Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1...
Transcript of Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1...
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Center of Emergent MaterialsNSF MRSEC – DMR
Quantum Spin Liquids:Signatures of Fractionalization
Nandini TrivediPhysics Department
The Ohio State University
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
Ru Rh
Re Os Ir
http://trivediresearch.org.ohio-state.edu/
Université Paris-Saclay, Zoom Seminar, July 1, 2020
![Page 3: Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1 magnons (bosons) into two S=1/2 neutral spinons (fermions) Lake et al Nat. Mat. 4,](https://reader036.fdocuments.us/reader036/viewer/2022071105/5fdef5ac4c8edd43be25e994/html5/thumbnails/3.jpg)
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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Evidence for QSL
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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)
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Topological entanglement entropy: Information resource
Ian Osbourneh/K
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Finite g implies the existence of topological order
à long range entanglement structureàQuantum dimension of excitations
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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
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Evidence for spinon Fermi surface
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Kitaev Model: spin structure factor
DMRG++ Open Source:https://web.ornl.gov/~gz1/dmrgPlusPlus/
Brillouin Zone: Momenta cuts
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Structure Factor S(k) – Intermediate phase
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S(k) → spinon Fermi surface
DMRG Results
ConjecturedFermi surface
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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
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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….
![Page 49: Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1 magnons (bosons) into two S=1/2 neutral spinons (fermions) Lake et al Nat. Mat. 4,](https://reader036.fdocuments.us/reader036/viewer/2022071105/5fdef5ac4c8edd43be25e994/html5/thumbnails/49.jpg)
1
1-spin flip spectral functions
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1
1-spin flip and 2 spin-flip spectral functions
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0.2 0.35
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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….
![Page 53: Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1 magnons (bosons) into two S=1/2 neutral spinons (fermions) Lake et al Nat. Mat. 4,](https://reader036.fdocuments.us/reader036/viewer/2022071105/5fdef5ac4c8edd43be25e994/html5/thumbnails/53.jpg)
Main ingredients for Kitaev materials
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Spin-Orbit coupled Mott Insulators
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Destructive interference between the two pathways generates bond-dependent interactions
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G. Jackeli and G. Khaliullin, PRL 102, 017205 (2009)
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Crystal structure of a-RuCl3à candidate Kitaev material
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![Page 59: Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1 magnons (bosons) into two S=1/2 neutral spinons (fermions) Lake et al Nat. Mat. 4,](https://reader036.fdocuments.us/reader036/viewer/2022071105/5fdef5ac4c8edd43be25e994/html5/thumbnails/59.jpg)
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: Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1 magnons (bosons) into two S=1/2 neutral spinons (fermions) Lake et al Nat. Mat. 4,](https://reader036.fdocuments.us/reader036/viewer/2022071105/5fdef5ac4c8edd43be25e994/html5/thumbnails/60.jpg)
Thermal Hall Conductivity
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What object carries heat? Electron? Spin?
Majorana edge mode
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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: Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1 magnons (bosons) into two S=1/2 neutral spinons (fermions) Lake et al Nat. Mat. 4,](https://reader036.fdocuments.us/reader036/viewer/2022071105/5fdef5ac4c8edd43be25e994/html5/thumbnails/64.jpg)
�xy =1
3
e2
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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: Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1 magnons (bosons) into two S=1/2 neutral spinons (fermions) Lake et al Nat. Mat. 4,](https://reader036.fdocuments.us/reader036/viewer/2022071105/5fdef5ac4c8edd43be25e994/html5/thumbnails/65.jpg)
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 ??
![Page 66: Quantum Spin Liquids: Signatures of Fractionalization...1d Quantum Spin Liquid Fractionalized S=1 magnons (bosons) into two S=1/2 neutral spinons (fermions) Lake et al Nat. Mat. 4,](https://reader036.fdocuments.us/reader036/viewer/2022071105/5fdef5ac4c8edd43be25e994/html5/thumbnails/66.jpg)