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arXiv:1905.01060 · Physical review. B./Physical review. B

Cluster functional renormalization group and absence of a bilinear spin liquid in the J1−J2 Heisenberg model

Dietrich Roscher, Nico Gneist, Michael M. Scherer, Simon Trebst, Sebastian Diehl

Published Sep 13, 2019• 20 citations
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Abstract

The pseudofermion functional renormalization group (pf-FRG) has been put forward as a semianalytical scheme that, for a given microscopic spin model, allows to discriminate whether the low-temperature states exhibit magnetic ordering or a tendency toward the formation of quantum spin liquids. However, the precise nature of the putative spin-liquid ground state has remained hard to infer from the original (single-site) pf-FRG scheme. Here, we introduce a cluster pf-FRG approach, which allows for a more stringent connection between a microscopic spin model and its low-temperature spin-liquid ground states. In particular, it allows to calculate spatially structured fermion bilinear expectation values on spatial clusters, which are formed by splitting the original lattice into several sublattices, thereby allowing for the positive identification of a family of bilinear spin-liquid states. As an application of this cluster pf-FRG approach, we consider the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)$ Heisenberg model on a square lattice, which is a paradigmatic example for a frustrated quantum magnet exhibiting quantum spin-liquid behavior for intermediate coupling strengths. In the well-established large-$N$ limit of this model, we show that our approach correctly captures the emergence of the $\ensuremath{\pi}$-flux spin-liquid state at low temperatures. For small $N$, where the precise nature of the ground state remains controversial, we focus on the widely studied case of $N=2$, for which we determine the low-temperature phase diagram near the strongly frustrated regime after implementing the fermion-number constraint by the flowing Popov-Fedotov method. Our results suggest that the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model does not support the formation of a fermion bilinear spin-liquid state.

DOI
10.1103/physrevb.100.125130

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The pseudofermion functional renormalization group (pf-FRG) has been put forward as a semianalytical scheme that, for a given microscopic spin model, allows to discriminate whether the low-temperature states exhibit magnetic ordering or a tendency toward the formation of quantum spin liquids. However, the precise nature of the putative spin-liquid ground state has remained hard to infer from the original (single-site) pf-FRG scheme. Here, we introduce a cluster pf-FRG approach, which allows for a more stringent connection between a microscopic spin model and its low-temperature spin-liquid ground states. In particular, it allows to calculate spatially structured fermion bilinear expectation values on spatial clusters, which are formed by splitting the original lattice into several sublattices, thereby allowing for the positive identification of a family of bilinear spin-liquid states. As an application of this cluster pf-FRG approach, we consider the J1\ensuremath-J2SU(N){J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)J1​\ensuremath-J2​SU(N) Heisenberg model on a square lattice, which is a paradigmatic example for a frustrated quantum magnet exhibiting quantum spin-liquid behavior for intermediate coupling strengths. In the well-established large-NNN limit of this model, we show that our approach correctly captures the emergence of the \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux spin-liquid state at low temperatures. For small NNN, where the precise nature of the ground state remains controversial, we focus on the widely studied case of N=2N=2N=2, for which we determine the low-temperature phase diagram near the strongly frustrated regime after implementing the fermion-number constraint by the flowing Popov-Fedotov method. Our results suggest that the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg model does not support the formation of a fermion bilinear spin-liquid state.
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 SU(N)\mathrm{SU}(N)SU(N)-spin systems with arbitrary NNN and demonstrate that the scheme becomes exact in the large-NNN limit. Numerically solving the generalized real-space renormalization group equations for arbitrary NNN, we can make a stringent connection between the physically most significant case of SU(2) spins and more accessible SU(N)\mathrm{SU}(N)SU(N) models. In a case study of the square-lattice SU(N)\mathrm{SU}(N)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 NNN. In a companion paper [Roscher et al., Phys. Rev. B 97, 064416 (2018)] we formulate a momentum-space pf-FRG approach for SU(N)\mathrm{SU}(N)SU(N) spin models that allows us to explicitly study the large-NNN limit and access the low-temperature spin liquid phase.
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)\mathrm{SU}(N)SU(N) Heisenberg model at large NNN, 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-NNN 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 [Buessen et al., Phys. Rev. B 97, 064415 (2018)] we present a different set of modifications of the pf-FRG scheme that allow us to study SU(N)\mathrm{SU}(N)SU(N) Heisenberg models (using a real-space RG approach) for arbitrary values of NNN, albeit only up to the phase transition towards spin liquid physics.
Evidence of valence bond solid and quantum paramagnetism in the J1J_1J1​--J2J_2J2​--J3J_3J3​ Heisenberg model from pseudofermion functional renormalization group
Motivated by recent tensor network studies reporting a quantum spin liquid (QSL) within the quantum paramagnetic (QP) regime, we investigate numerically the zero-temperature phase diagram of the J1J_1J1​--J2J_2J2​--J3J_3J3​ Heisenberg model on the square lattice using the pseudofermion functional renormalization group (pf-fRG). Thanks to improved frequency resolution and larger lattice truncations, we are able to explore a significantly larger parameter space than previous pf-fRG work to map out the phase boundary of an extended QP region. We further resolve possible bond order within the QP regime by measuring the dimer response to small deformations in J1J_1J1​, i.e. bond anisotropy that breaks the point-group symmetry of the Hamiltonian. The functional renormalization group flow of the dimer susceptibility suggests that the QP window splits into two: a region where it grows continuously down to the lowest energy scale to signal valence bond solid (VBS) order, and another region where it reaches a peak at a finite scale and is subsequently suppressed to indicate a QSL.
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 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice and to the antiferromagnetic nearest-neighbor Heisenberg model on the kagome lattice. For the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, we find that in the regime of maximal frustration a SU(2) $\ensuremath{\pi}$-flux state with Dirac spinons yields the largest mean-field amplitudes. For the kagome model, we identify a gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid with a small circular spinon Fermi surface and approximate Dirac cones at low but finite energies.
When sizable quantum fluctuations and strong frustration mechanisms act in concert to repel the formation of conventional long-range order in quantum magnets, they can make way for massively entangled spin liquid phases which may imbue the material with extraordinary properties. \nThe search for such curious phases of matter has proceeded for several decades, gaining extra momentum some fifteen years ago when Kitaev proposed an analytically solvable model for a quantum spin liquid with anyonic excitations on the basis of realistic microscopic spin exchange terms [Kitaev, Annals of Physics 321, 2 (2006)]. Attempts to identify different models or materials which harbor quantum spin liquid ground states have kept researchers -- experimentalists and theorists alike -- in suspense ever since. However, the simulation of quantum many-body systems poses a serious challenge even to modern numerical techniques, particularly in the case of frustrated quantum magnetism in three spatial dimensions. Such models evade tractability by many established approaches, leaving a methodological void. \n \nIn this thesis, we report on recent progress in cutting-edge implementations of the pseudo-fermion functional renormalization group (pf-FRG), which has originally been proposed by Reuther and Wölfle in the context of two-dimensional frustrated quantum magnetism with highly symmetric spin interactions [Reuther and Wölfle, Phys. Rev. B 81, 144410 (2010)]. Reflecting the growing interest in models with SU(2) symmetry-breaking spin exchange terms that has followed the unearthing of Kitaev's honeycomb model, we present a generalized implementation of the pf-FRG which is suited to numerically simulate arbitrary microscopic models with diagonal or off-diagonal two-spin interactions, even in three-dimensional frustrated quantum magnets. We provide insight into the inner workings of the method which has emerged over the course of the last couple of years, arguing that the pf-FRG formalism simultaneously combines aspects of a large-S expansion as well as a large-N expansion on equal footing, thus being able to resolve the subtle interplay between magnetic ordering tendencies and disruptive quantum fluctuations. \n \nMoreover, on a case by case basis we explore the stability of quantum spin liquids in paradigmatic models of frustrated quantum magnetism and elucidate the joint action of geometric frustration, exchange frustration, and quantum fluctuations to inhibit the formation of magnetic long-range order. Examples include: (i) a Heisenberg spin model on the three-dimensional diamond lattice where geometric frustration arises from antiferromagnetic next-nearest neighbor interactions. The additional competition with nearest neighbor interactions leads to an unusually large ground state degeneracy already on classical level. The quantum-to-classical transition is studied by systematically varying the spin length and the different roles of quantum fluctuations and thermal fluctuations are discussed. The theory is applied to interpret experiments on the spin-liquid candidate NiRh2O4; (ii) a Heisenberg antiferromagnet on the face centered cubic lattice which is augmented by competing bond-directional interactions, evolving around the concurrent manifestation of geometric frustration and exchange frustration. A situation is identified where both mechanisms collude to give rise to an unusually large degree of frustration over a wide parameter regime. We discuss related experimental findings on the highly frustrated iridate compound Ba2CeIrO6; (iii) a Heisenberg antiferromagnet on the kagome lattice with additional Dzyaloshinskii-Moriya interactions. The robustness of the unperturbed kagome antiferromagnet's spin liquid ground state against low-symmetry Dzyaloshinskii-Moriya interactions is investigated. Implications for the spin-liquid candidate herbertsmithite are discussed.
We use our recently developed functional renormalization group (FRG) approach for quantum spin systems to investigate the phase diagram of the frustrated J1J2J3{J}_{1}{J}_{2}{J}_{3}J1​J2​J3​ 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 44%4.