Pseudofermion Functional Renormalization Group Research Articles

Quantum paramagnetism and helimagnetic orders in the Heisenberg model on the body centered cubic lattice

We investigate the spin $S=1/2$ Heisenberg model on the body centered cubic lattice in the presence of ferromagnetic and antiferromagnetic nearest-neighbor $J_{1}$, second-neighbor $J_{2}$, and third-neighbor $J_{3}$ exchange interactions. The classical ground state phase diagram obtained by a Luttinger-Tisza analysis is shown to host six different (noncollinear) helimagnetic orders in addition to ferromagnetic, N\'eel, stripe and planar antiferromagnetic orders. Employing the pseudofermion functional renormalization group (PFFRG) method for quantum spins ($S=1/2$) we find an extended nonmagnetic region, and significant shifts to the classical phase boundaries and helimagnetic pitch vectors caused by quantum fluctuations while no new long-range dipolar magnetic orders are stabilized. The nonmagnetic phase is found to disappear for $S=1$. We calculate the magnetic ordering temperatures from PFFRG and quantum Monte Carlo methods, and make comparisons to available data

Characterization of quantum spin liquids and their spinon band structures via functional renormalization

We combine the pseudofermion functional renormalization group (PFFRG) method with a self-consistent Fock-like mean-field scheme to calculate low-energy effective theories for emergent spinon excitations in spin-1/2 quantum spin liquids. Using effective spin interactions from PFFRG as an input for the Fock equation and allowing for the most general types of free spinon ans\"atze as classified by the projective symmetry group (PSG) method, we are able to systematically determine spinon band structures for spin-liquid candidate systems beyond mean-field theory. We apply this approach to the antiferromagnetic $J_1$-$J_2$ Heisenberg model on the square lattice and to the antiferromagnetic nearest-neighbor Heisenberg model on the kagome lattice. For the $J_1$-$J_2$ model, we find that in the regime of maximal frustration a SU(2) $\pi$-flux state with Dirac spinons yields the largest mean-field amplitudes. For the kagome model, we identify a gapless $\mathbb{Z}_2$ spin liquid with a small circular spinon Fermi surface and approximate Dirac-cones at low but finite energies.

Quantum and Classical Phases of the Pyrochlore Heisenberg Model with Competing Interactions

We investigate the quantum Heisenberg model on the pyrochlore lattice for a generic spin S in the presence of nearest-neighbor J1 and second-nearest-neighbor J2 exchange interactions. By employing the pseudofermion functional renormalization group method, we find, for S=1/2 and S=1, an extended quantum-spin-liquid phase centered around J2=0, which is shown to be robust against the introduction of breathing anisotropy. The effects of temperature, quantum fluctuations, breathing anisotropies, and a J2 coupling on the nature of the scattering profile, and the pinch points, in particular, are studied. For the magnetic phases of the J1−J2 model, quantum fluctuations are shown to renormalize phase boundaries compared to the classical model and to modify the ordering wave vectors of spiral magnetic states, while no new magnetic orders are stabilized.18 MoreReceived 5 April 2018Revised 11 October 2018DOI:https://doi.org/10.1103/PhysRevX.9.011005Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasFrustrated magnetismQuantum spin liquidPhysical SystemsQuantum spin modelsTechniquesFunctional renormalization groupCondensed Matter, Materials & Applied Physics

Phase transition of the frustrated antiferromagntic J1-J2-J3 spin-1/2 Heisenberg model on a simple cubic lattice

We have comprehensively investigated the frustrated J1-J2-J3 Heisenberg model on a simple cubic lattice. This model allows three regimes of magnetic order, viz., (π, π, π), (0, π, π) and (0, 0, π,), denoted as AF1, AF2, and AF3, respectively. The effects of the interplay of neighboring couplings on the model are studied in the entire temperature range. The zero temperature magnetic properties of this model are discussed utilizing the linear spin wave (LSW) theory, nonlinear spin wave (NLSW) theory, and Green’s function (GF) method. The zero temperature phase diagrams evaluated by the LSW and NLSW methods are illustrated, and are observed to exhibit different parameter boundaries. In certain regions and along the parameter boundaries, the possible phase transformations driven by the parameters are discussed. The results obtained using the LSW, NLSW, and GF methods are compared with those obtained using the series expansion (SE) method, and are observed to be in good agreement when the value of J2 is not close to the parameter boundaries. The ground state energies obtained using the LSW and NLSW methods are close to that obtained using the SE method. At finite temperatures, only the GF method is employed to evaluate the magnetic properties, and the calculated phase diagram is observed to be identical to the classical phase diagram. The results indicate that at the parameter boundaries, a temperature-driven first-order phase transition between AF1 and AF2 may occur along the boundary line. Along the AF1-AF3 and AF2-AF3 boundary lines, AF3 is less stable than AF1 and AF2. Our calculated critical temperature agrees with that obtained using Monte Carlo simulations and pseudofermion functional renormalization group scheme.

Renormalization group analysis of dipolar Heisenberg model on square lattice

We present a detailed functional renormalization group analysis of spin-1/2 dipolar Heisenberg model on square lattice. This model is similar to the well known $J_1$-$J_2$ model and describes the pseudospin degrees of freedom of polar molecules confined in deep optical lattice with long-range anisotropic dipole-dipole interactions. Previous study of this model based on tensor network ansatz indicates a paramagnetic ground state for certain dipole tilting angles which can be tuned in experiments to control the exchange couplings. The tensor ansatz formulated on a small cluster unit cell is inadequate to describe the spiral order, and therefore the phase diagram at high azimuthal tilting angles remains undetermined. Here we obtain the full phase diagram of the model from numerical pseudofermion functional renormalization group calculations. We show that an extended quantum paramagnetic phase is realized between the N\'{e}el and stripe/spiral phase. In this region, the spin susceptibility flows smoothly down to the lowest numerical renormalization group scales with no sign of divergence or breakdown of the flow, in sharp contrast to the flow towards the long-range ordered phases. Our results provide further evidence that the dipolar Heisenberg model is a fertile ground for quantum spin liquids.

Effects of two-loop contributions in the pseudofermion functional renormalization group method for quantum spin systems

We implement an extension of the pseudofermion functional renormalization group method for quantum spin systems that takes into account two-loop diagrammatic contributions. An efficient numerical treatment of the additional terms is achieved within a nested graph construction which recombines different one-loop interaction channels. In order to be fully self-consistent with respect to self-energy corrections, we also include certain three-loop terms of Katanin type. We first apply this formalism to the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice and benchmark our results against the previous one-loop plus Katanin approach. Even though the renormalization group (RG) equations undergo significant modifications when including the two-loop terms, the magnetic phase diagram, comprising N\'eel ordered and collinear ordered phases separated by a magnetically disordered regime, remains remarkably unchanged. Only the boundary position between the disordered and the collinear phases is found to be moderately affected by two-loop terms. On the other hand, critical RG scales, which we associate with critical temperatures ${T}_{\text{c}}$, are reduced by a factor of $\ensuremath{\sim}2$ indicating that the two-loop diagrams play a significant role in enforcing the Mermin-Wagner theorem. Improved estimates for critical temperatures are also obtained for the Heisenberg ferromagnet on the three-dimensional simple cubic lattice where errors in ${T}_{\text{c}}$ are reduced by $\ensuremath{\sim}34%$. These findings have important implications for the quantum phase diagrams calculated within the previous one-loop plus Katanin approach which turn out to be already well converged.

Functional renormalization group approach to SU(N) Heisenberg models: Real-space renormalization group at arbitrary N

The pseudofermion functional renormalization group (pf-FRG) is one of the few numerical approaches that has been demonstrated to quantitatively determine the ordering tendencies of frustrated quantum magnets in two and three spatial dimensions. The approach, however, relies on a number of presumptions and approximations, in particular the choice of pseudofermion decomposition and the truncation of an infinite number of flow equations to a finite set. Here we generalize the pf-FRG approach to $\mathrm{SU}(N)$-spin systems with arbitrary $N$ and demonstrate that the scheme becomes exact in the large-$N$ limit. Numerically solving the generalized real-space renormalization group equations for arbitrary $N$, we can make a stringent connection between the physically most significant case of SU(2) spins and more accessible $\mathrm{SU}(N)$ models. In a case study of the square-lattice $\mathrm{SU}(N)$ Heisenberg antiferromagnet, we explicitly demonstrate that the generalized pf-FRG approach is capable of identifying the instability indicating the transition into a staggered flux spin liquid ground state in these models for large, but finite, values of $N$. In a companion paper [Roscher et al., Phys. Rev. B 97, 064416 (2018)] we formulate a momentum-space pf-FRG approach for $\mathrm{SU}(N)$ spin models that allows us to explicitly study the large-$N$ limit and access the low-temperature spin liquid phase.

Functional renormalization group approach to SU(N) Heisenberg models: Momentum-space renormalization group for the large- N limit

In frustrated magnetism, making a stringent connection between microscopic spin models and macroscopic properties of spin liquids remains an important challenge. A recent step towards this goal has been the development of the pseudofermion functional renormalization group approach (pf-FRG) which, building on a fermionic parton construction, enables the numerical detection of the onset of spin liquid states as temperature is lowered. In this work, focusing on the SU(N) Heisenberg model at large N, we extend this approach in a way that allows us to directly enter the low-temperature spin liquid phase, and to probe its character. Our approach proceeds in momentum space, making it possible to keep the truncation minimalistic, while also avoiding the bias introduced by an explicit decoupling of the fermionic parton interactions into a given channel. We benchmark our findings against exact mean-field results in the large-N limit, and show that even without prior knowledge the pf-FRG approach identifies the correct mean-field decoupling channel. On a technical level, we introduce an alternative finite temperature regularization scheme that is necessitated to access the spin liquid ordered phase. In a companion paper arXiv:1711.02182 we present a different set of modifications of the pf-FRG scheme that allow us to study SU(N) Heisenberg models (using a real-space RG approach) for arbitrary values of N, albeit only up to the phase transition towards spin liquid physics.

Signatures of a gearwheel quantum spin liquid in a spin- 12 pyrochlore molybdate Heisenberg antiferromagnet

We theoretically investigate the low-temperature phase of the recently synthesized Lu$_2$Mo$_2$O$_5$N$_2$ material, an extraordinarily rare realization of a $S=1/2$ three-dimensional pyrochlore Heisenberg antiferromagnet in which Mo$^{5+}$ are the $S=1/2$ magnetic species. Despite a Curie-Weiss temperature ($\Theta_{\rm CW}$) of $-121(1)$ K, experiments have found no signature of magnetic ordering $or$ spin freezing down to $T^*\approx0.5$ K. Using density functional theory, we find that the compound is well described by a Heisenberg model with exchange parameters up to third nearest neighbors. The analysis of this model via the pseudofermion functional renormalization group method reveals paramagnetic behavior down to a temperature of at least $T=|\Theta_{\rm CW}|/100$, in agreement with the experimental findings hinting at a possible three-dimensional quantum spin liquid. The spin susceptibility profile in reciprocal space shows momentum-dependent features forming a "gearwheel" pattern, characterizing what may be viewed as a molten version of a chiral noncoplanar incommensurate spiral order under the action of quantum fluctuations. Our calculated reciprocal space susceptibility maps provide benchmarks for future neutron scattering experiments on single crystals of Lu$_2$Mo$_2$O$_5$N$_2$.

Numerical treatment of spin systems with unrestricted spin length S : A functional renormalization group study

We develop a generalized pseudo-fermion functional renormalization group (PFFRG) approach that can be applied to arbitrary Heisenberg models with spins ranging from the quantum case $S=1/2$ to the classical limit $S\rightarrow\infty$. Within this framework, spins of magnitude $S$ are realized by implementing $M=2S$ copies of spin-1/2 degrees of freedom on each lattice site. We confirm that even without explicitly projecting onto the highest spin sector of the Hilbert space, ground states tend to select the largest possible local spin magnitude. This justifies the average treatment of the pseudo fermion constraint in previous spin-1/2 PFFRG studies. We apply this method to the antiferromagnetic $J_1$-$J_2$ honeycomb Heisenberg model with nearest neighbor $J_1>0$ and second neighbor $J_2>0$ interactions. Mapping out the phase diagram in the $J_2/J_1$-$S$ plane we find that upon increasing $S$ quantum fluctuations are rapidly decreasing. In particular, already at $S=1$ we find no indication for a magnetically disordered phase. In the limit $S\rightarrow\infty$, the known phase diagram of the classical system is exactly reproduced. More generally, we prove that for $S\rightarrow\infty$ the PFFRG approach is identical to the Luttinger-Tisza method.

Functional renormalization group analysis of Dzyaloshinsky-Moriya and Heisenberg spin interactions on the kagome lattice

We investigate the effects of Dzyaloshinsky-Moriya (DM) interactions on the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ kagome-Heisenberg model using the pseudofermion functional renormalization group (PFFRG) technique. In order to treat the off-diagonal nature of DM interactions, we develop an extended PFFRG scheme. We benchmark this approach in parameter regimes that have previously been studied with other methods and find good agreement of the magnetic phase diagram. Particularly, finite DM interactions are found to stabilize all types of noncollinear magnetic orders of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model ($\mathbf{q}=0, \sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$, and cuboc orders) and shrink the extents of magnetically disordered phases. We discuss our results in the light of the mineral herbertsmithite which has been experimentally predicted to host a quantum spin liquid at low temperatures. Our PFFRG data indicate that this material lies in close proximity to a quantum critical point. In parts of the experimentally relevant parameter regime for herbertsmithite, the spin-correlation profile is found to be in good qualitative agreement with recent inelastic-neutron-scattering data.

Competing magnetic orders and spin liquids in two- and three-dimensional kagome systems: Pseudofermion functional renormalization group perspective

Quantum magnets on kagome lattice geometries in two and three spatial dimensions are archetypal examples of spin systems in which geometric frustration inhibits conventional magnetic ordering and instead benefits the emergence of long-range entangled spin liquids at low temperature. Here we employ a recently developed pseudo-fermion functional renormalization group (pf-FRG) approach to study the low-temperature quantum magnetism of kagome and hyperkagome spin systems with exchange interactions beyond the nearest neighbor coupling. We find that next-nearest neighbor couplings stabilize a variety of magnetic orders as well as induce additional spin liquid regimes giving rise to rather rich phase diagrams, which we characterize in detail. On a technical level, we find that the pf-FRG approach is in excellent quantitative agreement with high-temperature series expansions over their range of validity and it exhibits a systematic finite-size convergence in the temperature regime below. We discuss notable advantages and some current limitations of the pf-FRG approach in the ongoing search for unconventional forms of quantum magnetism.

Intertwined nematic orders in a frustrated ferromagnet

We investigate the quantum phases of the frustrated spin-$\frac{1}{2}$ $J_1$-$J_2$-$J_3$ Heisenberg model on the square lattice with ferromagnetic $J_1$ and antiferromagnetic $J_2$ and $J_3$ interactions. Using the pseudo-fermion functional renormalization group technique, we find an intermediate paramagnetic phase located between classically ordered ferromagnetic, stripy antiferromagnetic, and incommensurate spiral phases. We observe that quantum fluctuations lead to significant shifts of the spiral pitch angles compared to the classical limit. By computing the response of the system with respect to various spin rotation and lattice symmetry-breaking perturbations, we identify a complex interplay between different nematic spin states in the paramagnetic phase. While retaining time-reversal invariance, these phases either break spin-rotation symmetry, lattice-rotation symmetry, or a combination of both. We therefore propose the $J_1$-$J_2$-$J_3$ Heisenberg model on the square lattice as a paradigmatic example where different intimately connected types of nematic orders emerge in the same model.

Functional renormalization group for three-dimensional quantum magnetism

We formulate a pseudofermion functional renormalization group (PFFRG) scheme to address frustrated quantum magnetism in three dimensions. In a scenario where many numerical approaches fail due to sign problem or small system size, three-dimensional (3D) PFFRG allows for a quantitative investigation of the quantum spin problem and its observables. We illustrate 3D PFFRG for the simple cubic ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ quantum Heisenberg antiferromagnet, and benchmark it against other approaches, if available.

Spin liquid nature in the HeisenbergJ1−J2triangular antiferromagnet

We investigate the spin-$\frac{1}{2}$ Heisenberg model on the triangular lattice in the presence of nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ antiferromagnetic couplings. Motivated by recent findings from density-matrix renormalization group (DMRG) claiming the existence of a gapped spin liquid with signatures of spontaneously broken lattice point group symmetry [Zhu and White, Phys. Rev. B 92, 041105 (2015); Hu, Gong, Zhu, and Sheng, Phys. Rev. B 92, 140403 (2015)], we employ the variational Monte Carlo (VMC) approach to analyze the model from an alternative perspective that considers both magnetically ordered and paramagnetic trial states. We find a quantum paramagnet in the regime $0.08\lesssim J_2/J_1\lesssim 0.16$, framed by $120^{\circ}$ coplanar (stripe collinear) antiferromagnetic order for smaller (larger) $J_2/J_1$. By considering the optimization of spin-liquid wave functions of a different gauge group and lattice point group content as derived from Abrikosov mean-field theory, we obtain the gapless $U(1)$ Dirac spin liquid as the energetically most preferable state in comparison to all symmetric or nematic gapped $\mathbb{Z}_{2}$ spin liquids so far advocated by DMRG. Moreover, by the application of few Lanczos iterations, we find the energy to be the same as the DMRG result within error-bars. To further resolve the intriguing disagreement between VMC and DMRG, we complement our methodological approach by the pseudofermion functional renormalization group (PFFRG) to compare the spin structure factors for the paramagnetic regime calculated by VMC, DMRG, and PFFRG. This model promises to be an ideal test-bed for future numerical refinements in tracking the long-range correlations in frustrated magnets.

Quantum phase transitions in frustrated magnetic systems

We review our recent work on quantum phase transitions in frustrated magnetic systems. In the first part a Pseudo Fermion Functional Renormalization Group (PFFRG) method is presented. By using an exact representation of spin 1/2 operators in terms of pseudofermions a quantum spin Hamiltonian may be mapped onto an interacting fermion system. For the latter an FRG treatment is employed. The results for the J1−J2 model and similar models of frustrated interaction show phase diagrams in agreement with those obtained by other methods, but give more detailed information on the nature of correlations, in particular in the non-magnetic phases. Applications of PFFRG to geometrically frustrated systems and to highly anisotropic Kitaev type models are also reported. In the second part the derivation of quantum spin models from the microscopic many-body Hamiltonian is discussed. The results for multiband systems with strong spin-orbit interaction encountered in the iridates class of compounds are shown to resolve some of the questions posed by experiment.

Renormalization group analysis of competing quantum phases in theJ1-J2Heisenberg model on the kagome lattice

Recent discoveries in neutron scattering experiments for Kapellasite and Herbertsmithite as well as theoretical calculations of possible spin liquid phases have revived interest in magnetic phenomena on the kagome lattice. We study the quantum phase diagram of the $S=1/2$ Heisenberg kagome model as a function of nearest neighbor coupling ${J}_{1}$ and second neighbor coupling ${J}_{2}$. Employing the pseudofermion functional renormalization group (PFFRG), we find four types of magnetic quantum order ($\mathbf{q}=0$ order, cuboc order,ferromagnetic order, and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order) as well as extended magnetically disordered regions by which we specify the possible parameter regime for Kapellasite. In the disordered regime $\frac{{J}_{2}}{{J}_{1}}\ensuremath{\ll}1$, the flatness of the magnetic susceptibility at the zone boundary which is observed for Herbertsmithite can be reconciled with the presence of small ${J}_{2}>0$ coupling. In particular, we analyze the dimer susceptibilities related to different valence-bond crystal (VBC) patterns, which are strongly inhomogeneous, indicating the rejection of VBC order in the RG flow.

Finite-temperature phase diagram of the Heisenberg-Kitaev model

We discuss the finite-temperature phase diagram of the Heisenberg-Kitaev model on the hexagonal lattice, which has been suggested to describe the spin-orbital exchange in the Mott-insulating iridate Na${}_{2}$IrO${}_{3}$. The model exhibits magnetically ordered ground states well beyond the isotropic Heisenberg limit as well as a gapless spin-liquid phase around the anisotropic Kitaev limit. Using a pseudofermion functional renormalization group (RG) approach we extract both the Curie-Weiss scale and the critical ordering scale from the RG flow of the magnetic susceptibility. The Curie-Weiss scale switches sign---indicating a transition of the dominant exchange from antiferromagnetic to ferromagnetic---deep in the magnetically ordered regime for which we find no significant frustration. We discuss our results in light of recent susceptibility measurements for Na${}_{2}$IrO${}_{3}$.