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Fingerprints of fractionalized excitations in scattering probes of quantum spin liquids
Natalia PerkinsUniversity of Minnesota
Spin Dynamics in the DIRAC Systems, SPICE June 8, 2017
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Collaborators
Brent Perreault (UMN)
Jeroen van den Brink(IFW, Dresden)
Gábor Halász (KITP,UCSB)
Fiona Burnell(UMN) Gia-Wei Chern
University of Virginia
Dima Kovrizhin Cambridge UK
Johannes Knolle Cambridge UK
Roderich Moessner MPIPKS, Dresden
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Quantum spin liquids
1973: Anderson proposes the “Resonating Valence Bond” state - a prototype of the modern QSLs
+ + … � =
QSL: State of interacting spins that breaks no rotational or translational symmetry and has only short range spin correlations.
QSLs are characterized by topological order and long range entanglement (difficult to probe experimentally).
QSLs supports excitations with fractional quantum numbers and statistics.
Unlike states with broken symmetry, QSLs are not characterized by any local order parameter.
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Quantum spin liquidsMain Question:
How to probe fractionalized quasiparticles in QSL and their statistics?
Take home message:
Signatures of quantum order are mainly in the excitations
excitation
Incident light, X-rays, neutrons … Scattered
light, X-rays, neutrons…
fractionalizedexcitations
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Since excitations carry fractional quantum numbers relative to the local degrees of freedom, only multiple quasiparticles can couple to external probes: Response from QSL is always a multi-particle continuum.
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KCuF3
B Lake et al, PRL 111, 137205 (2013)
Spinon excitations probed by neutrons:
The fractionalization was definitively identified by excellent quantitative agreement between experiments and exact calculation based on the Bethe Ansatz.
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Probing continua of excitations inKitaev spin liquids
• Spin liquid ground state and fractionalized excitations in 2D and 3D Kitaev models
• Raman response in 2D & 3D Kitaev model
• RIXS response in 2D & 3D Kitaev models
• Conclusions
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Kitaev model on the honeycomb lattice
A. Kitaev, Annals of Physics 321, 2 (2006)
H = �X
x�bonds
Jx
�x
j
�x
k
�X
y�bonds
Jy
�y
j
�y
k
�X
z�bonds
Jz
�z
j
�z
k
Mapping spins to Majorana fermions:
Exactly solvable 2D model
Spin liquid ground state
Fractionalized excitation
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Spin fractionalization and Majorana fermions
Quadratic Hamiltonian in each flux sector:
Large number of conserved quantities, local plaquette operators:
uzij
ci
cj
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Excitations in the 2D Kitaev spin liquid
Fig. from M. Hermanns et al, 2014
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3D Kitaev family
H<0> H<1> H<1>
Hyperhoneycomb lattices
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3D Kitaev family
M. Hermanns et al, 2015
HyperoctagonHyperhexagon
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Hyperhoneycomb lattice
M. Hermanns et al, 2015
closed line of Dirac nodes
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Hyperoctagon latticeMajorana metal
Fermi surfaces
M. Hermanns et al, 2015
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Hyperhexagon lattice
gapless Weyl points
M. Hermanns et al, 2015
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Experimental realizationsNa2IrO3
alpha-Li2IrO3
Y.Singh, P. Gegenwart, PRL 2010, 2011
K. Plumb et al, Phys. Rev. B (2014)A. Banerjee et al, Nature Materials (2016)
alpha-RuCl3
beta-Li2IrO3
T. Takayama et al, PRL (2015)
gamma-Li2IrO3
Modic,'Nature'Comminica1ons'5,'4203'(2014)'
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Spectroscopy of Kitaev Spin Liquids
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Raman scattering in a nutshell
Devereaux, Hackl RMP 2007
Photon induced spin exchange
Two-magnon scattering in Mott insulators: Loudon-Fleuri scattering vertex
Photon-in photon-out process
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Raman Scattering in Kitaev model
Raman vertex: diagonal in fluxes but creates two Majorana fermions
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Raman Scattering results (2D)isotropic point:
polarization independent
Knolle, Chern, Kovrizhin, Moessner, NP (PRL 2014)
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Raman Scattering results (2D)
anisotropic point: polarization dependence
Knolle, Chern, Kovrizhin, Moessner, NP (PRL 2014)
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computed perturbatively
Knolle, Chern, Kovrizhin, Moessner, NP (PRL 2014)
The Heisenberg contribution depends on the photon polarizations, δ-function peak at the four flux gap.
Raman Scattering results (2D)
Big ‘hump’ with fine features of the Majorana DOS. Salient signatures of fractionalization are visible! (comparison gives JK~8meV)
L. Sandilands,Y.J. Kim, K.S. BurchPhys. Rev. Lett. 114 (2015)
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The Raman response is polarization dependent!
Brent Perreault, Johannes Knolle, NBP, F. J. Burnell, PRB 2015
Raman Scattering results (3D)Glamazda, Lemmens, Do, Choi, Choi, Nature Comm. 7 (2016)
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RIXS in a nutshellIn L-edge experiments, the scattering typically, though not always, happens through a direct RIXS process.
L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, J. van den Brink, RMP (2011)
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RIXS from Ir4+
(Na,Li)2IrO3 with Ir4+ in 5d 5 configuration [L3 edge]:
5d
2p
Core hole
Kitaev model: H = HK
Non-magnetic vacancy
Incoming photon: Q, ε
Outgoing photon: Q’, ε’
0 f
k,!k k0,!k0
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Halász, Chalker, Moessner, PRB 2014
1
23
4
56
XXX
The Kitaev model with a single vacancy at site r= the original Kitaev model with switched off couplings around site r (exactly solvable)
We demand that the vacancy is always in the spin-up state.
d†r,#| "i = | "i
d†r,#| #i = 0
d†r,"| "i = 0
d†r,"| #i = | "i
Intermediate state with a vacancy
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RIXS amplitude for the Kitaev model
A↵�(m,q) =P
r,nr
hm|dr,↵|nrihnr|d†r,� |0i
⌦�En+i� eiq·r
I(!,q) =P
m |P
↵,� T↵�A↵�(m,q)|2 �(! � Em)
q ⌘ k� k0
The four fundamental RIXS channels are introduced by decomposingthe polarization tensor into
(a) Spin-conserving (SC) channel with
(b) three non spin-conserving (NSC) channels with
create two flux excitations
Kramers−Heisenberg formula
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Fast collision approximation
(Na,Li)2IrO3 and α-RuCl3 : Γ / Jx,y,z >> 1
t ~ 1 / Γ << 1 / Jx,y,z !
The lowest order RIXS amplitude is then
NSC channels recover INS amplitudes for infinite Γ
Flux creation: Finite gap, little dispersion
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Results: SC channel2D Kitaev model
Elastic response Inelastic response
xno flux and two fermion excitations
I0(!,q) /Z
BZd2k �(! � "k � "q�k) ["k � "q�k]
2 ��1� ei'k ei'q�k��2
interference between the two sublattices
A B
|0i
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RIXS response in SC channel
Reciprocal space
Γ
M K
K ~
M ~
BZ
EBZ
G. B. Halasz, NBP, J.van den Brink (PRL 2014)
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RIXS response in SC channel
Reciprocal space
Γ
M K
K ~
M ~
BZ
EBZ
Gapless response at a finite number of discrete points
Dirac points
K points
Gapless points
~ Γ, K, K points G. B. Halasz, NBP, J.van den Brink (PRL 2014)
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RIXS response in SC channel
Reciprocal space
Γ
M K
K ~
M ~
BZ
EBZ
20 ][),(~ kqkqk −−∝ εεI
the response actually vanishes at the and points due to the factor� K
G. B. Halasz, NBP, J.van den Brink (PRL 2014)
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RIXS response in SC channel
Reciprocal space
Γ
M K
K ~
M ~
BZ
EBZ
G. B. Halasz, NBP, J.van den Brink (PRL 2014)
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Results: SC channel in3D Kitaev models
closed line of Dirac nodes Fermi surfacesgapless Weyl points
For each model, the low-energy(gapless) response is determined by the nodal structure of the fermions.
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Hyperhoneycomb lattice
The Majorana fermions are gapless along a nodal line within the Γ-X-Y plane.
The response is thus gapless in most of the Γ-X-Y plane and also in most ofthe Z-A-T plane. However, it is still gapped at a generic point of the BZ.
G. B. Halasz, B. Perreault, NBP (1705.05894)
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Hyperhexagon lattice
The fermions are gapless at Weyl points.
The response is thus only gapless at particular points of the BZ.
G. B. Halasz, B. Perreault, NBP (1705.05894)
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Hyperoctagon lattice
The Majorana fermions are gapless on a Fermi surface.
The response is thus gapless in most of the BZ.G. B. Halasz, B. Perreault, NBP (1705.05894)
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Generic Kitaev spin liquidsTime-reversal-symmetric perturbations w.r.t. HK
[Song, You, Balents, PRL 2016]
NSC channels (i.e., INS response): Gap disappears
SC channel: Two-fermion response
∫ −−−∝BZ
20 )( d),( kqkkq εεωδωI
22 1 ][ kqkkqk
−−−× −ϕϕεε ii ee
Robust Robust Survives only for perturbations with
three-fold rotation symmetry
High-energy response is robust against perturbations, even beyond the phase transition into an ordered phase.Low-energy response of a generic KSL can be completely different from that of a pure Kitaev model.
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Thank you