility in the static limit. We also calculate the connected correlation function involving three different spin components (chiral non-linear susceptibility) of quantum Heisenberg magnets up to second order in the exchange couplings., Comment: 21 pages, 10 figures

We use the spin functional renormalization group to investigate the $J_1$-$J_2$ quantum Heisenberg model on a square lattice. By incorporating sum rules associated with the fixed length of the spin operators as well as the nontrivial quantum dynamics implied by the spin algebra, we are able to compute the ground state phase diagram for arbitrary spin $S$, including the quantum paramagnetic phase at strong frustration. Our prediction for the extent of this paramagnetic region for $ S = 1/2 $ agrees well with other approaches that are computationally more expensive. We find that the quantum paramagnetic phase disappears for $ S \gtrsim 5 $ due to the suppression of quantum fluctuations with increasing $S$., Comment: 14 pages, 4 figures

Using a functional renormalization group approach we derive the renormalization group (RG) flow of a dissipative variant of the Yukawa-Sachdev-Ye-Kitaev model describing $N$ fermions on a quantum dot which interact via a disorder-induced Yukawa coupling with $M$ bosons. The inverse Euclidean propagator of the bosons is assumed to exhibit a non-analytic term proportional to the modulus of the Matsubara frequency. We show that, to leading order in $1/N$ and $1/M$, the hierarchy of formally exact flow equations for the irreducible vertices of the disorder-averaged model can be closed at the level of the two-point vertices. We find that the RG flow exhibits a non-Fermi liquid fixed point characterized by a finite fermionic anomalous dimension $\eta$ which is related to the bosonic anomalous dimension $\gamma$ via the scaling law $2 = 2 \eta + \gamma$ with $ 0 < \eta < 1/2$. We explicitly calculate $\eta$ and the critical exponents characterizing the linearized RG flow in the vicinity of the fixed point as functions of $N/M$., Comment: 17 pages, 9 figures

Exact functional renormalization group (FRG) flow equations for quantum systems can be derived directly within an operator formalism without using functional integrals. This simple insight opens new possibilities for applying FRG methods to models for strongly correlated electrons with projected Hilbert spaces, such as quantum spin models, the $t$-$J$ model, or the Hubbard model at infinite on-site repulsion. By representing these models in terms of Hubbard X-operators, we derive exact flow equations for the time-ordered correlation functions of the X-operators (X-FRG), which allow us to calculate the electronic correlation functions in the projected Hilbert space of these models. The Hubbard-I approximation for the single-particle Green function of the Hubbard model is recovered from a trivial truncation of the flow equations where the two-point vertex is approximated by its atomic limit. We use our approach to calculate the quasi-particle residue and damping in the ``hidden Fermi liquid'' state of the Hubbard model at infinite on-site repulsion where the Hamiltonian consists only of the projected kinetic energy., Comment: 27 pages, 4 figures

We re-examine the effect of long-range Coulomb interactions on the collective amplitude and phase modes in the incommensurate charge-density-wave ground state of quasi-one-dimensional conductors. Using an effective action approach we show that the longitudinal acoustic phonon protects the gapless linear dispersion of the lowest phase mode in the presence of long-range Coulomb interactions. Moreover, in Gaussian approximation, amplitude fluctuations are not affected by long-range Coulomb interactions. We also calculate the collective mode dispersions at finite temperatures and compare our results with the measured energies of amplitude and phase modes in K$_{0.3}$MoO$_3$. With the exception of the lowest phase mode, the temperature dependence of the measured mode energies can be quantitatively described within a multi-phonon Fr\"{o}hlich model for generic electron-phonon interactions neglecting long-range Coulomb interactions.

We use our recently developed functional renormalization group (FRG) approach for quantum spin systems to investigate the phase diagram of the frustrated $J_{1}J_{2}J_{3}$ quantum Heisenberg model on a cubic lattice. From a simple truncation of the hierarchy of FRG flow equations for the irreducible spin-vertices which retains only static spin fluctuations and neglects the flow of the four-spin interaction, we can estimate the critical temperature with a similar accuracy as the numerically more expensive pseudofermion FRG. In the regime where the ground state exhibits either ferromagnetic or antiferromagnetic order, a more sophisticated truncation including the renormalization of the four-spin interaction as well as dynamic spin fluctuations reveals the underlying renormalization group fixed point and yields critical temperatures which deviate from the accepted values by at most 4 %., Comment: 18 pages, 12 figures, corrected mistakes described in erratum: 10.1103/PhysRevB.107.019904

We investigate dimerized quantum spin systems using the spin functional renormalization group approach proposed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] which directly focuses on the physical spin correlation functions and avoids the representation of the spins in terms of fermionic or bosonic auxiliary operators. Starting from decoupled dimers as initial condition for the renormalization group flow equations, we obtain the spectrum of the triplet excitations as well as the magnetization in the quantum paramagnetic, ferromagnetic, and thermally disordered phases at all temperatures. Moreover, we compute the full phase diagram of a weakly coupled dimerized spin system in three dimensions, including the correct mean field critical exponents at the two quantum critical points., Comment: 17 pages, 10 figures

The Holstein model with dispersionless Einstein phonons is one of the simplest models describing electron-phonon interactions in condensed matter. A naive extrapolation of perturbation theory in powers of the relevant dimensionless electron-phonon coupling $\lambda_0$ suggests that at zero temperature the model exhibits a Pomeranchuk instability characterized by a divergent uniform compressibility at a critical value of $\lambda_0$ of order unity. In this work, we re-examine this problem using modern functional renormalization group (RG) methods. For dimensions $d > 3$ we find that the RG flow of the Holstein model indeed exhibits a tricritical fixed point associated with a Pomeranchuk instability. This non-Gaussian fixed point is ultraviolet stable and is closely related to the well-known ultraviolet stable fixed point of $\phi^3$-theory above six dimensions. To realize the Pomeranchuk critical point in the Holstein model at fixed density both the electron-phonon coupling $\lambda_0$ and the adiabatic ratio $\omega_0 / \epsilon_F$ have to be fine-tuned to assume critical values of order unity, where $\omega_0$ is the phonon frequency and $\epsilon_F$ is the Fermi energy. On the other hand, for dimensions $d \leq 3$ we find that the RG flow of the Holstein model does not have any critical fixed points. This rules out a quantum critical point associated with a Pomeranchuk instability in $d \leq 3$., Comment: 17 pages, 7 figures

We calculate the dynamic structure factor $S (\boldsymbol{k},\omega)$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures $T$ close to the critical temperature $T_c$ using our recently developed functional renormalization group approach to quantum spin systems. In $d=3$ dimensions we find that for small momenta $\boldsymbol{k}$ and frequencies $\omega$ the dynamic structure factor assumes the scaling form $S(\boldsymbol{k},\omega) = (\tau T G (\boldsymbol{k})/\pi)\Phi (k\xi, \omega\tau)$, where $ G (\boldsymbol{k})$ is the static spin-spin correlation function, $\xi$ is the correlation length, and the characteristic time-scale $\tau$ is proportional to $\xi^{5/2}$. We explicitly calculate the dynamic scaling function $\Phi (x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\xi = \infty$ our result for the dynamic structure factor can be written as $S (\boldsymbol{k},\omega) = (\pi\omega_k)^{-1} T_c G (\boldsymbol{k}) \Psi_c (\omega/\omega_k)$, where $\omega_k \propto k^{5/2}$. We find that $\Psi_c(\nu)$ vanishes as $\nu^{-13/5}$ for large $\nu$, and as $\nu^{3/5}$ for small $\nu$. While the large-frequency behavior of $\Psi_c (\nu)$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\epsilon = 6-d$, our result for small frequencies disagrees with previous calculations. We argue that up until now neither experiments nor numerical simulations are sufficiently accurate to determine the low-frequency behavior of $\Psi_c (\nu)$. We also calculate the low-temperature behavior of $S ( \boldsymbol{k},\omega)$ in one- and two dimensional ferromagnets and find that it satisfies dynamic scaling with exponent $z=2$ and exhibits a pseudogap for small frequencies., Comment: 19 pages, 18 figures

We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B $\mathbf{99}$, 060403(R) (2019)] to calculate the spin-spin correlation function $G (\boldsymbol{k}, \omega )$ of quantum Heisenberg magnets at infinite temperature. For small wavevectors $\boldsymbol{k} $ and frequencies $\omega$ we find that $G ( \boldsymbol{k}, \omega )$ assumes in dimensions $d > 2$ the diffusive form predicted by hydrodynamics. In three dimensions our result for the spin-diffusion coefficient ${\cal{D}}$ is somewhat smaller than previous theoretical predictions based on the extrapolation of the short-time expansion, but is still about $30 \%$ larger than the measured high-temperature value of ${\cal{D}}$ in the Heisenberg ferromagnet Rb$_2$CuBr$_4\cdot$2H$_2$O. In reduced dimensions $d \leq 2$ we find superdiffusion characterized by a frequency-dependent complex spin-diffusion coefficient ${\cal{D}} ( \omega )$ which diverges logarithmically in $d=2$, and as a power-law ${\cal{D}} ( \omega ) \propto \omega^{-1/3}$ in $d=1$. Our result in one dimension implies scaling with dynamical exponent $z =3/2$, in agreement with recent calculations for integrable spin chains. Our approach is not restricted to the hydrodynamic regime and allows us to calculate the dynamic structure factor $S ( \boldsymbol{k} , \omega )$ for all wavevectors. We show how the short-wavelength behavior of $S ( \boldsymbol{k}, \omega )$ at high temperatures reflects the relative sign and strength of competing exchange interactions., Comment: 30 pages, 11 figures

Using a functional renormalization group approach we show that at low temperatures and in a certain range of magnetic fields the longitudinal dynamic structure factor of quantum Heisenberg ferromagnets in dimensions $D\leq 2$ exhibits a well-defined quasi-particle peak with linear dispersion that we identify as zero-magnon sound. In $D>2$ the larger phase space available for the decay into transverse spin waves leads only to a broad hump centered at zero frequency whose width scales linearly in momentum., Comment: 6 pages, 3 figures

We determine the global renormalization group (RG) flow of the Sachdev-Ye-Kitaev (SYK) model. This flow allows for an understanding of the surprising role of critical slowing down at a quantum first-order transition in strongly-correlated electronic systems. From a simple truncation of the infinite hierarchy of the exact functional RG flow equations we identify several fixed points: Apart from a stable fixed point, associated with the celebrated non-Fermi liquid state of the model, we find another stable fixed point related to an integer-valence state. These stable fixed points are separated by a discontinuity fixed point with one relevant direction, describing a quantum first-order transition. Most notably, the fermionic spectrum continues to be quantum critical even at the discontinuity fixed point. This rules out a description of this quantum first-order transition in terms of a local effective Ising variable that is established for classical transitions. It reveals that quantum phase coexistence can be a genuine critical state of matter., Comment: 6 pages, 4 figures

We use the spin functional renormalization group recently developed by two of us [J. Krieg and P. Kopietz, Phys. Rev. B $\bf{99}$, 060403(R) (2019)] to calculate the magnetization $M ( H , T )$ and the damping of magnons due to classical longitudinal fluctuations of quantum Heisenberg ferromagnets. In order to guarantee that for vanishing magnetic field $H \rightarrow 0$ the magnon spectrum is gapless when the spin rotational invariance is spontaneously broken, we use a Ward identity to express the magnon self-energy in terms of the magnetization. In two dimensions our approach correctly predicts the absence of long-range magnetic order for $H=0$ at finite temperature $T$. The magnon spectrum then exhibits a gap from which we obtain the transverse correlation length. We also calculate the wave-function renormalization factor of the magnons. As a mathematical by-product, we derive a recursive form of the generalized Wick theorem for spin operators in frequency space which facilitates the calculation of arbitrary time-ordered connected correlation functions of an isolated spin in a magnetic field., Comment: 26 pages, 6 figures

We show how the one-loop "poor man's scaling" equations for the Kondo model with arbitrary impurity spin can be obtained within the framework of the functional renormalization group approach for quantum spin systems recently developed by Krieg and Kopietz [arXiv:1807.02524]. We argue that our method supersedes the "poor man's scaling" approach and can also be used to study the strong coupling regime of the Kondo model., Comment: 7 pages, 2 figures; added references, corrected typos

We show that the diagrammatic approach to quantum spin systems developed in a seminal work by Vaks, Larkin, and Pikin [Sov. Phys. JETP 26, 188 (1968)] can be embedded in the framework of the functional renormalization group. The crucial insight is that the generating functional of the time-ordered connected spin correlation functions of an arbitrary quantum spin system satisfies an exact renormalization group flow equation which resembles the corresponding flow equation of a system of interacting bosons. The $SU (2)$ spin algebra is implemented via a non-trivial initial condition for the renormalization group flow. Our method is rather general and offers a new non-perturbative approach to quantum spin systems., Comment: 6 pages, 1 figure, 1 table; revised main text and added Supplemental Material

We investigate the effect of long-range Coulomb interaction on the quasiparticle properties and the dielectric function of clean three-dimensional Weyl semimetals at zero temperature using a functional renormalization group (FRG) approach. The Coulomb interaction is represented via a bosonic Hubbard-Stratonovich field which couples to the fermionic density. We derive truncated FRG flow equations for the fermionic and bosonic self-energies and for the three-legged vertices with two fermionic and one bosonic external legs. We consider two different cutoff schemes --- cutoff in fermionic or bosonic propagators --- in order to calculate the renormalized quasiparticle velocity and the dielectric function for an arbitrary number of Weyl nodes and the interaction strength. If we approximate the dielectric function by its static limit, our results for the velocity and the dielectric function are in good agreement with that of A. A. Abrikosov and S. D. Beneslavski{\u{i}} [Sov. Phys. JETP \textbf{32}, 699 (1971)] exhibiting slowly varying logarithmic momentum dependence for small momenta. We extend their result for an arbitrary number of Weyl nodes and finite frequency by evaluating the renormalized velocity in the presence of dynamic screening and calculate the wave function renormalization., Comment: 15 pages, 11 figures and 1 table. Accepted for publication in Phys. Rev. B

In magnets with non-collinear spin configuration the expectation value of the conventionally defined spin current operator contains a contribution which renormalizes an external magnetic field and hence affects only the precessional motion of the spin polarization. This term, which has been named angular spin current by Sun and Xie [Phys. Rev B 72, 245305 (2005)], does not describe the translational motion of magnetic moments. We give a prescription how to separate these two types of spin transport and show that the translational movement of the spin is always polarized along the direction of the local magnetization. We also show that at vanishing temperature the classical magnetic order parameter in magnetic insulators cannot carry a translational spin current, and elucidate how this affects the interpretation of spin supercurrents., Comment: 8 pages, 3 figures

We reexamine the effect of long-range Coulomb interactions on the quasiparticle velocity in graphene. Using a nonperturbative functional renormalization group approach with partial bosonization in the forward scattering channel and momentum transfer cutoff scheme, we calculate the quasiparticle velocity, $v (k)$, and the quasiparticle residue, $Z$, with frequency-dependent polarization. One of our most striking results is that $v ( k ) \propto \ln [ C_k (\alpha) / k ]$ where the momentum- and interaction-dependent cutoff scale $C_k (\alpha) $ vanishes logarithmically for $k \rightarrow 0$. Here $k$ is measured with respect to one of the charge neutrality (Dirac) points and $\alpha=2.2$ is the strength of dimensionless bare interaction. Moreover, we also demonstrate that the so-obtained multilogarithmic singularity is reconcilable with the perturbative expansion of $v (k)$ in powers of the bare interaction., Comment: 7 pages, 4 figures. Accepted for publication in Phys. Rev. B

We study the high magnetic field regime of the antiferromagnetic insulator Cs$_2$CuCl$_4$ by expressing the spin-1/2 operators in the relevant Heisenberg model in terms of hard-core bosons and implementing the hard-core constraint via an infinite on-site interaction. We focus on the case where the external magnetic field exceeds the saturation field $B_{c}\approx8.5\;\mathrm{T}$ and is oriented along the crystallographic $a$ axis perpendicular to the lattice plane. Because in this case the excited states are separated by an energy gap from the ground state, we may use the self-consistent ladder approximation to take the strong correlations due to the hard-core constraint into account. In Cs$_2$CuCl$_4$ there are additional interactions besides the hard-core interaction which we treat in self-consistent Hartree-Fock approximation. We calculate the spectral function of the hard-core bosons from which we obtain the in-plane components of the dynamic structure factor, the magnetic susceptibility, and the specific heat. Our results for the specific heat are in good agreement with the available experimental data. We conclude that the self-consistent ladder approximation in combination with a self-consistent Hartree-Fock decoupling of the non-hard-core interactions gives an accurate description of the physical properties of gapped hard-core bosons in two dimensions at finite temperatures., Comment: 14 pages, 15 figures

Using a non-perturbative functional renormalization group approach we calculate the renormalized quasi-particle velocity $v (k)$ and the static dielectric function $\epsilon ( k )$ of suspended graphene as functions of an external momentum $k$. Our numerical result for $v (k )$ can be fitted by $v ( k ) / v_F = A + B \ln ( \Lambda_0 / k )$, where $v_F$ is the bare Fermi velocity, $\Lambda_0$ is an ultraviolet cutoff, and $A = 1.37$, $B =0.51$ for the physically relevant value ($e^2/v_F =2.2$) of the coupling constant. In contrast to calculations based on the static random-phase approximation, we find that $\epsilon (k )$ approaches unity for $k \rightarrow 0$. Our result for $v (k )$ agrees very well with a recent measurement by Elias et al. [Nat. Phys. 7, 701 (2011)]., Comment: 5 pages, 4 figures

We present a simple method for summing so-called parquet diagrams of fermionic many-body systems with competing instabilities using the functional renormalization group. Our method is based on partial bosonization of the interaction utilizing multi-channel Hubbard-Stratonovich transformations. A simple truncation of the resulting flow equations, retaining only the frequency-independent parts of the two-point and three-point vertices amounts to solving coupled Bethe-Salpeter equations for the effective interaction to leading logarithmic order. We apply our method by revisiting the X-ray problem and deriving the singular frequency dependence of the X-ray response function and the particle-particle susceptibility. Our method is quite general and should be useful in many-body problems involving strong fluctuations in several scattering channels., Comment: Bibliography has been revised and updated 9 pages, 4 figures

Spin dimer systems are a promising playground for the detailed study of quantum phase transitions. Using the magnetic field as the tuning parameter it is in principle possible to observe a crossover from the characteristic scaling near critical points to the behavior of a finite temperature phase transition. In this work we study two-dimensional coupled spin dimer systems by comparing numerical quantum Monte Carlo simulations with analytical calculations of the susceptibility, the magneto-caloric effect, and the helicity modulus. The magneto-caloric behavior of the magnetization with temperature can be used to determine the critical fields with high accuracy, but the critical scaling does not show the expected logarithmic corrections. The zeros of the cooling rate are an excellent indicator of the competition between quantum criticality and vortex physics, but they are not directly associated with the quantum phase transition or the finite temperature Berezinsky-Kosterlitz-Thouless transition. The results give a unified picture of the full quantum and finite temperature phase diagram., Comment: 9 pages with 10 figures. New version includes updated figures, data, and references. For more information and the latest version see http://www.physik.uni-kl.de/eggert/papers/index.html

We show that in a weak external magnetic field H the quasi-particle residue and the renormalized electron Lande factor of two-dimensional Fermi liquids exhibit a non-analytic magnetic field dependence proportional to |H| which is due to electron-electron interactions. We explicitly calculate the corresponding prefactors to second order in the interaction and show that they are determined by low-energy scattering processes involving only momenta close to the Fermi surface. Experimentally, these non-analytic terms can be detected from measurements of the magnetic field dependence of the density of states and the magnetoconductivity., Comment: 11 pages, 5 figures

For temperatures below 0.6 K the geometrically frustrated layered quantum antiferromagnet Cs$_2$CuCl$_4$ in a magnetic field perpendicular to the layers orders magnetically in a so-called cone state, where the magnetic moments have a finite component in the field direction while their projection onto the layers forms a spiral. Modeling this system by a two-dimensional spatially anisotropic quantum Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interaction, we find that even for vanishing temperature the usual spin-wave expansion is plagued by infrared divergencies which are due to the coupling between longitudinal and transverse spin fluctuations in the cone state. Similar divergencies appear also in the ground state of the interacting Bose gas in two and three dimensions. Using known results for the correlation functions of the interacting Bose gas, we present a non-perturbative expression for the dynamic structure factor in the cone state of Cs$_2$CuCl$_4$. We show that in this state the spectral line shape of spin fluctuations exhibits singular scattering continua which can be understood in terms of the well-known anomalous longitudinal fluctuations in the ground state of the two-dimensional Bose gas., Comment: 13 pages, 7 figures

We calculate the effects of the spin-lattice coupling on the magnon spectrum of thin ferromagnetic films consisting of the magnetic insulator yttrium-iron garnet. The magnon-phonon hybridisation generates a characteristic minimum in the spin dynamic structure factor which quantitatively agrees with recent Brillouin light scattering experiments. We also show that at room temperature the phonon contribution to the magnon damping exhibits a rather complicated momentum dependence: In the exchange regime the magnon damping is dominated by Cherenkov type scattering processes, while in the long-wavelength dipolar regime these processes are subdominant and the magnon damping is two orders of magnitude smaller. We supplement our calculations by actual measurements of the magnon relaxation in the dipolar regime. Our theory provides a simple explanation of a recent experiment probing the different temperatures of the magnon and phonon gases in yttrium-iron garnet., Comment: 15 pages, 9 figures

The low-temperature behavior of the magnetic insulator Cs2CuCl4 can be modeled by an anisotropic triangular lattice spin-1/2 Heisenberg antiferromagnet with two different exchange couplings J and J' = J/3. We show that in a wide range of magnetic fields the experimentally observed field dependence of the crossover temperature T_c for spin-liquid behavior can be explained within a mean-field theory based on the representation of spin operators in terms of Majorana fermions. We also show that for small magnetic fields the specific heat and the spin susceptibility both exhibit a maximum as a function of temperature at T_c = J/2. In the spin-liquid regime, the Majorana fermions can only propagate along the direction of the strongest bond, in agreement with the dimensional reduction scenario advanced by Balents [Nature (London) 464, 199 (2010)]., Comment: 9 pages, 8 figures

We show that the functional renormalization group is a numerically cheap method to obtain the low-energy behavior of the Anderson impurity model describing a localized interacting electron coupled to a bath of conduction electrons. Our approach uses an external magnetic field as flow parameter, partial bosonization of the transverse spin fluctuations, and frequency-independent interaction vertices which are fixed by Ward identities. We calculate the quasi-particle residue and the spin susceptibility in the particle-hole symmetric case and obtain excellent agreement with the Bethe ansatz results for arbitrary strengths of the interaction., Comment: 5 pages, 4 figures; version published in PRB Rapid Communications

Using a nonequilibrium functional renormalization group (FRG) approach we calculate the time evolution of the momentum distribution of a magnon gas in contact with a thermal phonon bath. As a cutoff for the FRG procedure we use a hybridization parameter {\Lambda} giving rise to an artificial damping of the phonons. Within our truncation of the FRG flow equations the time evolution of the magnon distribution is obtained from a rate equation involving cutoff-dependent nonequilibrium self-energies, which in turn satisfy FRG flow equations depending on cutoff-dependent transition rates. Our approach goes beyond the Born collision approximation and takes the feedback of the magnons on the phonons into account. We use our method to calculate the thermalization of a quasi two-dimensional magnon gas in the magnetic insulator yttrium-iron garnet after a highly excited initial state has been generated by an external microwave field. We obtain good agreement with recent experiments., Comment: 16 pages, 6 figures, final version

Using functional renormalization group methods, we study an effective low-energy model describing the Ising-nematic quantum critical point in two-dimensional metals. We treat both gapless fermionic and bosonic degrees of freedom on equal footing and explicitly calculate the momentum and frequency dependent effective interaction between the fermions mediated by the bosonic fluctuations. Following earlier work by S.-S. Lee for a one-patch model, Metlitski and Sachdev [Phys. Rev. B {\bf{82}}, 075127] recently found within a field-theoretical approach that certain three-loop diagrams strongly modify the one-loop results, and that the conventional 1/N expansion breaks down in this problem. We show that the singular three-loop diagrams considered by Metlitski and Sachdev are included in a rather simple truncation of the functional renormalization group flow equations for this model involving only irreducible vertices with two and three external legs. Our approximate solution of these flow equations explicitly yields the vertex corrections of this problem and allows us to calculate the anomalous dimension $\eta_{\psi}$ of the fermion field., Comment: 21 pages, 10 figures

We generalize the spin-wave expansion in powers of the inverse spin to time-dependent quantum spin models describing rotating magnets or magnets in time-dependent external fields. We show that in these cases, the spin operators should be projected onto properly defined rotating reference frames before the spin components are bosonized using the Holstein-Primakoff transformation. As a first application of our approach, we calculate the reorganization of the magnetic state due to Bose-Einstein condensation of magnons in the magnetic insulator yttrium-iron garnet; we predict a characteristic dip in the magnetization which should be measurable in experiments., Comment: 6 pages, 5 figures, final version published in PRB

We show that a chain of Heisenberg spins interacting with long-range dipolar forces in a magnetic field h perpendicular to the chain exhibits a quantum critical point belonging to the two-dimensional Ising universality class. Within linear spin-wave theory the magnon dispersion for small momenta k is [Delta^2 + v_k^2 k^2]^{1/2}, where Delta^2 \propto |h - h_c| and v_k^2 \propto |ln k|. For fields close to h_c linear spin-wave theory breaks down and we investigate the system using density-matrix and functional renormalization group methods. The Ginzburg regime where non-Gaussian fluctuations are important is found to be rather narrow on the ordered side of the transition, and very broad on the disordered side., Comment: 6 pages, 5 figures

In an external magnetic field perpendicular to the plane of the layers, the quasi two-dimensional frustrated antiferromagnet Cs_2CuCl_4 exhibits a magnetically ordered "cone state" at low temperatures. In this state the component of the magnetic moments in field direction is finite, while their projection onto the plane of the layers forms a spiral. We present both theoretical and experimental results for the magnetic field dependence of the elastic constants and the ultrasonic attenuation rate in the cone state. Our theoretical analysis is based on the usual spin-wave expansion around the classical ground state of a Heisenberg model on an anisotropic triangular lattice with Dzyaloshinskii-Moriya interactions. Magnon-phonon interactions are modeled by expanding the exchange interactions up to second order in powers of the phonon coordinates. As long as the external magnetic field is not too close to the critical field where the cone state becomes unstable, we obtain reasonable agreement between theory and experiment, suggesting that at least in this regime magnons behave as well-defined quasiparticles. We also show that the assumption of well-defined magnons implies that at long wavelengths the ultrasonic attenuation rate in the cone state of Cs_2CuCl_4 is proportional to the fourth power of the phonon momentum., Comment: 20 pages, 7 figures, final version published in PRB

We use the Gross-Pitaevskii equation to determine the spatial structure of the condensate density of interacting bosons whose energy dispersion epsilon_k has two degenerate minima at finite wave-vectors q. We show that in general the Fourier transform of the condensate density has finite amplitudes for all integer multiples of q. If the interaction is such that many Fourier components contribute, the Bose condensate is localized at the sites of a one-dimensional lattice with spacing 2pi/q; in this case Bose-Einstein condensation resembles the transition from a liquid to a crystalline solid. We use our results to investigate the spatial structure of the Bose condensate formed by magnons in thin films of ferromagnets with dipole-dipole interactions., Comment: 9 pages, 4 figures, final version publicated in EPJD

Using the functional renormalization group (FRG) and the numerical renormalization group (NRG), we calculate the spectral function of the Anderson impurity model at zero and finite temperatures. In our FRG scheme spin fluctuations are treated non-perturbatively via a suitable Hubbard-Stratonovich field, but vertex corrections are neglected. A comparison with our highly accurate NRG results shows that this FRG scheme gives a quantitatively good description of the spectral line-shape at zero and finite temperatures both in the weak and strong coupling regimes, although at zero temperature the FRG is not able to reproduce the known exponential narrowing of the Kondo resonance at strong coupling., Comment: 6 pages, 3 figures; new references added

Using functional methods and the exact renormalization group we derive Ward identities for the Anderson impurity model. In particular, we present a non-perturbative proof of the Yamada-Yosida identities relating certain coefficients in the low-energy expansion of the self-energy to thermodynamic particle number and spin susceptibilities of the impurity. Our proof underlines the relation of the Yamada-Yosida identities to the U(1) x U(1) symmetry associated with particle number and spin conservation in a magnetic field., Comment: 8 pages, corrected statements about infintite flatband limit

The time evolution of magnons subject to a time-dependent microwave field is usually described within the so-called "S-theory", where kinetic equations for the distribution function are obtained within the time-dependent Hartree-Fock approximation. To explain the recent observation of "Bose-Einstein condensation of magnons" in an external microwave field [Demokritov et al., Nature 443, 430 (2006)], we extend the "S-theory" to include the Gross-Pitaevskii equation for the time-dependent expectation values of the magnon creation and annihilation operators. We explicitly solve the resulting coupled equations within a simple approximation where only a single condensed mode is retained. We also re-examine the usual derivation of an effective boson model from a realistic spin model for yttrium-iron garnet films and argue that in the parallel pumping geometry (where both the static and the time-dependent magnetic field are parallel to the macroscopic magnetization) the time-dependent Zeemann energy cannot give rise to magnon condensation., Comment: 8 pages, 2 figures, final version (minor changes, included references)

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-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., Comment: 5 pages, 3 figures

Motivated by recent experiments on thin films of the ferromagnetic insulator yttrium-iron garnet (YIG), we have developed an efficient microscopic approach to calculate the spin-wave spectra of these systems. We model the experimentally relevant magnon band of YIG using an effective quantum Heisenberg model on a cubic lattice with ferromagnetic nearest neighbor exchange and long-range dipole-dipole interactions. After a bosonization of the spin degrees of freedom via a Holstein-Primakoff transformation and a truncation at quadratic order in the bosons, we obtain the spin-wave spectra for experimentally relevant parameters without further approximation by numerical diagonalization, using efficient Ewald summation techniques to carry out the dipolar sums. We compare our numerical results with two different analytic approximations and with predictions based on the phenomenological Landau-Lifshitz equation., Comment: 10 pages, 8 figures, final version (minor corrections, references updated)

We develop a functional renormalization group approach which describes the low-energy single-particle properties of the Anderson impurity model up to intermediate on-site interactions $U \lesssim 15 \Delta$, where $\Delta$ is the hybridization in the wide-band limit. Our method is based on a generalization of a method proposed by Sch\"{u}tz, Bartosch and Kopietz [Phys. Rev. B 72, 035107 (2005)], using two independent Hubbard-Stratonovich fields associated with transverse and longitudinal spin fluctuations. Although we do not reproduce the exponentially small Kondo scale in the limit $U \to \infty$, the spin fluctuations included in our approach remove the unphysical Stoner instability predicted by mean-field theory for $U > \pi \Delta$. We discuss different decoupling schemes and show that a decoupling which manifestly respects the spin-rotational invariance of the problem gives rise to the lowest quasiparticle weight. To obtain a closed flow equation for the fermionic self-energy we also propose a new truncation scheme of the functional renormalization group flow equations using Dyson-Schwinger equations to express bosonic vertex functions in terms of fermionic ones., Comment: 15 pages, 16 figures

We reexamine the 1/S-correction to the self-energy of the gapless magnon of a D-dimensional quantum Heisenberg antiferromagnet in a uniform magnetic field h using a hybrid approach between 1/S-expansion and non-linear sigma model, where the Holstein-Primakoff bosons are expressed in terms of Hermitian field operators representing the uniform and the staggered components of the spin-operators [N. Hasselmann and P. Kopietz, Europhys. Lett. {\bf{74}}, 1067 (2006)]. By integrating over the field associated with the uniform spin-fluctuations we obtain the effective action for the staggered spin-fluctuations on the lattice, which contains fluctuations on all length scales and does not have the cutoff ambiguities of the non-linear sigma model. We show that in dimensions D <= 3 the magnetic field dependence of the spin-wave velocity c(h) is non-analytic in h^2, with c(h) - c(0) proportional to h^2 \ln | h | in D=3, and c(h) - c(0) proportional to | h | in D=2. The frequency dependent magnon self-energy is found to exhibit an even more singular magnetic field dependence, implying a strong momentum dependence of the quasi-particle residue of the gapless magnon. We also discuss the problem of spontaneous magnon decay and show that in D > 1 dimensions the damping of magnons with momentum k is proportional to |k|^{2D -1} if spontaneous magnon decay is kinematically allowed., Comment: 19 pages, 4 figures

We calculate the dynamic structure factor S (omega, q) of spinless fermions in one dimension with quadratic energy dispersion k^2/2m and long range density-density interaction whose Fourier transform f_q is dominated by small momentum-transfers q << q_0 << k_F. Here q_0 is a momentum-transfer cutoff and k_F is the Fermi momentum. Using functional bosonization and the known properties of symmetrized closed fermion loops, we obtain an expansion of the inverse irreducible polarization to second order in the small parameter q_0 / k_F. In contrast to perturbation theory based on conventional bosonization, our functional bosonization approach is not plagued by mass-shell singularities. For interactions which can be expanded as f_q = f_0 + f_0^{2} q^2/2 + O (q^4) with finite f_0^{2} we show that the momentum scale q_c = 1/ | m f_0^{2} | separates two regimes characterized by a different q-dependence of the width gamma_q of the collective zero sound mode and other features of S (omega, q). For q_c << q << k_F we find that the line-shape in this regime is non-Lorentzian with an overall width gamma_q of order q^3/(m q_c) and a threshold singularity at the lower edge., Comment: 33 Revtex pages, 17 figures

We calculate the effect of order parameter fluctuations on the fermionic single-particle excitations in the superfluid state of neutral fermions interacting with short range attractive forces. We show that in dimensions D \leq 3 the singular effective interaction between the fermions mediated by the gapless Bogoliubov-Anderson mode prohibits the existence of well-defined quasi-particles. We explicitly calculate the single-particle spectral function in the BEC regime in D=3 and show that in this case the quasi-particle residue and the density of states are logarithmically suppressed., Comment: 4 RevTex pages, 3 figures; title changed, new Figure 1, added references. We argue that in the entire regime of the BCS-BEC crossover the quasi-particle picture breaks down in D <=3 for neutral fermions (but NOT for charged fermions)

We present a functional renormalization group calculation of the effect of strong interactions on the shape of the Fermi surface of weakly coupled metallic chains. In the regime where the bare interchain hopping is small, we show that scattering processes involving large momentum transfers perpendicular to the chains can completely destroy the warping of the true Fermi surface, leading to a confined state where the renormalized interchain hopping vanishes and a coherent motion perpendicular to the chains is impossible., Comment: 4 RevTex pages, 5 figures,final version as published by PRB

The emergence of a finite staggered magnetization in quantum Heisenberg antiferromagnets subject to a uniform magnetic field can be viewed as Bose-Einstein condensation of magnons. Using non-perturbative results for the infrared behavior of the interacting Bose gas, we present exact results for the staggered spin-spin correlation functions of quantum antiferromagnets in a magnetic field at zero temperature. In particular, we show that in dimensions 1 < D <= 3 the longitudinal dynamic structure factor S_parallel(q,omega) describing staggered spin fluctuations in the direction of the staggered magnetization exhibits a critical continuum whose weight can be controlled experimentally by varying the magnetic field., Comment: 4 pages, 1 figure, revised version. To appear in Physical Review Letters

We show that in two dimensions (2D) a systematic expansion of the self-energy and the effective interaction of the dilute electron gas in powers of the two-body T-matrix T_0 can be generated from the exact hierarchy of functional renormalization group equations for the one-particle irreducible vertices using the chemical potential as flow parameter. Due to the interference of particle-particle and particle-hole channels at order T_0^2, in 2D the ladder approximation for the self-energy is not reliable beyond the leading order in T_0. We also discuss two-body scattering in vacuum in arbitrary dimensions from the renormalization group point of view and argue that the singular interaction proposed by Anderson [Phys. Rev. Lett. 65, 2306 (1990)] cannot be ruled out on the basis of the ladder approximation., Comment: 4 pages, 1 figure, typos corrected

Using a non-perturbative functional renormalization group approach involving both fermionic and bosonic fields we calculate the interaction-induced change of the Fermi surface of spinless fermions moving on two chains connected by weak interchain hopping t_{bot}. We show that interchain backscattering can strongly reduce the distance Delta between the Fermi momenta associated with the bonding and the antibonding band, corresponding to a large reduction of the effective interchain hopping t_{bot}^{*} A self-consistent one-loop approximation neglecting marginal vertex corrections and wave-function renormalizations predicts a confinement transition for sufficiently large interchain backscattering, where the renormalized t_{bot}^{*} vanishes. However, a more accurate calculation taking vertex corrections and wave-function renormalizations into account predicts only weak confinement in the sense that 0< | t_{bot}^{*} | << | t_{bot} |. Our method can be applied to other strong-coupling problems where the dominant scattering channel is known., Comment: 15 RevTex pages, 11 figures; final version as published in Phys. Rev. B; changes: figure with weak coupling RG flow added; more extensive discussion of previous work added; some references added

This review is a summary of my work (partially in collaboration with Kurt Schoenhammer) on higher-dimensional bosonization during the years 1994-1996. It has been published as a book entitled "Bosonization of interacting fermions in arbitrary dimensions" by Springer Verlag (Lecture Notes in Physics m48, Springer, Berlin, 1997). I have NOT revised this review, so that there is no reference to the literature after 1996. However, the basic ideas underlying the functional bosonization approach outlined in this review are still valid today., Comment: 287 pages, 33 figues. This review has been published in the lecture note series by Springer Verlag, but it is currently out of print. It will be available again via Springer Verlag in the future, see http://www.springerlink.com/openurl.asp?genre=journal&issn=1616-6361 and http://www.springerlink.com/openurl.asp?genre=journal&issn=0940-7677 Permission by Springer Verlag to publish this review in this archive has been obtained

We calculate the damping gamma_q of collective density oscillations (zero sound) in a one-dimensional Fermi gas with dimensionless forward scattering interaction F and quadratic energy dispersion k^2 / 2 m at zero temperature. For wave-vectors | q| /k_F small compared with F we find to leading order gamma_q = v_F^{-1} m^{-2} Y (F) | q |^3, where v_F is the Fermi velocity, k_F is the Fermi wave-vector, and Y (F) is proportional to F^3 for small F. We also show that zero-sound damping leads to a finite maximum proportional to |k - k_F |^{-2 + 2 eta} of the charge peak in the single-particle spectral function, where eta is the anomalous dimension. Our prediction agrees with photoemission data for the blue bronze K_{0.3}MoO_3., Comment: final version as published; with more technical details; we have added a discussion of recent work which appeared after our initial cond-mat posting; 13 pages, 5 figures

We generalize our recently developed super-field functional renormalization group (RG) method involving both Fermi and Bose fields [F. Schuetz, L. Bartosch, and P. Kopietz, Phys. Rev. B 72, 035105 (2005)] to include the possibility that some bosonic components of the field have a finite vacuum expectation value. We derive an exact hierarchy of flow equations for the one-line irreducible vertices and the vacuum expectation value of the field. We apply our method to an interacting Fermi system where the interaction can be decoupled in the zero-sound channel and is then mediated by a collective bosonic field. The vacuum expectation value of the zero-frequency and zero-momentum component of the bosonic field is then closely related to the fermionic density. This can be exploited to calculate the compressibility of the interacting system. By using a cutoff in the bosonic propagator, the RG can be set up such that the self-consistent Hartree approximation is imposed as initial condition for the RG flow and the corrections to this approximation are generated as the remaining degrees of freedom are successively eliminated by the RG procedure., Comment: 15 pages, 2 figures; published version, some typos corrected; special issue "Renormalization Group 2005" of JPA. (proc. of Helsinki conference)