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Center of Emergent MaterialsNSF MRSEC – DMR
Quantum Spin Liquids:Signatures of Fractionalization
Nandini TrivediPhysics Department
The Ohio State University
[email protected]
Ru Rh
Re Os Ir
http://trivediresearch.org.ohio-state.edu/
Université Paris-Saclay, Zoom Seminar, July 1, 2020
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Quantum Spin Liquids:Signatures of Fractionalization
Nandini TrivediPhysics Department
The Ohio State University
[email protected]
Ru Rh
Re Os Ir
http://trivediresearch.org.ohio-state.edu/
Université Paris-Saclay, Zoom Seminar, July 1, 2020
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Roadmap
ØBig picture
Ø2D: Kitaev Modelv Discovery of a new gapless QSL with a spinon Fermi
surface
v Spectrum of 1 spin flip and 2 spin flip excitations
ØQSL Materials
ØHow do you detect a QSL?
ØGoing forward….
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Big PictureWhat is a QSL?Why is it interesting? Important?
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First Signatures of a QSLUV
IR ??
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First signatures of a QSL: large frustration parameter
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Spontaneously broken time reversal symmetry
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First Signatures of a QSL
oMott insulator (odd number of electrons in unit cell)
oLocal moments (typically S=1/2 or Jeff=1/2)
oStrongly interacting momentsqCW~100 K
oNo magnetic ordering f= q /Tc ~ 104
UV
IR ??
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First Signatures of a QSL
o Mott insulator (odd number of electrons in unit cell)
o Local moments (typically S=1/2 or Jeff=1/2)
o Strongly interacting momentsq~100 K
o No magnetic ordering f= q /Tc ~ 104
So why is a paramagnet interesting?
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Quantum MatterqLandau paradigm: spontaneously broken symmetryà
local order parameter mbosonic excitations: magnons (for continuous spins)
qTopological Paradigm
“IQHE”Topological InsulatorsTopological SuperconductorsTopological Weyl and Dirac SemimetalsTopological magnons
“FQHE”Quantum Spin Liquids
Possess Topological Order
§ Ground state degeneracy§ Long range entanglement§ Fractionalized Excitations
Review: Savary and Balents, Repts. on Progress in Physics 80, 016502 (2017)
Wen, X.-G. (1989) PRB 40, 7387Wen, X.-G. and Niu, Q. (1990) PRB 41, 9377
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Quantum MatterqLandau paradigm: spontaneously broken symmetryà
local order parameter mbosonic excitations: magnons (for continuous spins)
qTopological Paradigm
“IQHE”Topological InsulatorsTopological SuperconductorsTopological Weyl and Dirac SemimetalsTopological magnons
“FQHE”Quantum Spin Liquids
Possess Topological Order
§ Ground state degeneracy§ Long range entanglement§ Fractionalized Excitations
Important for storing information non-locally; robust against decoherence
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Singletor valence bond
valence bond solid
Resonating valence bond-- candidate quantum spin liquid
Anderson 1973
Picture of a QSL
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Singletor valence bond
valence bond solid
Resonating valence bond-- candidate quantum spin liquid
Anderson 1973
deconfined S=1/2 spinons
Fermionic excitations
Excitations of a QSLContrast with ordered magnet:Magnons: S=1 (bosons)
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Why would a bunch of interacting spins not order at T=0 ?
(1) Low spin(2) Low dimensionality(3) Frustration[Geometric, Interactions, …]
Quantum Fluctuations
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Fractionalization of excitationsin quantum spin liquids
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1d Quantum Spin Liquid Fractionalized S=1 magnons (bosons)into two S=1/2 neutral spinons (fermions)
Lake et al Nat. Mat. 4, 329 (2005)
Inelastic neutron scattering S(q,w)KCuF3
Broad spectrum indicates fractionalization of magnons
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T=6K T=50 K
T=150 K T=300 K
Black dotted: multi-spinon continuum predicted at T=0 (Muller ansatz equation)
Ordering~ 10K
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Roadmap
ØBig picture
Ø2D: Kitaev Modelv Discovery of a new gapless QSL with a spinon Fermi
surface
v Spectrum of 1 spin flip and 2 spin flip excitations
ØQSL Materials
ØHow do you detect a QSL?
ØGoing forward….
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Kitaev Model
Honey comb latticeBipartite lattice; no geometric frustration
put qubit on each site
A. Kitaev, Annals of Physics 321, 2-111 (2006)
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Kitaev Model: bond-dependent interactions
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Kitaev Model: bond-dependent interactions
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Kitaev Model: bond-dependent interactions
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Ground State:
c-majoranafermions have a Dirac dispersion
All plaquettes have zero flux
000
0000
0 0 0
0
0 0 0
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Excitations:(1)Gapped flux excitation (visons) (2) Gapless majorana fermions
Gapless Z2 Quantum Spin Liquid
000
0p00
0 0 0
0
0 0 0
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Now add a magnetic field…
Focus on here: AF Kitaev interactions and field along
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Non-abelian gapped Kitaev spin liquid:
h
• Majorana fermions get gapped
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Our Main Results: Kitaev Model in a Magnetic Field
Ds: single spin flip energyDp: 2-spin flip energy
Results based on exact diagonalization ED and Density matrix renormalization group (DMRG)
Gapless
Gapless QSL with spinonfermi surface
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David Ronquillo Adu Vengal Nirav Patel Subhasree Pradhan Field-orientation-dependent spin dynamics of the Kitaev honeycomb model Phys. Rev. B 99, 140413 (2019)
Magnetic field induced intermediate gapless spin-liquid phase with a spinonFermi surfacePNAS 201821406 (2019)
PRB 101, 180401 (2020)
Two-Magnon Bound States in the KitaevModel in a [111]-Field
Related work:Z. Zhu, et al., Phys. Rev. B 97, 241110 (2018) M. Gohlke, et al., Phys. Rev. B 98, 014418 (2018) C. Hickey and S. Trebst, Nat. Comm. 10, 530 (2019)H.C. Jiang et al. arXiv 1809.08247 Y. Motome and J. Nasu, JPSJ 89, 012002 (2020)
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Evidence for TWO phase transitions
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Kitaev Model + Magnetic field: magnetization
4
Density Matrix Renormalization Group calculations with 160 spins
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Kitaev Model + Magnetic field: susceptibility
4
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Evidence for gapless intermediate phase
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Energy Spectra in a field [111]<latexit sha1_base64="nrSpGZ3GVjXR6xdLJdRBTfulZpE=">AAAB7HicbVBNSwMxEJ34WetX1aOXYBE8lU0R9Fjw4rGC2xa2S8mm2TY0m12SrFCW/gYvHhTx6g/y5r8xbfegrQ8GHu/NMDMvyqQw1vO+0cbm1vbObmWvun9weHRcOzntmDTXjPsslanuRdRwKRT3rbCS9zLNaRJJ3o0md3O/+8S1Eal6tNOMhwkdKRELRq2T/IAQEg5qda/hLYDXCSlJHUq0B7Wv/jBlecKVZZIaExAvs2FBtRVM8lm1nxueUTahIx44qmjCTVgsjp3hS6cMcZxqV8rihfp7oqCJMdMkcp0JtWOz6s3F/7wgt/FtWAiV5ZYrtlwU5xLbFM8/x0OhObNy6ghlWrhbMRtTTZl1+VRdCGT15XXSaTaI1yAP1/VWs4yjAudwAVdA4AZacA9t8IGBgGd4hTek0At6Rx/L1g1UzpzBH6DPH7eBjec=</latexit><latexit sha1_base64="nrSpGZ3GVjXR6xdLJdRBTfulZpE=">AAAB7HicbVBNSwMxEJ34WetX1aOXYBE8lU0R9Fjw4rGC2xa2S8mm2TY0m12SrFCW/gYvHhTx6g/y5r8xbfegrQ8GHu/NMDMvyqQw1vO+0cbm1vbObmWvun9weHRcOzntmDTXjPsslanuRdRwKRT3rbCS9zLNaRJJ3o0md3O/+8S1Eal6tNOMhwkdKRELRq2T/IAQEg5qda/hLYDXCSlJHUq0B7Wv/jBlecKVZZIaExAvs2FBtRVM8lm1nxueUTahIx44qmjCTVgsjp3hS6cMcZxqV8rihfp7oqCJMdMkcp0JtWOz6s3F/7wgt/FtWAiV5ZYrtlwU5xLbFM8/x0OhObNy6ghlWrhbMRtTTZl1+VRdCGT15XXSaTaI1yAP1/VWs4yjAudwAVdA4AZacA9t8IGBgGd4hTek0At6Rx/L1g1UzpzBH6DPH7eBjec=</latexit><latexit sha1_base64="nrSpGZ3GVjXR6xdLJdRBTfulZpE=">AAAB7HicbVBNSwMxEJ34WetX1aOXYBE8lU0R9Fjw4rGC2xa2S8mm2TY0m12SrFCW/gYvHhTx6g/y5r8xbfegrQ8GHu/NMDMvyqQw1vO+0cbm1vbObmWvun9weHRcOzntmDTXjPsslanuRdRwKRT3rbCS9zLNaRJJ3o0md3O/+8S1Eal6tNOMhwkdKRELRq2T/IAQEg5qda/hLYDXCSlJHUq0B7Wv/jBlecKVZZIaExAvs2FBtRVM8lm1nxueUTahIx44qmjCTVgsjp3hS6cMcZxqV8rihfp7oqCJMdMkcp0JtWOz6s3F/7wgt/FtWAiV5ZYrtlwU5xLbFM8/x0OhObNy6ghlWrhbMRtTTZl1+VRdCGT15XXSaTaI1yAP1/VWs4yjAudwAVdA4AZacA9t8IGBgGd4hTek0At6Rx/L1g1UzpzBH6DPH7eBjec=</latexit><latexit sha1_base64="nrSpGZ3GVjXR6xdLJdRBTfulZpE=">AAAB7HicbVBNSwMxEJ34WetX1aOXYBE8lU0R9Fjw4rGC2xa2S8mm2TY0m12SrFCW/gYvHhTx6g/y5r8xbfegrQ8GHu/NMDMvyqQw1vO+0cbm1vbObmWvun9weHRcOzntmDTXjPsslanuRdRwKRT3rbCS9zLNaRJJ3o0md3O/+8S1Eal6tNOMhwkdKRELRq2T/IAQEg5qda/hLYDXCSlJHUq0B7Wv/jBlecKVZZIaExAvs2FBtRVM8lm1nxueUTahIx44qmjCTVgsjp3hS6cMcZxqV8rihfp7oqCJMdMkcp0JtWOz6s3F/7wgt/FtWAiV5ZYrtlwU5xLbFM8/x0OhObNy6ghlWrhbMRtTTZl1+VRdCGT15XXSaTaI1yAP1/VWs4yjAudwAVdA4AZacA9t8IGBgGd4hTek0At6Rx/L1g1UzpzBH6DPH7eBjec=</latexit>
✓ ⇠ |~h|/K<latexit sha1_base64="EvQfYUSU8bt+QQM0540Qkg1SMPQ=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJS7AInmpSBD0WvAheKtgPaELZbCft0s0Hu5NCSQv+FS8eFPHq7/Dmv3Hb5qCtDwYe780wM89PBFdo299GYW19Y3OruF3a2d3bPzAPj5oqTiWDBotFLNs+VSB4BA3kKKCdSKChL6DlD29nfmsEUvE4esRxAl5I+xEPOKOopa554uIAkLqKhxN3BCwbTCeX912zbFfsOaxV4uSkTHLUu+aX24tZGkKETFClOo6doJdRiZwJmJbcVEFC2ZD2oaNpRENQXjY/f2qda6VnBbHUFaE1V39PZDRUahz6ujOkOFDL3kz8z+ukGNx4GY+SFCFii0VBKiyMrVkWVo9LYCjGmlAmub7VYgMqKUOdWEmH4Cy/vEqa1YpjV5yHq3KtmsdRJKfkjFwQh1yTGrkjddIgjGTkmbySN+PJeDHejY9Fa8HIZ47JHxifP8V4lfA=</latexit><latexit sha1_base64="EvQfYUSU8bt+QQM0540Qkg1SMPQ=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJS7AInmpSBD0WvAheKtgPaELZbCft0s0Hu5NCSQv+FS8eFPHq7/Dmv3Hb5qCtDwYe780wM89PBFdo299GYW19Y3OruF3a2d3bPzAPj5oqTiWDBotFLNs+VSB4BA3kKKCdSKChL6DlD29nfmsEUvE4esRxAl5I+xEPOKOopa554uIAkLqKhxN3BCwbTCeX912zbFfsOaxV4uSkTHLUu+aX24tZGkKETFClOo6doJdRiZwJmJbcVEFC2ZD2oaNpRENQXjY/f2qda6VnBbHUFaE1V39PZDRUahz6ujOkOFDL3kz8z+ukGNx4GY+SFCFii0VBKiyMrVkWVo9LYCjGmlAmub7VYgMqKUOdWEmH4Cy/vEqa1YpjV5yHq3KtmsdRJKfkjFwQh1yTGrkjddIgjGTkmbySN+PJeDHejY9Fa8HIZ47JHxifP8V4lfA=</latexit><latexit sha1_base64="EvQfYUSU8bt+QQM0540Qkg1SMPQ=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJS7AInmpSBD0WvAheKtgPaELZbCft0s0Hu5NCSQv+FS8eFPHq7/Dmv3Hb5qCtDwYe780wM89PBFdo299GYW19Y3OruF3a2d3bPzAPj5oqTiWDBotFLNs+VSB4BA3kKKCdSKChL6DlD29nfmsEUvE4esRxAl5I+xEPOKOopa554uIAkLqKhxN3BCwbTCeX912zbFfsOaxV4uSkTHLUu+aX24tZGkKETFClOo6doJdRiZwJmJbcVEFC2ZD2oaNpRENQXjY/f2qda6VnBbHUFaE1V39PZDRUahz6ujOkOFDL3kz8z+ukGNx4GY+SFCFii0VBKiyMrVkWVo9LYCjGmlAmub7VYgMqKUOdWEmH4Cy/vEqa1YpjV5yHq3KtmsdRJKfkjFwQh1yTGrkjddIgjGTkmbySN+PJeDHejY9Fa8HIZ47JHxifP8V4lfA=</latexit><latexit sha1_base64="EvQfYUSU8bt+QQM0540Qkg1SMPQ=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJS7AInmpSBD0WvAheKtgPaELZbCft0s0Hu5NCSQv+FS8eFPHq7/Dmv3Hb5qCtDwYe780wM89PBFdo299GYW19Y3OruF3a2d3bPzAPj5oqTiWDBotFLNs+VSB4BA3kKKCdSKChL6DlD29nfmsEUvE4esRxAl5I+xEPOKOopa554uIAkLqKhxN3BCwbTCeX912zbFfsOaxV4uSkTHLUu+aX24tZGkKETFClOo6doJdRiZwJmJbcVEFC2ZD2oaNpRENQXjY/f2qda6VnBbHUFaE1V39PZDRUahz6ujOkOFDL3kz8z+ukGNx4GY+SFCFii0VBKiyMrVkWVo9LYCjGmlAmub7VYgMqKUOdWEmH4Cy/vEqa1YpjV5yHq3KtmsdRJKfkjFwQh1yTGrkjddIgjGTkmbySN+PJeDHejY9Fa8HIZ47JHxifP8V4lfA=</latexit>
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Spin-Spin Correlations in Intermediate phase
5
Distinct power law decay of real-space spin-spin correlations!
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Entanglement Entropy for a Gapped QSL
AB
“Area Law” Entanglement in a gapped system
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Topological Entanglement Entropy g
B A
Kitaev-Preskill Constructionto extract g
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Energy Spectra
N = 18<latexit sha1_base64="DJEy+3Ef+ynoJIVHEpwCKMNNqLI=">AAAB63icdVDLSsNAFJ3UV62vqks3g0UQF2ESQ1sXQsGNK6lgH9CGMplO2qGTSZiZCCX0F9y4UMStP+TOv3HSVlDRAxcO59zLvfcECWdKI/RhFVZW19Y3ipulre2d3b3y/kFbxakktEViHstugBXlTNCWZprTbiIpjgJOO8HkKvc791QqFos7PU2oH+GRYCEjWOfSzaVTH5QryL6oV12vCpGNUM1xnZy4Ne/cg45RclTAEs1B+b0/jEkaUaEJx0r1HJRoP8NSM8LprNRPFU0wmeAR7RkqcESVn81vncETowxhGEtTQsO5+n0iw5FS0ygwnRHWY/Xby8W/vF6qw7qfMZGkmgqyWBSmHOoY5o/DIZOUaD41BBPJzK2QjLHERJt4SiaEr0/h/6Tt2g6ynVuv0jhbxlEER+AYnAIH1EADXIMmaAECxuABPIFnK7IerRfrddFasJYzh+AHrLdPa2WNwA==</latexit><latexit sha1_base64="DJEy+3Ef+ynoJIVHEpwCKMNNqLI=">AAAB63icdVDLSsNAFJ3UV62vqks3g0UQF2ESQ1sXQsGNK6lgH9CGMplO2qGTSZiZCCX0F9y4UMStP+TOv3HSVlDRAxcO59zLvfcECWdKI/RhFVZW19Y3ipulre2d3b3y/kFbxakktEViHstugBXlTNCWZprTbiIpjgJOO8HkKvc791QqFos7PU2oH+GRYCEjWOfSzaVTH5QryL6oV12vCpGNUM1xnZy4Ne/cg45RclTAEs1B+b0/jEkaUaEJx0r1HJRoP8NSM8LprNRPFU0wmeAR7RkqcESVn81vncETowxhGEtTQsO5+n0iw5FS0ygwnRHWY/Xby8W/vF6qw7qfMZGkmgqyWBSmHOoY5o/DIZOUaD41BBPJzK2QjLHERJt4SiaEr0/h/6Tt2g6ynVuv0jhbxlEER+AYnAIH1EADXIMmaAECxuABPIFnK7IerRfrddFasJYzh+AHrLdPa2WNwA==</latexit><latexit sha1_base64="DJEy+3Ef+ynoJIVHEpwCKMNNqLI=">AAAB63icdVDLSsNAFJ3UV62vqks3g0UQF2ESQ1sXQsGNK6lgH9CGMplO2qGTSZiZCCX0F9y4UMStP+TOv3HSVlDRAxcO59zLvfcECWdKI/RhFVZW19Y3ipulre2d3b3y/kFbxakktEViHstugBXlTNCWZprTbiIpjgJOO8HkKvc791QqFos7PU2oH+GRYCEjWOfSzaVTH5QryL6oV12vCpGNUM1xnZy4Ne/cg45RclTAEs1B+b0/jEkaUaEJx0r1HJRoP8NSM8LprNRPFU0wmeAR7RkqcESVn81vncETowxhGEtTQsO5+n0iw5FS0ygwnRHWY/Xby8W/vF6qw7qfMZGkmgqyWBSmHOoY5o/DIZOUaD41BBPJzK2QjLHERJt4SiaEr0/h/6Tt2g6ynVuv0jhbxlEER+AYnAIH1EADXIMmaAECxuABPIFnK7IerRfrddFasJYzh+AHrLdPa2WNwA==</latexit><latexit sha1_base64="DJEy+3Ef+ynoJIVHEpwCKMNNqLI=">AAAB63icdVDLSsNAFJ3UV62vqks3g0UQF2ESQ1sXQsGNK6lgH9CGMplO2qGTSZiZCCX0F9y4UMStP+TOv3HSVlDRAxcO59zLvfcECWdKI/RhFVZW19Y3ipulre2d3b3y/kFbxakktEViHstugBXlTNCWZprTbiIpjgJOO8HkKvc791QqFos7PU2oH+GRYCEjWOfSzaVTH5QryL6oV12vCpGNUM1xnZy4Ne/cg45RclTAEs1B+b0/jEkaUaEJx0r1HJRoP8NSM8LprNRPFU0wmeAR7RkqcESVn81vncETowxhGEtTQsO5+n0iw5FS0ygwnRHWY/Xby8W/vF6qw7qfMZGkmgqyWBSmHOoY5o/DIZOUaD41BBPJzK2QjLHERJt4SiaEr0/h/6Tt2g6ynVuv0jhbxlEER+AYnAIH1EADXIMmaAECxuABPIFnK7IerRfrddFasJYzh+AHrLdPa2WNwA==</latexit>
N = 24<latexit sha1_base64="HrFa2p5YY5Z01JL5W7hOuba47+U=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRZBPJTdUqgXoeDFk1SwH9AuJZtm29AkuyRZoSz9C148KOLVP+TNf2O23YO2Phh4vDfDzLwg5kwb1/12ChubW9s7xd3S3v7B4VH5+KSjo0QR2iYRj1QvwJpyJmnbMMNpL1YUi4DTbjC9zfzuE1WaRfLRzGLqCzyWLGQEm0y6v6nVh+WKW3UXQOvEy0kFcrSG5a/BKCKJoNIQjrXue25s/BQrwwin89Ig0TTGZIrHtG+pxIJqP13cOkcXVhmhMFK2pEEL9fdEioXWMxHYToHNRK96mfif109MeO2nTMaJoZIsF4UJRyZC2eNoxBQlhs8swUQxeysiE6wwMTaekg3BW315nXRqVc+teg/1SvMqj6MIZ3AOl+BBA5pwBy1oA4EJPMMrvDnCeXHenY9la8HJZ07hD5zPHwbLjXs=</latexit><latexit sha1_base64="HrFa2p5YY5Z01JL5W7hOuba47+U=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRZBPJTdUqgXoeDFk1SwH9AuJZtm29AkuyRZoSz9C148KOLVP+TNf2O23YO2Phh4vDfDzLwg5kwb1/12ChubW9s7xd3S3v7B4VH5+KSjo0QR2iYRj1QvwJpyJmnbMMNpL1YUi4DTbjC9zfzuE1WaRfLRzGLqCzyWLGQEm0y6v6nVh+WKW3UXQOvEy0kFcrSG5a/BKCKJoNIQjrXue25s/BQrwwin89Ig0TTGZIrHtG+pxIJqP13cOkcXVhmhMFK2pEEL9fdEioXWMxHYToHNRK96mfif109MeO2nTMaJoZIsF4UJRyZC2eNoxBQlhs8swUQxeysiE6wwMTaekg3BW315nXRqVc+teg/1SvMqj6MIZ3AOl+BBA5pwBy1oA4EJPMMrvDnCeXHenY9la8HJZ07hD5zPHwbLjXs=</latexit><latexit sha1_base64="HrFa2p5YY5Z01JL5W7hOuba47+U=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRZBPJTdUqgXoeDFk1SwH9AuJZtm29AkuyRZoSz9C148KOLVP+TNf2O23YO2Phh4vDfDzLwg5kwb1/12ChubW9s7xd3S3v7B4VH5+KSjo0QR2iYRj1QvwJpyJmnbMMNpL1YUi4DTbjC9zfzuE1WaRfLRzGLqCzyWLGQEm0y6v6nVh+WKW3UXQOvEy0kFcrSG5a/BKCKJoNIQjrXue25s/BQrwwin89Ig0TTGZIrHtG+pxIJqP13cOkcXVhmhMFK2pEEL9fdEioXWMxHYToHNRK96mfif109MeO2nTMaJoZIsF4UJRyZC2eNoxBQlhs8swUQxeysiE6wwMTaekg3BW315nXRqVc+teg/1SvMqj6MIZ3AOl+BBA5pwBy1oA4EJPMMrvDnCeXHenY9la8HJZ07hD5zPHwbLjXs=</latexit><latexit sha1_base64="HrFa2p5YY5Z01JL5W7hOuba47+U=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRZBPJTdUqgXoeDFk1SwH9AuJZtm29AkuyRZoSz9C148KOLVP+TNf2O23YO2Phh4vDfDzLwg5kwb1/12ChubW9s7xd3S3v7B4VH5+KSjo0QR2iYRj1QvwJpyJmnbMMNpL1YUi4DTbjC9zfzuE1WaRfLRzGLqCzyWLGQEm0y6v6nVh+WKW3UXQOvEy0kFcrSG5a/BKCKJoNIQjrXue25s/BQrwwin89Ig0TTGZIrHtG+pxIJqP13cOkcXVhmhMFK2pEEL9fdEioXWMxHYToHNRK96mfif109MeO2nTMaJoZIsF4UJRyZC2eNoxBQlhs8swUQxeysiE6wwMTaekg3BW315nXRqVc+teg/1SvMqj6MIZ3AOl+BBA5pwBy1oA4EJPMMrvDnCeXHenY9la8HJZ07hD5zPHwbLjXs=</latexit>
| Gi = | "" . . . "i<latexit sha1_base64="O6W3nP6N6HLJ72KcIGQr1hVnMLM=">AAACH3icbVDLSgMxFM34rPU16tJNsAjiosyIqBuh4EKXFewDOsOQSTNtaCYTkoxSpv0TN/6KGxeKiLv+jWk7FG09EHJy7rlJ7gkFo0o7zshaWl5ZXVsvbBQ3t7Z3du29/bpKUolJDScskc0QKcIoJzVNNSNNIQmKQ0YaYe9mXG88Eqlowh90XxA/Rh1OI4qRNlJgXww8oWiQ3Q49iXiHkeuBlwokZfI029uJVrPD1BXYJafsTAAXiZuTEshRDexvcw1OY8I1ZkiplusI7WdIaooZGRa9VBGBcA91SMtQjmKi/Gwy3xAeG6UNo0SaxTWcqL87MhQr1Y9D44yR7qr52lj8r9ZKdXTlZ5SLVBOOpw9FKYM6geOwYJtKgjXrG4KwpOavEHeRRFibSIsmBHd+5EVSPyu7Ttm9Py9VTvM4CuAQHIET4IJLUAF3oApqAINn8ArewYf1Yr1Zn9bX1Lpk5T0H4A+s0Q+/7qUu</latexit><latexit sha1_base64="O6W3nP6N6HLJ72KcIGQr1hVnMLM=">AAACH3icbVDLSgMxFM34rPU16tJNsAjiosyIqBuh4EKXFewDOsOQSTNtaCYTkoxSpv0TN/6KGxeKiLv+jWk7FG09EHJy7rlJ7gkFo0o7zshaWl5ZXVsvbBQ3t7Z3du29/bpKUolJDScskc0QKcIoJzVNNSNNIQmKQ0YaYe9mXG88Eqlowh90XxA/Rh1OI4qRNlJgXww8oWiQ3Q49iXiHkeuBlwokZfI029uJVrPD1BXYJafsTAAXiZuTEshRDexvcw1OY8I1ZkiplusI7WdIaooZGRa9VBGBcA91SMtQjmKi/Gwy3xAeG6UNo0SaxTWcqL87MhQr1Y9D44yR7qr52lj8r9ZKdXTlZ5SLVBOOpw9FKYM6geOwYJtKgjXrG4KwpOavEHeRRFibSIsmBHd+5EVSPyu7Ttm9Py9VTvM4CuAQHIET4IJLUAF3oApqAINn8ArewYf1Yr1Zn9bX1Lpk5T0H4A+s0Q+/7qUu</latexit><latexit sha1_base64="O6W3nP6N6HLJ72KcIGQr1hVnMLM=">AAACH3icbVDLSgMxFM34rPU16tJNsAjiosyIqBuh4EKXFewDOsOQSTNtaCYTkoxSpv0TN/6KGxeKiLv+jWk7FG09EHJy7rlJ7gkFo0o7zshaWl5ZXVsvbBQ3t7Z3du29/bpKUolJDScskc0QKcIoJzVNNSNNIQmKQ0YaYe9mXG88Eqlowh90XxA/Rh1OI4qRNlJgXww8oWiQ3Q49iXiHkeuBlwokZfI029uJVrPD1BXYJafsTAAXiZuTEshRDexvcw1OY8I1ZkiplusI7WdIaooZGRa9VBGBcA91SMtQjmKi/Gwy3xAeG6UNo0SaxTWcqL87MhQr1Y9D44yR7qr52lj8r9ZKdXTlZ5SLVBOOpw9FKYM6geOwYJtKgjXrG4KwpOavEHeRRFibSIsmBHd+5EVSPyu7Ttm9Py9VTvM4CuAQHIET4IJLUAF3oApqAINn8ArewYf1Yr1Zn9bX1Lpk5T0H4A+s0Q+/7qUu</latexit><latexit sha1_base64="O6W3nP6N6HLJ72KcIGQr1hVnMLM=">AAACH3icbVDLSgMxFM34rPU16tJNsAjiosyIqBuh4EKXFewDOsOQSTNtaCYTkoxSpv0TN/6KGxeKiLv+jWk7FG09EHJy7rlJ7gkFo0o7zshaWl5ZXVsvbBQ3t7Z3du29/bpKUolJDScskc0QKcIoJzVNNSNNIQmKQ0YaYe9mXG88Eqlowh90XxA/Rh1OI4qRNlJgXww8oWiQ3Q49iXiHkeuBlwokZfI029uJVrPD1BXYJafsTAAXiZuTEshRDexvcw1OY8I1ZkiplusI7WdIaooZGRa9VBGBcA91SMtQjmKi/Gwy3xAeG6UNo0SaxTWcqL87MhQr1Y9D44yR7qr52lj8r9ZKdXTlZ5SLVBOOpw9FKYM6geOwYJtKgjXrG4KwpOavEHeRRFibSIsmBHd+5EVSPyu7Ttm9Py9VTvM4CuAQHIET4IJLUAF3oApqAINn8ArewYf1Yr1Zn9bX1Lpk5T0H4A+s0Q+/7qUu</latexit>
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TEE
Ronquillo, Vengal, Trivedi, PRB 99, 140413(R) (2019)
Page 39
Topological entanglement entropy: Information resource
Ian Osbourneh/K
Page 40
Finite g implies the existence of topological order
à long range entanglement structureàQuantum dimension of excitations
Page 41
Gapped Non abelian KSL 1 ⇠ d1 = 1
✏ ⇠ d✏ = 1
v ⇠ dv =p2 > 1
! D =qd21 + d2✏ + d2v
=p1 + 1 + 2
= 2
! � = logD = log2
Vacuum
Fermion
Vortex
abelian
Non abelian
Page 42
Evidence for spinon Fermi surface
Page 43
Kitaev Model: spin structure factor
DMRG++ Open Source:https://web.ornl.gov/~gz1/dmrgPlusPlus/
Brillouin Zone: Momenta cuts
Page 44
Structure Factor S(k) – Intermediate phase
Page 46
S(k) → spinon Fermi surface
DMRG Results
ConjecturedFermi surface
Page 47
Singularities at all the M points related by C3 Rotations and Translations
“Fermi Surface” of spinons in a Mott insulator!M
G
Test using VMC on projected wave function
Page 48
Roadmap
ØBig picture
Ø2D: Kitaev Modelv Discovery of a new gapless QSL with a spinon Fermi
surface
v Spectrum of 1 spin flip and 2 spin flip excitations
ØQSL Materials
ØHow do you detect a QSL?
ØGoing forward….
Page 49
1
1-spin flip spectral functions
Page 50
1
1-spin flip and 2 spin-flip spectral functions
Page 52
Roadmap
ØBig picture
Ø2D: Kitaev Modelv Discovery of a new gapless QSL with a spinon Fermi
surface
v Spectrum of 1 spin flip and 2 spin flip excitations
ØQSL Materials
ØHow do you detect a QSL?
ØGoing forward….
Page 53
Main ingredients for Kitaev materials
Page 54
Spin-Orbit coupled Mott Insulators
Page 55
Destructive interference between the two pathways generates bond-dependent interactions
Page 56
G. Jackeli and G. Khaliullin, PRL 102, 017205 (2009)
Page 57
Crystal structure of a-RuCl3à candidate Kitaev material
Page 59
Roadmap
ØBig picture
Ø2D: Kitaev Modelv Discovery of a new gapless QSL with a spinon Fermi
surface
v Spectrum of 1 spin flip and 2 spin flip excitations
ØQSL Materials
ØHow do you detect a QSL?
ØGoing forward….
Page 60
Thermal Hall Conductivity
Page 62
What object carries heat? Electron? Spin?
Majorana edge mode
Page 63
Signatures of a QSL: quantized thermal Hall conductance
Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, Sixiao Ma, K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Motome,T. Shibauchi & Y. Matsuda, Nature 559, 227 (2018).
2Dxy =
1
2
�⇡
6
k2B~ T
2Dxy = xyd
Page 64
�xy =1
3
e2
h<latexit sha1_base64="mtLfEhgOLq86JSZC18MVeRxEYw4=">AAACDHicbVC9TsMwGHT4LeUvwMhiUZCYqqQgAQNSBQtjkQit1ITIcZ3WqvMj20FEVl6AhVdhYQDEygOw8Ta4bQZoOcnS+e4+2d8FKaNCWta3MTe/sLi0XFmprq6tb2yaW9u3Isk4Jg5OWMI7ARKE0Zg4kkpGOiknKAoYaQfDy5Hfvidc0CS+kXlKvAj1YxpSjKSWfHPfFbQfIV895MW5st1Eh+FRoRS5axST26DwzZpVt8aAs8QuSQ2UaPnml9tLcBaRWGKGhOjaVio9hbikmJGi6maCpAgPUZ90NY1RRISnxtsU8EArPRgmXJ9YwrH6e0KhSIg8CnQyQnIgpr2R+J/XzWR46ikap5kkMZ48FGYMygSOqoE9ygmWLNcEYU71XyEeII6w1AVWdQn29MqzxGnUz+rW9XGteVG2UQG7YA8cAhucgCa4Ai3gAAwewTN4BW/Gk/FivBsfk+icUc7sgD8wPn8AXj+b8Q==</latexit><latexit sha1_base64="mtLfEhgOLq86JSZC18MVeRxEYw4=">AAACDHicbVC9TsMwGHT4LeUvwMhiUZCYqqQgAQNSBQtjkQit1ITIcZ3WqvMj20FEVl6AhVdhYQDEygOw8Ta4bQZoOcnS+e4+2d8FKaNCWta3MTe/sLi0XFmprq6tb2yaW9u3Isk4Jg5OWMI7ARKE0Zg4kkpGOiknKAoYaQfDy5Hfvidc0CS+kXlKvAj1YxpSjKSWfHPfFbQfIV895MW5st1Eh+FRoRS5axST26DwzZpVt8aAs8QuSQ2UaPnml9tLcBaRWGKGhOjaVio9hbikmJGi6maCpAgPUZ90NY1RRISnxtsU8EArPRgmXJ9YwrH6e0KhSIg8CnQyQnIgpr2R+J/XzWR46ikap5kkMZ48FGYMygSOqoE9ygmWLNcEYU71XyEeII6w1AVWdQn29MqzxGnUz+rW9XGteVG2UQG7YA8cAhucgCa4Ai3gAAwewTN4BW/Gk/FivBsfk+icUc7sgD8wPn8AXj+b8Q==</latexit><latexit sha1_base64="mtLfEhgOLq86JSZC18MVeRxEYw4=">AAACDHicbVC9TsMwGHT4LeUvwMhiUZCYqqQgAQNSBQtjkQit1ITIcZ3WqvMj20FEVl6AhVdhYQDEygOw8Ta4bQZoOcnS+e4+2d8FKaNCWta3MTe/sLi0XFmprq6tb2yaW9u3Isk4Jg5OWMI7ARKE0Zg4kkpGOiknKAoYaQfDy5Hfvidc0CS+kXlKvAj1YxpSjKSWfHPfFbQfIV895MW5st1Eh+FRoRS5axST26DwzZpVt8aAs8QuSQ2UaPnml9tLcBaRWGKGhOjaVio9hbikmJGi6maCpAgPUZ90NY1RRISnxtsU8EArPRgmXJ9YwrH6e0KhSIg8CnQyQnIgpr2R+J/XzWR46ikap5kkMZ48FGYMygSOqoE9ygmWLNcEYU71XyEeII6w1AVWdQn29MqzxGnUz+rW9XGteVG2UQG7YA8cAhucgCa4Ai3gAAwewTN4BW/Gk/FivBsfk+icUc7sgD8wPn8AXj+b8Q==</latexit>
Kitaev (2006) Jackeli, Khaliullin (2009)
Frustration from bond-dependent
interactions
Ronquillo, Vengal, Trivedi, PRB 99, 140413 (2019)Patel & Trivedi, PNAS 116,12199 (2019)Pradhan, Patel, Trivedi, PRB 101, 180401 (2020)
cf: FQHE
Chiral spinon edge mode à Quantized thermal Hall conductance
64
h/K (field)
Gapless Z2 QSL Gapped Z2 QSL
Gapless U(1) QSL
Magnon bound states
h=0 hc1 hc2
Fermi surface ofneutral, gapless spinons
in an insulator!xx ⇠ T
Predictions for Kitaev magnets
Page 65
Going forward…..
1. Predictions for Raman scattering to observe magnon bound states
2. Spin and heat transport
3. Observation of neutral spinon Fermi surfaces
4. Doped QSLs à ??