# Non-Abelian statistics in light scattering processes across interacting
Haldane chains

Vladimir Gnezdilov  B. Verkin Inst. for Low Temperature Physics and
Engineering, NASU, 61103 Kharkiv, Ukraine  Inst. for Condensed Matter Physics,
TU Braunschweig, D-38106 Braunschweig, Germany  Vladimir Kurnosov  B. Verkin
Inst. for Low Temperature Physics and Engineering, NASU, 61103 Kharkiv,
Ukraine  Yurii Pashkevich  O.O. Galkin Donetsk Inst. for Physics and
Engineering, NASU, Kyiv - Kharkiv 03028, Ukraine  Anup Kumar Bera  Helmholtz-
Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany  Solid State
Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India  A.T.M.
Nazmul Islam  Helmholtz-Zentrum Berlin für Materialien und Energie, 14109
Berlin, Germany  Bella Lake  Helmholtz-Zentrum Berlin für Materialien und
Energie, 14109 Berlin, Germany  Bodo Lobbenmeier  Inst. for Condensed Matter
Physics, TU Braunschweig, D-38106 Braunschweig, Germany  Dirk Wulferding
CCES, Inst. for Basic Science, Dept. Physics and Astronomy, Seoul Nat. Univ.,
Seoul 08826, Republic of Korea  Inst. for Condensed Matter Physics, TU
Braunschweig, D-38106 Braunschweig, Germany  Lab. of Emerging Nanometrology
LENA, D-38106 Braunschweig, Germany  Peter Lemmens  Inst. for Condensed Matter
Physics, TU Braunschweig, D-38106 Braunschweig, Germany  Lab. of Emerging
Nanometrology LENA, D-38106 Braunschweig, Germany

###### Abstract

The $S=1$ Haldane state is constructed from a product of local singlet dimers
in the bulk and topological states at the edges of a chain. It is a
fundamental representative of topological quantum matter. Its well-known
representative, the quasi-one-dimensional SrNi2V2O8 shows both conventional as
well as unconventional magnetic Raman scattering. The former is observed as
one- and two-triplet excitations with small linewidths and energies
corresponding to the Haldane gap $\Delta_{H}$ and the exchange coupling
$J_{c}$ along the chain, respectively. Well-defined magnetic quasiparticles
are assumed to be stabilized by interchain interactions and uniaxial single-
ion anisotropy. Unconventional scattering exists as broad continua of
scattering with an intensity $I(T)$ that shows a mixed bosonic / fermionic
statistic. Such a mixed statistic has also been observed in Kitaev spin
liquids and could point to a non-Abelian symmetry. As the ground state in the
bulk of SrNi2V2O8 is topologically trivial, we suggest its fractionalization
to be due to light-induced interchain exchange processes. These processes are
supposed to be enhanced due to a proximity to an Ising ordered state with a
quantum critical point. A comparison with SrCo2V2O8, the $S=1/2$ analogue to
our title compound, supports these statements.

Topological effects in condensed matter physics are an attractive field of
study as they combine very interesting concepts of fundamental physics with
exotic effects, such as anomalous transport in magnetic fields, nonlocal
interactions, and fractionalization of excitations haldane . Prominent
examples include relativistic Dirac electrons, quantum spin liquids, and the
fractional quantum Hall effect.

One of the fundamental aspects of topological systems is that the exchange or
braiding of quasiparticles in certain classes of systems leads to an
unconventional phase, different from 0 or $\pi$, known from bosons and
fermions. The resulting non-Abelian statistics has first been realized for the
fractional Hall effect at 5/2 filling wen91 ; moore91 . A planar system with a
dimensionality D$<$3 has to be chosen. Furthermore, weakly interacting
quasiparticles should exist that are degenerate in energy for large distances.
This leads to the emergent quality of non-Abelian systems.

Periodically driving a system by an electric field is another approach to
generate topological states Khemani16 ; Moritomo16 ; Wang13 ; Esin18 . The so
called Floquet Hamiltonian leads to energetically shifted replicas of
electron-photon states. Here, the question appears whether the response to a
Laser field may simultaneously induce a non-Abelian statistics by the exchange
of quasiparticles joel09 ; bishara09 ; Vogl-20 . Recently, it has been shown
that such driving may enhance correlations Peronaci-20 ; ESQFT  and transfer
chain systems from Luttinger liquids to charge density wave instabilities
Kennes-18 . In principle, light-induced exchange processes are well known from
insulating spin systems and described by the Loudon-Fleury Hamiltonian
Loudon68 . However, to our knowledge, the topological implications of light-
induced spin exchange simultaneously with the Raman scattering process have
not been addressed so far.

Figure 1: (Color online) (a) Temperature dependent Raman spectra of SrNi2V2O8
obtained in ($ac$) polarization. The blue and red arrows mark the one- and
two-triplet modes, respectively. For comparison, the 7-K spectrum in ($cc$)
polarization is plotted in black. (b) Integrated intensity ($I(T)$) of the
magnetic scattering contributions as a function of temperature. The blue line
corresponds to a fit using a Bose factor. (c) Effect of a transverse magnetic
field on the assigned one- and two-triplet modes. Zero field positions are
marked by dashed lines. Transverse magnetic field effect on inelastic neutron
scattering (full line), ESR (dashed line) bera-13 ; bera-15 ; bera-15b ;
wang-13  and our one-triplet Raman mode (diamonds). (d) Four twisted Ni chains
along the $c$ axis in two different projections.

In the following report we focus on SrNi2V2O8, a rather well-known
representative of weakly interacting Haldane spin chains with $S$=1 states.
This system has a singlet ground state with a spin gap in the bulk and
topological states at the surface haldane . No resonance effects exist in
Raman scattering. Therefore, selection rules of conventional Raman scattering
should be valid and follow the dominance of 1D correlations along the spin
chains, i.e. appreciable intensity should only be observed with a polarization
parallel to the chain direction (c axis). Nevertheless, finite interchain
interactions and single ion anisotropies lead to Ising fluctuations and
magnetic field induced long range order. With our work we follow the concept
of inducing a topological state of an interacting chain system using the Raman
scattering process and probe for possible non-Abelian statistics by the
measurement process itself. A deconfinement of spinon and holon excitations
with non-Abelian statistics has been proposed to exist for s=1
antiferromagnetic systems greiner09 .

In SrNi2V2O8 skewed spin chains are formed by edge-sharing NiO6 (Ni2+, $S=1$)
octahedra along the $c$ axis, see Fig. 1 (d). They are coordinated by
VO${}_{4}^{3-}$ (V5+, $S=0$) tetrahedra that weakly hybridize with the Ni-O
coordinations. There are four chains per unit cell bera-12 .

All important model parameters are well-known bera-13 ; bera-15 ; bera-15b ;
wang-13 ; bera-12 ; pahari-06 ; he-08  and the magnetic properties can be
summarized as follows: (i) a Haldane gap of $\Delta=3.6$ meV (29 cm-1) at the
AFM zone center ($k=q_{\mathrm{chain}}=\pi$) separates the singlet, spin-
liquid ground state from excited triplet states. The relation of the gap and
the exchange coupling along the chain gives $\Delta_{0}$ = 0.41J = 0.41
$\cdot$ 8.7 meV = 3.57 meV (28.6 cm-1).

From ESR two gaps are derived, $\Delta_{\perp}$ = 20.8 cm-1 and
$\Delta_{\parallel}$ = 12.7 cm-1, with a splitting induced by anisotropy. (ii)
The triplet excitations show a dispersion up to a maximum energy of $\sim
2.4J$ ($\sim 6.7\Delta$) at the zone boundary ($q_{\mathrm{chain}}=\pi/2$ and
3$\pi/2$). (iii) Interchain interactions and anisotropies are given by
$J_{\mathrm{inter}}\approx 0.26$ meV and $D=-0.29$ meV bera-15b . (iv) At the
AF zone center the resulting splitted branches are at 1.56 $\pm$ 0.1 meV
(12.64 $\pm$ 0.81 cm-1) and 2.58 $\pm$ 0.1 meV (20.90 $\pm$ 0.81 cm-1),
respectively bera-13 ; bera-15 ; bera-15b .

Figure 2: (Color online) (a) Raman spectra of SrNi2V2O8 taken perpendicular to
the Haldane chain direction in ($aa$) scattering geometry at 7 K and 390 K
together with low-temperature spectra in ($LL$), ($LR$) and ($ab$)
polarization. The yellow-shaded backgrounds represent the unconventional
scattering continuum with two maxima, $C1$ and $C2$. (b) Integrated
intensities of the $C1$, $C2$, quasielastic scattering (QES, E$\approx$0), and
their sum. The lines are guides to the eye. (c) Decomposition of the sum from
(b) (squares) into contributions with Fermionic and Bosonic statistics (full
red line). Individual components are given as shaded regions.

Experimental details of the SrNi2V2O8 single crystal growth and
characterization have been reported earlier bera-15 . Raman scattering
experiments were performed in a quasi-backscattering geometry on $ab$ and $ac$
surfaces with the excitation wavelength $\lambda=532.1$ nm and a power level
$P=5$ mW in parallel [($aa$), ($cc$)], crossed [($ab$), ($ac$)], and circular
[(LR), (LL)] light polarizations. Single crystal orientation has been
independently assured using the anisotropy of phonon Raman scattering and Laue
diffraction method. Raman spectra were measured in a variable temperature
closed cycle cryostat and a 10 T superconducting solenoid with the magnetic
field along the $c$ axis of the crystal.

Polarized Raman spectra of SrNi2V2O8 show all expected phonon modes and their
small linewidth and well-defined selection rules indicate the high sample
quality. There are no significant phonon anomalies as function of
temperatures, and therefore no evidence for a structural instability Phonons .

At low temperatures (T$<$30K) and in (ac) scattering geometry we notice two
modes, at 22 cm-1 (blue arrow) and at 220 cm-1 (red arrow), see Fig. 1(a) and
a summary in Table 1. The ($ac$) component with one polarization component
along the chain direction corresponds to $E$ symmetry. The first mode has a
rather narrow linewidth and its intensity $I(T)$ is critically depressed with
increasing temperatures, see Fig. 1(b), blue symbols. Also an applied
transverse magnetic field leads to a characteristic shift, see Fig. 1(c). This
shift corresponds very well to previous neutron scattering and ESR data
bera-13 ; bera-15 ; bera-15b ; wang-13  of triplet excitations in magnetic
fields. Therefore we assign this mode to a one-triplet excitation.

The second mode has a factor 6 larger linewidth. It does not show a pronounced
frequency shift with temperature and $I(T)$ is only fully depleted for
temperatures above $\sim 30$ K, i.e. it is less sensitive to thermal
fluctuations. The mode’s energy is larger than the maximum energy of the
neutron triplet dispersion ($\sim 23$ meV, 185 cm-1) bera-15 . Nevertheless,
it fits well to $3J=210$ cm-1.

A similar mode at $3J$ has also been observed in the Haldane chain compound
Y2BaNiO5 sulewski-95 . The factor $3$ can be rationalized in an Ising-like
spin system considering breaking spin bonds by a spin exchange process. This
scheme leads to a typical energy $E_{\mathrm{max}}=(z\cdot 2S-1)J$ hayes-78 ;
Cottam-Lockwood . With $S=1$ and the coordination number, $z=2$,
$E_{\mathrm{max}}=3J$ is derived. The good agreement of such a heuristic
picture of an Ising-like, Néel ground state to an excitation in a quantum
disordered system is far from trivial. In $1D$ or $2D$ spin liquids a
considerable renormalization to lower energies and increasing linewidth is
observed lemmens-03 . This could imply well-defined quasiparticles and weak
fluctuation corrections in this Haldane system. However, it might also mean
that these excitations are stabilized by some other mechanism, as discussed
later.

We use $I(T)\propto(1-e^{\Delta/k_{B}T})$, with singlet-triplet gap
$\Delta=3.6$ meV, to fit the one-triplet intensity (blue symbols), see solid
curve in Fig. 1 (b). This fit corresponds to a depression of $I(T)$ due to
scattering on thermally populated triplets across this gap, i.e. a bosonic
statistics. Such a model works very well for SrCu2(BO3)2 with orthogonal spin
dimers and strongly localized excited triplets lemmens00 .

The two-triplet scattering involves double exchange processes averaging over a
regime of $k$ space around the zone boundary. Therefore a simple analytic
expression of $I(T)$ does not exist Cottam-Lockwood . The observed smoother
dependence is rationalized by a superposition of scattering processes with a
dispersing singlet-triplet gap. A second scenario that is relevant for low-
dimensional systems considers the importance of local spin-spin correlations.
Therefore thermally induced triplet states do not scatter efficiently
lemmens-03 ; lemmens00 .

An unconventional Raman signal is observed between 100 and 280 cm-1 with broad
maxima and a linewidth of $\approx$200 cm-1. Its onset energy is around 60
cm-1 and a high energy cut-off exists at $\approx 350$ cm-1, see Fig. 2 (a).
This signal is only observed in ($LL$) and ($aa$) polarization and not in
($LR$), ($ab$), and ($ac$). Such a strict selection rule points to an
intrinsic origin and is not compatible with fluorescence or phonon density of
states. Therefore we attribute the continuum to some sort of
magnetic/electronic scattering.

After subtracting spectral contributions due to phonons we have identified
three major contributions, i.e. lower and higher energy maxima ($C1$,$C2$) and
quasielastic scattering ($QES$). The respective integrated intensities and
their sum are given in Fig. 2 (b). Spectral weight from the high energy $C2$
to lower energy $C1$ contribution is transferred into QES with increasing
temperatures. At lowest temperatures, comparable to the Haldane gap
($\Delta~{}\approx$ 42 K), there is a general drop of intensity. Furthermore,
a magnetic field leads to a suppression of the scattering intensity, see
Appendix. For larger magnetic fields SrNi2V2O8 shows a quantum phase
transition into a Néel-ordered phase.

We notice that the sum of these intensities reveals no appreciable temperature
dependence, at least in the temperature range 50-230 K. This is in sharp
contrast to $I(T)$ of one-triplet and two-triplet scattering (Fig. 1 (b)).
However, Raman scattering on quantum spin liquids based on the Heisenberg
Kagome wulferding-10  and the Kitaev honeycomb lattice sandilands-15 ;
glamazda-16  show a very similar weak temperature dependence. The latter is
attributed to their inherent entanglement that make thermal fluctuations
ineffective.

Also the observation of this mode in ($aa$) polarization with a high energy
cut-off is rather unexpected. This polarization has no component along the
chain direction with large exchange coupling. The Loudon-Fleury approach
implies that the spatial correlations of the exchange integral are probed by
the selected light polarization vectors. Due to the small interchain coupling
two-particle scattering probed in polarizations perpendicular to the chains
should have an energy scale of only 2$\Delta_{\perp}\approx$ 40 cm-1. On the
other hand SrNi2V2O8 is rather close to a phase boundary to Ising order due to
anisotropy and interchain interaction. Strong fluctuations at such a boundary
could favor a more complex mechanism of light scattering beyond the basic
Loudon-Fleury approach.

Therefore, we propose an alternative, Floquet-like scenario that consists of a
two step process: In a first step, the incident light drives the system into
an excited state where spins between neighboring chains are exchanged [e.g.,
along $J_{\mathrm{inter}}$, see Fig. 1(d)]. These states consists of defects
within the topological string order. In the second step these states relax
along the chain leading to a high energy scale, comparable to the energy of
dispersing triplets. This scenario introduces ”edge” states within the
topologically protected bulk due to the light-induced spin exchange. The first
step can also be replaced by a thermally induced spin-flip. This could be the
origin of the spectral weight shift in the light scattering process at
elevated temperatures becker-17 . Such a complex exchange-induced, topological
Raman process may also lead to a temperature dependence different from the
expected Bose statistics.

To determine the intensity $I(T)$ of the broader continuum we subtract the
phonon contributions and fit the remaining spectral weight to a sum of
Gaussians, see large squares in Fig. 2(c). The resulting temperature
dependence with a minimum at around 50 K and an increase both towards higher
and lower temperatures is completely different from conventional magnetic
Raman scattering (as discussed below). Motivated by this difference and the
above discussed Raman scattering process we have deconvoluted $I(T)$ into a
sum of a two-Fermion and a single Bose contribution. This corresponds to an
approach used for fractionalized modes in spin liquid states of Kitaev systems
nasu-16 . It takes into account that spin liquids carry topological defects
due to the creation/annihilation of pairs of Fermions (Majorana Fermions). The
bosonic contribution corresponds to their binding by residual interactions or
to additional low-energy triplet. For the Kitaev spin liquid materials
$\alpha$-RuCl3 and different phases of Li2IrO3 such a deconvolution leads to a
remarkable description of $I(T)$ nasu-16 ; sandilands-15 ; knolle-13 ;
knolle-14 ; perreault-15 ; nasu-14 .

In detail, the deconvolution consists of a sum of two-Fermion
$[(1-f(\omega_{F})]^{2}$ and Bose contributions
($1+n(\omega_{B}))=1/[1-\exp(-\hbar\omega_{B}/k_{B}T)]$, with
$\omega{{}_{F}}=7.8\pm 1$ meV and $\omega{{}_{B}}=6.7\pm 1$ meV, see red line
in Fig. 2(c). The respective weighting factors are proportional to the height
of the two shaded areas, see dashed lines. Both characteristic energies,
$\omega{{}_{F}}$ and $\omega_{B}$, are of the order of the doubled Haldane gap
of SrNi2V2O8 ($\Delta=3.6$ meV) giving evidence for their intrinsic origin.
Their order of magnitude is comparable to the ones of spin liquid materials
mentioned in the previous paragraph.

If we want to rationalize this deconvolution it should be noted that the bulk
of the Haldane phase is based on a very local product state with a gap and no
long range entanglement. In contrast to defects and edges, the bulk does not
support fractionalized excitations or edge modes with fermionic statistics. As
we have no evidence for defects from magnetic susceptibility data we propose
that the measurement process itself (exchange light scattering) and finite
temperatures dynamically induce the defects and resulting novel states. This
scenario is reminiscent of the concept of excited state quantum phase
transitionsESQFT . Both extracted energy scales, $\omega_{F}$ and
$\omega_{B}$, strongly support this assumption. Furthermore, the high cut-off
energy $E_{\mathrm{cut-off}}$ and the large linewidth of the order of 5-6$J$
are not compatible with the coordination number $z=2$ of a 1D chain.

Within the scheme of exchange-induced, broken bonds a maximum at about 3$J$
would be expected, in excellent agreement with the observed broad maxima. The
strict selection rules of the continuum with all components perpendicular to
the chain evidence that the existing correlations are rather two-dimensional
contrasting the dominating chain exchange that exists in the ground state. On
the other side, recent Floquet modelling of periodically driven Mott
insulators show nonequilibrium steady states with an enhancement of
correlations Peronaci-20 .

Finally, we note that our Raman experiments on the isostructural $S=1/2$
analogue SrCo2V2O8 shows an in-chain, spinon continuum in good agreement with
neutron scattering data Bera-17 . There is no evidence for high energy
scattering with out-of chain polarizations. The corresponding data is shown in
the Appendix.

Table 1: Summary of the observed magnetic excitations including energy at the maximum of scattering intensity, polarization, linewidth ($\Gamma$). In addition we give the polarization of quasielastic scattering. Nr |  $E_{\mathrm{max}}$ (cm-1) |  $\Gamma$ (cm-1) | Polarization | Parameters | Assignment, comments
---|---|---|---|---|---
1 | 22 | 2.5 | ($ac$) |  $\Delta=3.6$ meV (29 cm-1) | 1-triplet, B-dep.
2 | 220 | 15 | ($ac$) |  $E_{\mathrm{max}}\approx$$3J$ | 2-triplet, no B-dep.
3 | 100 - 280 |  $\approx$ 200 | (aa, $LL$, $ac$) | 0-6$J$, $\omega{{}_{F}}$=7.8 meV | unconventional continuum
4 |  $\approx$0 | - | ($aa$) | - | quasielastic

We summarize the properties of all three modes attributed to magnetic Raman
scattering in Table 1. They clearly differ with respect to energy, linewidth,
polarization, and in their dependence on temperature, as well as on an
external magnetic field.

##  I Summary

The presented polarization-resolved Raman spectra of single crystalline
SrNi2V2O8 provide a view of the manifold of excited states in a quasi-one-
dimensional $S=1$ Haldane system close to a critical state. Besides a one-
triplet and a two-triplet mode, a broad continuum is observed that we
attribute to a novel exited state Raman process induced by the exchange
process itself. The proposed scattering mechanism involves the generation of a
light-induced topological interchain-defect that relaxes via a high energy,
intrachain process. We have discussed the high cut-off energy of this process
to be related to enhanced dimensionality (2D) or a Floquet-like enhancement of
electronic correlations.

###### Acknowledgements.

We acknowledge important discussions with W. Brenig, J. Knolle, and S. R.
Manmana. This research was funded by the DFG Excellence Cluster
QuantumFrontiers, EXC 2123, as well as by DFG Le967/16-1, DFG-RTG 1952/1, and
the Quantum- and Nano-Metrology (QUANOMET) initiative within project NL-4.
D.W. acknowledges support by the Institute for Basic Science (IBS-R009-Y3).

##  II Appendix

Figure 3: (Color online) (a) Raman data obtained at $T=6$ K and in ($ab$)
polarization using various laser energies. (b) Magnetic field ($B$) dependence
of the scattering continuum in ($ab$) polarization. The inset plots its
integrated intensity over $B$. (c) Polarization-resolved Raman data of the
iso-structural $S=1/2$ system SrCo2V2O8.

In this appendix we investigate whether resonance effects or the skewed spin
chains formed by edge-sharing NiO6 octahedra along the $c$ axis are
responsible for the broad continua of magnetic/electronic Raman scattering.
Therefore we did detailed investigations using Lasers of different photon
energies. Furthermore, we studied the related, Co based SrCo2V2O8 system with
similar local coordinations.

In Fig. 3 (a) four data sets are given with the indicated wavelength of the
incident Laser radiation. The general shape and spectral weight continuum does
not evidence any effect. This supports our assumption that it is of intrinsic
origin and not related to other (e.g. fluorescence) processes. We also note
that the continuum’s intensity shows no significant laser energy dependence.
In Fig. 3(b) the intensity of the continuum as a function of magnetic field is
plotted. Here the magnetic field is aligned along the crystallographic $c$
axis, parallel to the chain direction. Since the mode is sensitive to external
magnetic fields, we can assume that it is of magnetic origin. Furthermore, its
intensity is strongly decreased for fields approaching 12 T, which corresponds
to the critical field. At larger fields, full spin polarization is achieved
bera-15b .

In the iso-structural sister compound SrCo2V2O8 we find no evidence for a
comparable broad continuum with a polarization perpendicular to the chain
direction [see Fig. 3(c)]. SrCo2V2O8, with comparable intrinsic energy scales,
is the $S=1/2$ analogue to our title compound. Hence, its excitation spectrum
is gapless and no topology-related edge states exist, in accordance with our
data. Taken together, these observations strongly support our interpretation
of the broad signal emerging in Raman scattering experiments on SrNi2V2O8.

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