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arXiv:1409.4034

Comment on "Topological excitations and the dynamic structure factor of spin liquids on the kagome lattice" (Punk, M., Chowdhury, D. & Sachdev, S. Nature Physics 10, 289-293 (2014))

Vasily R. Shaginyan, M. Ya. Amusia, J. W. Clark, G. S. Japaridze, ALFRED Z. MSEZANE, Konstantin G. Popov, V. A. Stephanovich, MIKHAIL V. ZVEREV, VICTOR A. KHODEL

Published Sep 14, 2014• 0 citations
arXivHTMLPDF

Abstract

The authors of a recent paper evidently take the view that the whole of progress made toward a theoretical understanding of the physics of quantum spin liquids (QSL) is associated with models of the kind proposed and applied in their present work. As motivation for this work, they observe that in contrast to existing theoretical models of both gapped and gapless spin liquids, which give rise to sharp dispersive features in the dynamic structure factor, the measured dynamic structure factor reveals an excitation continuum that is remarkably flat as a function of frequency. They go on to assert that "so far, the only theoretical model for a spin liquid state on the kagome lattice which naturally gives rise to a flat excitation band at low energies consists of the $Z_2$ spin liquids". Here we point out that there already exists a different and demonstrably successful approach to the QSL problem that does naturally feature a flat band.

DOI
10.48550/arxiv.1409.4034

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Dynamical Spectral Function of the Kagome Quantum Spin Liquid

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Quantum spin liquids (QSLs) host exotic fractionalized magnetic and gauge-field excitations whose microscopic origins and experimental verification remain frustratingly elusive. In the absence of static magnetic order, the spin excitation spectrum constitutes the crucial probe of QSL behavior, but its theoretical computation is a serious challenge. Here we employ state-of-the-art tensor-network methods to obtain the full dynamical spectral function of the $J_1$-$J_2$ kagome Heisenberg model and benchmark our results by tracking their evolution across the magnetically ordered and QSL phases. Reducing $|J_2|/J_1$ causes increasingly strong spin-wave renormalization, flattening these modes then merging them into a continuum characteristic of deconfined spinons at all finite energies in the QSL. The low-energy continuum and the occurrence of gap closure at multiple high-symmetry points identify this gapless QSL as the U(1) Dirac spin liquid. These results establish a unified understanding of spin excitations in highly frustrated quantum magnets and provide clear spectral fingerprints for experimental detection in candidate kagome QSL materials.

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Quantum spin liquids exhibit fractionalized spin excitations as a consequence of strong quantum many-body effects. The kagome antiferromagnetic Heisenberg model is a promising candidate for a quantum spin-liquid ground state; however, the nature of its excitation spectrum remains controversial, particularly regarding the presence of a spin gap and the gauge structure coupled to fractional quasiparticles. To address these issues, parton approaches have been extensively employed, where spin operators are represented in terms of fermionic or bosonic quasiparticles within the Abrikosov fermion and Schwinger boson frameworks. Thus far, these approaches have been pursued independently, and it has remained unclear how the results obtained from these frameworks compare, particularly with respect to the spin dynamics and gauge structure of the kagome antiferromagnet. Here, we investigate the dynamical spin structure factor of the antiferromagnetic Heisenberg model with a Dzyaloshinskii-Moriya interaction on the kagome lattice, relevant to herbertsmithite, by employing both approaches. We find that the dynamical spin structure factor obtained from the Abrikosov fermion mean-field theory exhibits dome-shaped features, and that its continuum structure significantly depends on the gauge structure of the spin-liquid ansatz. On the other hand, the Schwinger boson mean-field theory yields a concave-down structure in the low-energy region, distinct from that obtained using the Abrikosov fermion approach. Moreover, incorporating many-body effects beyond the mean-field approximation substantially reduces the low-energy gap and enhances the low-energy spectral weight, consistent with experimental observations. Our results suggest the importance of many-body effects in the Schwinger boson theory for capturing the low-energy spin dynamics of kagome antiferromagnets.

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Theory of Quantum Spin Liquids in Quasi 1-D Systems

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Ever since the introduction of resonating valence bond states by P.\\ W.\\ Anderson in 1970s, the search for exotic many--body quantum behavior in peculiar states of matter known as quantum spin liquids has been a central challenge in the field of condensed matter physics. An important searching ground for such \\emph{quantum--ness} is the low energy physics of frustrated magnets. In a quantum system, frustration---the inability of a physical system to achieve a global state that consistently minimizes its energy locally---enables the unavoidable quantum fluctuations to hybridize an extensive number of classical states into highly entangled quantum states at low temperatures. It can be argued that this is the most practical approach to achieve genuine many--body quantum behavior at large scales. The fundamental source of frustration in many physical systems, is the existence of local non--commuting terms. 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I illustrate that the standard Heisenberg antiferromagnet Hamiltonian over kagome strip is an extended gapless quantum phase, that is well characterized by two fermionic/bosonic gapless modes and power--law decaying spin and bond--energy correlations. I also demonstrate that the correlation functions oscillate at tunably incommensurate wave vectors. It turns out that this phase can be identified by a particular marginal instability of a two-band spinon Fermi surface coupled to an emergent U(1) gauge field. This interpretation is supported by analytic Abelian bosonization and with extensive numerical large--scale density matrix renormalization group study as well as variational Monte Carlo calculations on Gutzwiller ans\\"atz wave functions. This intriguing result is the first numerical demonstration of emergent fermionic spinons in a simple SU(2) invariant nearest-neighbor Heisenberg model beyond the strictly 1D (Bethe chain) limit. The unexpected success of the fermionic spinons in describing the low energy physics of kagome antiferromagnet, reintroduces the questions about the validity of spinon physics as an effective theory of quantum spin liquids. However, numerical methods are still far behind the theoretical advances. And still even today, an accurate description of the quantum spin liquid states using tensor network methods is notoriously challenging. It is known that for large quasi-1D systems, the density matrix renormalization group and related methods usually require significant computational resources and sometimes fail to converge to a satisfactory state. On the other hand, variational wavefunctions acquired from the Gutzwiller projection of gaussian fermionic theories have long served as both a theoretical starting point for the construction of such spin liquid states and as an inspiration for numerical variational Monte Carlo (VMC) to calculate observables of interest. Noting this observation I examine a different method by exploring the possibility of constructing a matrix product state (MPS) representation for a Gutzwiller--projected state from two given MPS representations of gaussian fermionic theories in the \\ref{chap:gutz} chapter. I investigate the complexity of different approaches to achieve Gutzwiller projection for MPSs and introduce the novel algorithm which we call the Guzwiller zipper method. The performance of the algorithm is tested against two copies of a single half-filled band of spin-1/2 fermionic spinons. In a successful attempt to describe the nature of spin liquid states on quasi-1D strips of triangular and kagome-like lattices, I apply this method to two MPS of multi-band fermionic spinon theories and compare with the complexity of the traditional VMC approach. In particular, we methodically disprove the conjecture of a spinon fermi surface spin liquid for the triangular lattice. Finally, I conclude this thesis by laying out a bird's eye view of the current and future of spin liquid physics in kagome and triangular lattices and pointing out numerous possible applications of the novel Gutzwiller zipper method. In addition, I would also discuss its possible extensions to more complicated tensor networks as well as versions for higher spatial dimensions.

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We calculate dynamical spin structure factors for gapped chiral spin liquid states in the spin-1/2 Heisenberg antiferromagnet on the kagome lattice using Schwinger-boson mean-field theory. In contrast to static (equal-time) structure factors, the dynamical structure factor shows clear signatures of time-reversal symmetry breaking for chiral spin liquid states. In particular, momentum inversion $\mathbit{k}\ensuremath{\rightarrow}\ensuremath{-}\mathbit{k}$ symmetry as well as the sixfold rotation symmetry around the $\mathrm{\ensuremath{\Gamma}}$ point are lost. We highlight other interesting features, such as a relatively flat onset of the two-spinon continuum for the cuboc1 state. Our work is based on the projective symmetry group classification of time-reversal symmetry breaking Schwinger-boson mean-field states by Messio, Lhuillier, and Misguich.

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The nature of the quantum spin-liquid state in Herbertsmithite

Matthias Punk, Debanjan Chowdhury, Subir Sachdev A

Recent neutron scattering experiments on the layered spin-1/2 kagome lattice antiferromagnet Herbertsmithite revealed the first signature of fractionalized excitations in a quantum spin liquid state. The precise nature of this state remains unclear, however. Indeed, mean-field models of gapped as well as gapless spin liquids exhibit sharp features in the dynamic structure factor, none of which have been observed in experiment.

Bulletin of the American Physical Society•Mar 6, 2014•0 citations
The authors of a recent paper evidently take the view that the whole of progress made toward a theoretical understanding of the physics of quantum spin liquids (QSL) is associated with models of the kind proposed and applied in their present work. As motivation for this work, they observe that in contrast to existing theoretical models of both gapped and gapless spin liquids, which give rise to sharp dispersive features in the dynamic structure factor, the measured dynamic structure factor reveals an excitation continuum that is remarkably flat as a function of frequency. They go on to assert that "so far, the only theoretical model for a spin liquid state on the kagome lattice which naturally gives rise to a flat excitation band at low energies consists of the Z2Z_2Z2​ spin liquids". Here we point out that there already exists a different and demonstrably successful approach to the QSL problem that does naturally feature a flat band.
Quantum spin liquids (QSLs) host exotic fractionalized magnetic and gauge-field excitations whose microscopic origins and experimental verification remain frustratingly elusive. In the absence of static magnetic order, the spin excitation spectrum constitutes the crucial probe of QSL behavior, but its theoretical computation is a serious challenge. Here we employ state-of-the-art tensor-network methods to obtain the full dynamical spectral function of the J1J_1J1​-J2J_2J2​ kagome Heisenberg model and benchmark our results by tracking their evolution across the magnetically ordered and QSL phases. Reducing ∣J2∣/J1|J_2|/J_1∣J2​∣/J1​ causes increasingly strong spin-wave renormalization, flattening these modes then merging them into a continuum characteristic of deconfined spinons at all finite energies in the QSL. The low-energy continuum and the occurrence of gap closure at multiple high-symmetry points identify this gapless QSL as the U(1) Dirac spin liquid. These results establish a unified understanding of spin excitations in highly frustrated quantum magnets and provide clear spectral fingerprints for experimental detection in candidate kagome QSL materials.
Quantum spin liquids exhibit fractionalized spin excitations as a consequence of strong quantum many-body effects. The kagome antiferromagnetic Heisenberg model is a promising candidate for a quantum spin-liquid ground state; however, the nature of its excitation spectrum remains controversial, particularly regarding the presence of a spin gap and the gauge structure coupled to fractional quasiparticles. To address these issues, parton approaches have been extensively employed, where spin operators are represented in terms of fermionic or bosonic quasiparticles within the Abrikosov fermion and Schwinger boson frameworks. Thus far, these approaches have been pursued independently, and it has remained unclear how the results obtained from these frameworks compare, particularly with respect to the spin dynamics and gauge structure of the kagome antiferromagnet. Here, we investigate the dynamical spin structure factor of the antiferromagnetic Heisenberg model with a Dzyaloshinskii-Moriya interaction on the kagome lattice, relevant to herbertsmithite, by employing both approaches. We find that the dynamical spin structure factor obtained from the Abrikosov fermion mean-field theory exhibits dome-shaped features, and that its continuum structure significantly depends on the gauge structure of the spin-liquid ansatz. On the other hand, the Schwinger boson mean-field theory yields a concave-down structure in the low-energy region, distinct from that obtained using the Abrikosov fermion approach. Moreover, incorporating many-body effects beyond the mean-field approximation substantially reduces the low-energy gap and enhances the low-energy spectral weight, consistent with experimental observations. Our results suggest the importance of many-body effects in the Schwinger boson theory for capturing the low-energy spin dynamics of kagome antiferromagnets.
Ever since the introduction of resonating valence bond states by P.\\ W.\\ Anderson in 1970s, the search for exotic many--body quantum behavior in peculiar states of matter known as quantum spin liquids has been a central challenge in the field of condensed matter physics. An important searching ground for such \\emph{quantum--ness} is the low energy physics of frustrated magnets. In a quantum system, frustration---the inability of a physical system to achieve a global state that consistently minimizes its energy locally---enables the unavoidable quantum fluctuations to hybridize an extensive number of classical states into highly entangled quantum states at low temperatures. It can be argued that this is the most practical approach to achieve genuine many--body quantum behavior at large scales. The fundamental source of frustration in many physical systems, is the existence of local non--commuting terms. In the case of antiferromagnet spin theories the simplest examples are triangles of nearest--neighboring sites. Consequently, lattices with triangles as building blocks have long been studied in hopes of discovering spin liquids physics. In particular the kagome lattice, which is composed of corner--sharing triangles, has been the center of at least four decades of spin liquid research. This is due to the underconstrained nature of system made out of corner--sharing triangles. After the first three chapters, introduction in chapter~\\ref{chap:intro}, preliminary physics in chapter~\\ref{chap:prelim}, and a brief introduction to spin--liquids in chapter~\\ref{chap:QSLs}, in chapter~\\ref{chap:kagome} I tackle a simpler version of the long-standing problem of spin--1/21/21/2 Heisenberg antiferromagnets by considering a quasi--1D lattice consisting entirely of corner--sharing triangles, \\emph{kagome strip}. I illustrate that the standard Heisenberg antiferromagnet Hamiltonian over kagome strip is an extended gapless quantum phase, that is well characterized by two fermionic/bosonic gapless modes and power--law decaying spin and bond--energy correlations. I also demonstrate that the correlation functions oscillate at tunably incommensurate wave vectors. It turns out that this phase can be identified by a particular marginal instability of a two-band spinon Fermi surface coupled to an emergent U(1) gauge field. This interpretation is supported by analytic Abelian bosonization and with extensive numerical large--scale density matrix renormalization group study as well as variational Monte Carlo calculations on Gutzwiller ans\\"atz wave functions. This intriguing result is the first numerical demonstration of emergent fermionic spinons in a simple SU(2) invariant nearest-neighbor Heisenberg model beyond the strictly 1D (Bethe chain) limit. The unexpected success of the fermionic spinons in describing the low energy physics of kagome antiferromagnet, reintroduces the questions about the validity of spinon physics as an effective theory of quantum spin liquids. However, numerical methods are still far behind the theoretical advances. And still even today, an accurate description of the quantum spin liquid states using tensor network methods is notoriously challenging. It is known that for large quasi-1D systems, the density matrix renormalization group and related methods usually require significant computational resources and sometimes fail to converge to a satisfactory state. On the other hand, variational wavefunctions acquired from the Gutzwiller projection of gaussian fermionic theories have long served as both a theoretical starting point for the construction of such spin liquid states and as an inspiration for numerical variational Monte Carlo (VMC) to calculate observables of interest. Noting this observation I examine a different method by exploring the possibility of constructing a matrix product state (MPS) representation for a Gutzwiller--projected state from two given MPS representations of gaussian fermionic theories in the \\ref{chap:gutz} chapter. I investigate the complexity of different approaches to achieve Gutzwiller projection for MPSs and introduce the novel algorithm which we call the Guzwiller zipper method. The performance of the algorithm is tested against two copies of a single half-filled band of spin-1/2 fermionic spinons. In a successful attempt to describe the nature of spin liquid states on quasi-1D strips of triangular and kagome-like lattices, I apply this method to two MPS of multi-band fermionic spinon theories and compare with the complexity of the traditional VMC approach. In particular, we methodically disprove the conjecture of a spinon fermi surface spin liquid for the triangular lattice. Finally, I conclude this thesis by laying out a bird's eye view of the current and future of spin liquid physics in kagome and triangular lattices and pointing out numerous possible applications of the novel Gutzwiller zipper method. In addition, I would also discuss its possible extensions to more complicated tensor networks as well as versions for higher spatial dimensions.
We calculate dynamical spin structure factors for gapped chiral spin liquid states in the spin-1/2 Heisenberg antiferromagnet on the kagome lattice using Schwinger-boson mean-field theory. In contrast to static (equal-time) structure factors, the dynamical structure factor shows clear signatures of time-reversal symmetry breaking for chiral spin liquid states. In particular, momentum inversion $\mathbit{k}\ensuremath{\rightarrow}\ensuremath{-}\mathbit{k}$ symmetry as well as the sixfold rotation symmetry around the $\mathrm{\ensuremath{\Gamma}}$ point are lost. We highlight other interesting features, such as a relatively flat onset of the two-spinon continuum for the cuboc1 state. Our work is based on the projective symmetry group classification of time-reversal symmetry breaking Schwinger-boson mean-field states by Messio, Lhuillier, and Misguich.
Understanding the nature of the excitation spectrum in quantum spin liquids is of fundamental importance, in particular for the experimental detection of candidate materials.However, current theoretical and numerical techniques have limited capabilities, especially in obtaining the dynamical structure factor, which gives a crucial characterization of the ultimate nature of the quantum state and may be directly assessed by inelastic neutron scattering.In this work, we investigate the low-energy properties of the S ¼ 1=2 Heisenberg model on the triangular lattice, including both nearest-neighbor J 1 and next-nearestneighbor J 2 superexchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models.For J 2 ¼ 0, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations at K points (i.e., at the corners of the Brillouin zone).The strong renormalization of the magnon branch (also including rotonlike minima around the M points, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a nontrivial magnetic π flux threading half of the triangular plaquettes.When increasing the frustrating ratio J 2 =J 1 , we detect a progressive softening of the magnon branch at M, which eventually becomes gapless within the spin-liquid phase.This feature is captured by the band structure of the unprojected wave function (with two Dirac points for each spin component).In addition, we observe an intense signal at low energies around the K points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids.