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arXiv:1311.5569 · New Journal of Physics

Signatures of spin-triplet excitations in optical conductivity of valence bond solids

Kyusung Hwang, Subhro Bhattacharjee, Yong Baek Kim

Published Dec 2, 2014• 7 citations
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Abstract

We show that the optical responses below the Mott gap can be used to probe the spin-triplet excitations in valence bond solid (VBS) phases in Mott insulators. The optical conductivity in this regime arises due to the electronic polarization mechanism via virtual electron hopping processes. We apply this mechanism to the Hubbard model with spin-orbit couplings and/or the corresponding spin model with significant Dzyaloshinskii-Moriya (DM) interactions, and compute the optical conductivity of VBS states on both ideal and deformed Kagome lattices. In case of the deformed Kagome lattice, we study the antiferromagnet, Rb$_2$Cu$_3$SnF$_{12}$ with the pinwheel VBS state. In case of the ideal Kagome lattice, we explore the optical conductivity signatures of the spin-triplet excitations for three VBS states with (1) a 12-site unit cell, (2) a 36-site unit cell with six-fold rotation symmetry, and (3) a 36-site unit cell with three-fold rotation symmetry, respectively. We find that increasing the DM interactions generally leads to broad and smooth features in the optical conductivity with interesting experimental consequences. The optical conductivity reflects the features of the spin-triplet excitations that can be measured in future experiments.

DOI
10.1088/1367-2630/16/12/123009

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We show that the optical responses below the Mott gap can be used to probe the spin-triplet excitations in valence bond solid (VBS) phases in Mott insulators. The optical conductivity in this regime arises due to the electronic polarization mechanism via virtual electron hopping processes. We apply this mechanism to the Hubbard model with spin-orbit couplings and/or the corresponding spin model with significant Dzyaloshinskii-Moriya (DM) interactions, and compute the optical conductivity of VBS states on both ideal and deformed Kagome lattices. In case of the deformed Kagome lattice, we study the antiferromagnet, Rb2_22​Cu3_33​SnF12_{12}12​ with the pinwheel VBS state. In case of the ideal Kagome lattice, we explore the optical conductivity signatures of the spin-triplet excitations for three VBS states with (1) a 12-site unit cell, (2) a 36-site unit cell with six-fold rotation symmetry, and (3) a 36-site unit cell with three-fold rotation symmetry, respectively. We find that increasing the DM interactions generally leads to broad and smooth features in the optical conductivity with interesting experimental consequences. The optical conductivity reflects the features of the spin-triplet excitations that can be measured in future experiments.
We consider Z2{Z}_{2}Z2​ spin liquids on the kagome lattice on the verge of a valence bond solid (VBS) transition, where vortex excitations carrying Z2{Z}_{2}Z2​ magnetic flux---so-called visons---condense. We show that these vison excitations can couple directly to the external electromagnetic field, even though they carry neither spin nor charge. This is possible via a magnetoelastic coupling mechanism recently identified. [Potter, Senthil, and Lee, arXiv:1301.3495; Hao, Phys. Rev. B 85, 174432 (2012)] For the case of transitions to a 36-site unit cell VBS state, the corresponding finite ac conductivity has a specific power-law frequency dependence, which is related to the crossover exponent of the quantum critical point. The visons' contribution to the optical conductivity at transitions to VBS states with a 12-site unit cell vanishes, however.
One of the most promising candidate ground states for the quantum antiferromagnetic Heisenberg model on the kagome lattice is the valence bond solid (VBS) with a 36-site unit cell. We present a theory of triplet excitation spectra about this ground state using bond operator formalism. In particular we obtain dispersions of all 18 triplet modes in the reduced Brillouin zone. In the bond operator mean-field theory, it is found that a large number of triplet modes are nondispersive. In particular, the lowest triplet excitation is nondispersive and degenerate with a dispersive mode at the zone center. Away from the zone center, the lowest triplet is separated from two other flat modes by a small energy gap. Quantum fluctuations are considered by taking into account scattering processes of two triplets and their bound-state formation, which leads to a downward renormalization of the lowest spin triplet gap. The dispersion of the lowest triplet excitation in the VBS state is compared with the dispersive lower bound of the triplet continuum expected in competing spin liquid phases. Implications to future neutron-scattering experiments are discussed.
Motivated by a recent experiment on Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​, where spin-1/2 Cu2+{\text{Cu}}^{2+}Cu2+ moments reside on the layers of kagome-like lattices, we investigate quantum ground states of the antiferromagnetic Heisenberg model on a series of deformed kagome lattices. The deformation is characterized by a weaker exchange coupling $\ensuremath{\alpha}J$ on certain lattice links appropriate for Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​ with $\ensuremath{\alpha}=1$ corresponding to the ideal kagome lattice. In particular, we study possible valence bond solid phases using the perturbation theory around isolated dimer limits, dimer series expansion, and self-consistent bond operator mean-field theory. It is shown that the valence bond solid phase with a 36-site unit cell of the ideal kagome lattice is quite sensitive to a small lattice distortion as the kind discovered in Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​. As a result, we find that a more likely quantum ground state in Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​ is the valence bond solid phase with a 12-site unit cell, where six dimers form a pinwheel structure, leading to strong modification of the elementary triplet and singlet excitation spectra in the deformed kagome lattices.
A proposed ground state for the kagome lattice Heisenberg model consists of a valence bond crystal (VBC) with a 36-site unit cell. We calculate the low-lying triplet and singlet excitations in the VBC phase for the infinite-lattice model and for the 36-site cluster. For the infinite lattice, the lowest triplet excitation is found to have a spin gap of approximately $0.08\ifmmode\pm\else\textpm\fi{}0.02J$ and a bandwidth of only about 0.01J0.01J0.01J. For the 36-site cluster, which consists of a single unit cell with periodic boundary conditions, there are substantial finite-size effects: the spin gap there is estimated to be approximately 0.2J0.2J0.2J, which is close to the exact diagonalization result of 0.164J0.164J0.164J. The triplet excitations attract one another and form many bound states in the spin-singlet channel. We find a large number of such bound states for the 36-site cluster, many of which appear to lie below the spin gap, which is again in agreement with the results from exact diagonalization.
The spin-1/2 quantum antiferromagnet on the kagome lattice provides a quintessential example in strongly correlated electron physics in which the effects of both geometric frustration and quantum fluctuation are pushed to their limit. Among possible nonmagnetic ground states, the valence bond solid (VBS) with a 36-site unit cell is one of the most promising candidates. A natural theoretical framework for the analysis of such VBS order is to consider quantum states on a bond connecting the nearest-neighboring sites as fundamental quantum modes of the system and treat them as effectively independent ``bond particles.'' While correctly describing the VBS order in the ground state, this approach, known as the bond operator theory, significantly overestimates the lowest spin excitation energy. To overcome this problem, we take a next logical step in this paper to improve the bond operator theory and consider extended spin clusters as fundamental building blocks of the system. Depending on two possible configurations of the VBS order, various spin clusters are considered: (i) in the VBS order with staggered hexagonal resonance, we consider one spin cluster for a star of David shape and two spin clusters with each composed of a perfect hexagon and three attached dimers, and (ii) in the VBS order with uniform hexagonal resonance, we consider one spin cluster composed of a star of David and three attached dimers. It is shown that the majority of low-energy spin excitations are nearly or perfectly flat in energy. With most of its weight coming from the star of David, the lowest spin excitation has a gap much lower than the previous value obtained by the bond operator theory, narrowing the difference against exact diagonalization results. Despite the better agreement, however, the fact that the lowest spin excitation mostly comes from the star-of-David structure is at odds with the bond operator theory as well as previous series expansion studies. Expecting that the true low-energy spin excitations have contributions from both the star of David and its surroundings, we propose using two superclusters of spin in future work, treating spin excitations from both contributions on a more equal footing.
We propose that a valence-bond-solid (VBS) order can be stabilized in certain two-dimensional antiferromagnets due to spin-lattice coupling. In contrast to the VBS state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type, the spin-lattice coupling-induced VBS state can occur when 2S2S2S is not an integer multiple of the coordination number zzz. S=2S=2S=2 spins on the triangular lattice with z=6z=6z=6 is discussed as an example. Within the Schwinger boson mean-field theory the ground state is derived as the direct product of states, one of which represents the conventional long-range ordered spins, and the other given by the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ modulation of the valence bond amplitudes. Excitation spectrum for the modulated valence bond state is worked out within the single-mode approximation. The spectrum offers a new collective mode, distinct from the spin wave excitations of the magnetically ordered ground state, and observable in neutron scattering.