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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., 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 neglecting long-range Coulomb interactions.

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., 17 pages, 10 figures

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 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 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

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

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

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

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., 20 pages, 7 figures, final version published in PRB

We calculate the average conductivity sigma (omega) of interacting electrons in one dimension in the presence of a long-range random potential (forward scattering disorder). Taking the curvature of the energy dispersion into account, we show that weak disorder leads to a transport scattering rate that vanishes as omega^2 for small frequency omega. This implies that the real part of the conductivity remains finite for omega -> 0, while the imaginary part diverges. These effects are lost within the usual bosonization approach, which relies on the linearization of the energy dispersion. We discuss our result in the light of a recent experiment., 5 RevTex pages; more careful comparison with experiments and discussion of interchain hopping added; some references added; to appear in Phys. Rev. B

We calculate the nuclear relaxation rate of oxygen in the undoped quasi two-dimensional quantum Heisenberg antiferromagnet Sr_2CuO_2Cl_2 above the Neel temperature. The calculation is performed at two-loop order with the help of the Dyson-Maleev formulation of the spin-wave expansion, taking all scattering processes involving two and three magnons into account. At low temperatures T we find 1 / T_1 = c_1 T^3 + c_2 T^4 + O (T^5), and give explicit expressions for the coefficients c_1 (two-magnon scattering) and c_2 (three magnon scattering). We compare our result with a recent experiment by Thurber et al. and show that this experiment directly probes the existence of short-wavelength magnons in a two-dimensional antiferromagnet., Final version, to appear in Phys. Rev. B (1 August 1997)

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., 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