Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin...
Transcript of Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin...
![Page 1: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/1.jpg)
Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular
and the Kagome Lattices
H. KAWAMURA Osaka Univerisity
Collaborators: K. WATANABE, H. NAKANO, T. SAKAI, T. SHIMOKAWA
NQS 2014, Nov. 5, 2014 KIKEN, Kyoto
![Page 2: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/2.jpg)
Possible quantum spin liquid state in frustrated magnets
RVB state [P.W. Anderson (‘73)]
Realizable in frustrated systems?
Novel liquid-like quantum spin state without magnetic long-range order ?
Long quest for the quantum spin liquid
![Page 3: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/3.jpg)
Resonating Valence Bond (RVB) state
Resoting state of quantum-mechanical singlet state generally with a finite gap
long-range)RVB possibly gapless
“gapless” or “gapped”
![Page 4: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/4.jpg)
Ground state of the simplest nearest- neighbor bilinear Heisenberg model
AF magnetic LRO at T=0 even for S=1/2
[ 120 degree structure ]
+ +・・・
Triangular lattice
Kagome lattice
Liquid-like ground state without magnetic long-range order nor the spin freezing
Z2 spin liquid U(1) spin liquid, valence bond crystal, etc
![Page 5: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/5.jpg)
Experimental discovery of the “quantum spin liquid” state
Quantum spin liquid states observed in certain S=1/2 frustrated AFs
κ-(ET)2Cu2(CN)3
EtMe3Sb[Pd(dmit)2]2
S=1/2 organic salts Mott insulator
What is its nature ? → still remains controvertial
Triangular lattice
Kagome lattice
herbersmithite: ZnCu3(OH)6Cl2
[K. Kanoda, Y.Shimizu, R.Kato, M.Tamura et al]
[D.G. Nocera et al]
![Page 6: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/6.jpg)
S=1/2 organic triangular AF I
κ-(ET)2Cu2(CN)3 slightly distorted triangular lattice
Gapless spin liquid
No magnetic LRO down to 32mK
NMR spectrum
Specific heat susceptibility
[Y. Shimizu, K. Kanoda et al ‘03]
[κ-ET]
![Page 7: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/7.jpg)
S=1/2 organic triangular AF II EtMe3Sb[Pd(dmit)2]2 [T.Itou , S. Maegawa et al ’08, ‘10]
Transition-like anomaly in the spin-liquid state
NMR spectrum
Gapless spin-liquid-like behavior
NMR relaxation rate T1-1
![Page 8: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/8.jpg)
S=1/2 kagome AF
Herbertsmithite:
T
T--1
[A.Olariu, P.Mendels et al `08]
ZnCu3(OH)6Cl2
17O NMR shift
Structurally perfect, but ~15% Zn2+ is randomly replaced by Cu2+
Gapless spin-liquid behavior observed
NMR relaxation rate
[M.P. Shores et al `05]
[D.E. Freedman et al `10]
![Page 9: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/9.jpg)
Inelastic neutron scattering on a single-crystal herbertsmithite
[T.H. Hang et al, 2012]
Broad and extended gapless spectrum without sharp structures nor dispersion
Single-crystal ZnCu3(OD)6Cl2
![Page 10: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/10.jpg)
Origin of the observed quantum spin-liquid-like behaviors
Randomness might be relevant
random singlet phase (valence bond glass)
Triangular organic salts Kagome herbertsmithite
Freezing of dielectric degrees of freedom self-generates the effective quenched randomness for the spin degrees of Freedom at low T.
Random substitution of magnetic Cu2+
by nonmagnetic Zn2+ (and vice versa) together with the possible Jahn-Teller distortion induces the effective quenched randomness.
![Page 11: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/11.jpg)
I Introduction
II S =1/2 random triangular AF
III S =1/2 random kagome AF
IV Related systems
Unfrustrated S =1/2 random AF
Classical random triangular AF
] [K. Watanabe, H. Kawamura, H. Nakano and T. Sakai, JPSJ 83, 034714 (2014)]
] [H. Kawamura, K. Watanabe and T. Shimokawa, JPSJ 83, 103704 (2014)]
![Page 12: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/12.jpg)
Quantum spin-liquid-like behavior in the S =1/2 random triangular-lattice Heisenberg AF
S =1/2 organic salts
κ-(ET)2Cu2(CN)3
EtMe3Sb[Pd(dmit)2]2
![Page 13: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/13.jpg)
Strong coupling between the spin and the charge (polarization)
κ-(BEDT-TTF)2Cu2(CN)3
Intradimer charge imbalance ?
Effective randomness in the exchange interaction between spins
[ M. A. Jawad, et al. Phys.Rev.B (2010) ]
AC dielectric constant
Random freezing of electric polarization at low T
![Page 14: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/14.jpg)
N=9(500),12(500),21(250), 27(80) [rhombus] 15(500),18(250),24(160),30(24) ・ periodic B.C. ・ sample # 200~500(80) ・ TITPACK: Lanczos method
uniform distribution
0 Exact diagonalization method
N=9(500),12(500) [rhombus], 15(80),18(40) ・ periodic B.C. ・ sample # 500 ・ TITPACK: Householder method
T =0 T >0
no randomness maximal randomness
Δ : randomness parameter (0 ≦ Δ ≦ 1 )
Bond-random S=1/2 AF Heisenberg model
on the triangular lattice
![Page 15: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/15.jpg)
AF Neel order
0 ≦ Δ ≦ Δc ⇒ AF LRO
Δc ≦ Δ ≦ 1 ⇒ No AF LRO Δc ~ 0.6
< sublattice mag. ms2 >
Spin-wave (Δ = 0)
From spin-wave theory
Extrapolation with c2= 0
Neel order disappears at Δ > Δc
Sublattice magnetization
![Page 16: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/16.jpg)
Spin glass order
Δc’ ≦ Δ ≦ 1 ⇒ No AF nor SG LRO, Spin-Liquid state
Δc’ ~ Δc~ 0.6
Spin liquid state appears at Δ > Δc’
Extrapolation with c2 ≠ 0
< Spin freezing parameter q >
From spin-wave theory,
Spin freezing parameter
![Page 17: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/17.jpg)
Chiral order
Spin is noncoplanar locally even at T =0 ⇒ quantum fluctuations But no chiral LRO
< Chiral freezing parameter qχ >
< Local scalar chirality amplitude χlocal >
[Scalar chirality]
Chiral freezing parameter
Coplanar or noncoplanar ?
![Page 18: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/18.jpg)
Magnetization curve
Near linear magnetization curve for strong randomness.
The 1/3 -plateau disappears
magnetization
< N = 30 >
1/3 plateau
[Exp. dmit] [D. Watanabe et al ‘12]
T=0
![Page 19: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/19.jpg)
Energy gap
Gapless (or very small gap) in the spin-liquid regime at Δ≧Δc
Ground state is a spin singlet for most of samples, with a small fraction of triplets.
First excited state is a spin triplet for most of samples.
Energy gap Rate of triplet ground states
1st excited states
![Page 20: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/20.jpg)
Specific heat
< Overall >
T –linear low-T specific heat in the spin-liquid regime at Δ≧Δc
T - Linear
γ-term calc. ~ 20 mJK-2 exp. κ-ET ~ 12 mJK-2
[Exp: κ-ΕΤ]
T
Specific heat
![Page 21: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/21.jpg)
Susceptibility
[Y.Shimizu, et al . PRL, 2003 ]
Susceptibility exhibits a gapless behavior.
For sufficiently strong randomness, An intrinsic Curie tail is observed.
Susceptibility
![Page 22: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/22.jpg)
NMR relaxation rate T1-1
NMR relaxation rate T1-1
Gapless behavior at low T characterized by an exponent 1.5~2
[Exp. dmit]
∝T1.5
∝T2
![Page 23: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/23.jpg)
A weak finite-T anomaly : Its origin ?
Experimentally, an anomaly in NMR T1
-1
(in specific heat) observed
[Y.Shimizu et al ‘03]
[S.Yamashita et al ‘08]
[T.Itou et al ’08,’ 10] [Y. Shimizu et al ‘03]
NMR relaxation rate T1-1
Broad peak appears for L>18
Some cooperative effect
[Exp. dmit]
![Page 24: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/24.jpg)
A possible candidate of the weak finite-T anomaly
characterized by a parity-like two-valued topological quantum number : no distinction between “R ” & ”L” (no “circulation” in the usual sense)
Vortex formed by chirality vectors
Topological transition (or a crossover) within the spin-liquid state
κ
Z2 vortex
vortex binding-unbinding
[]
[H.K. & S. Miyashita, ‘84]
T < Tv T > Tv
?
![Page 25: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/25.jpg)
Random Singlet Phase ( Valence Bond Glass )
Nature of the quantum spin liquid state
Gapless behavior reflecting the distribution of singlet binding energy due to the distribution of Jij . Nearly free spins can exist as `orphan‘ spins.
Rubust against perturbations. No QCP !
![Page 26: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/26.jpg)
Relation to experiments
1. Stabilization of quantum spin-liquid state
2. Gapless behavior, including the T –linear low-T specific heat, the power-law behavior of the NMR relaxation rate, and gapless (occasionally Curie-like) low-T susceptibility. 3. Near T -linear magnetization curve
4. Robustness against various perturbations e.g., magnetic fields, deuteration, pressure etc. 5. Intrinsic inhomogeniety ! weak inhomogeneous moment induced by fields
NMR [Shimizu et al ’06]
µSR [Nakajima et al ’12] ! `microscopic phase separation’ suggested
![Page 27: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/27.jpg)
Summary (triangular) * S=1/2 random Heisenberg AF on the triangular lattice exhibits a randomness-induced quantum spin-liquid ground state for sufficiently strong randomness.
Random Singlet Phase ( Valence Bond Glass )
* The state is gapless (or nearly gapless), with a T –linear low-T specific heat .
* The spin-liquid state realized here is a “random-singlet” state or a “valence bond glass (VBG) ” state, rather than the RVB state. The state is robust against various perturbations, with no direct relevance to quantum criticality. * The random-singlet phase picture seems to explain various features of experimental results on organic κ-ET and dmit salts.
![Page 28: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/28.jpg)
Quantum spin-liquid-like behavior in the S =1/2 random kagome-lattice Heisenberg AF
herbertsmithite
ZnCu3(OH)6Cl2
![Page 29: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/29.jpg)
Random-bond kagome model as a minimal model of herbertsmithite
Bond-random modulation of the effective exchange coupling J on the kagome plane
[D.E. Freedman et al `10] Zn2+ on the triangular layer replaced by Cu2+
Bond-randomkagome model
Jahn-Teller distortion
![Page 30: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/30.jpg)
N=12,16,18,21,24(100s),27(50s), 30(12s), ・ periodic B.C. ・ TITPACK: Lanczos method
uniform distribution
0 Exact diagonalization method
N= 12,15(100s), 18(20s) ・ periodic B.C ・ TITPACK: Householder method
T =0 T >0
no randomness maximal randomness
Δ : randomness parameter (0 ≦ Δ ≦ 1 )
Bond-random S=1/2 AF Heisenberg model
on the kagome lattice
![Page 31: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/31.jpg)
AF Neel and SG orders (T =0)
Neither Neel or SG order at any Δ
Sublattice magnetization
ms2
Spin liquid state for any Δ !
![Page 32: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/32.jpg)
Chiral order
Spin is noncoplanar locally even at T =0 ⇒ quantum fluctuations But no chiral LRO at any Δ
< Chiral freezing parameter qχ >
< Local scalar chirality amplitude χlocal >
[Scalar chirality] Chiral freezing parameter
![Page 33: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/33.jpg)
Specific heat
T - Linear
T
C
Change of behavior around Δc ~ 0.4 within the nonmagnetic state
N=12 & 18
[exp.] ∝T
∝T2/3
The low-T peak (structure) gone for Δ>Δc T-linear low-T specific heat
[J.S. Helton et al. 2007]
Regular kagome (Δ=0)
[S.Sugiura and A.Shimizu, 2013]
![Page 34: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/34.jpg)
Susceptibility
χ
T
gapless susceptibility with a Curie-tail for stronger randomness Change of behavior around Δc ~ 0.4
within the nonmagnetic state
T=0 [J.S. Helton et al, 2010]
[exp.]
![Page 35: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/35.jpg)
“Phase transition” within the non-magnetic state
R = [total number of samples]
[Number of samples with triplet ground states]
R becomes nonzero for Δ > Δc ~0.4, suggesting some sort of transition
AF
Random Singlet
Random Singlet
Z2 (or U(1) ?) spin liquid
![Page 36: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/36.jpg)
Dynamical structure factor S(q,ω) At Γ point T = 0
Δ = 1
(Calc.)
(Exp.) single crystal Very broad
intensity
No gap
Δ = 0
[T.-H. Han et al, ’12]
[A.M. Lauchli et al, 2009 ]
For details, see poster by T.Shimokawa
magnetic
![Page 37: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/37.jpg)
Summary (kagome) * S=1/2 random Heisenberg AF on the kagome lattice exhibits within the non-magnetic state a phase transition from the randomness-irrelevant to the randomness-relevant quantum spin-liquid state with increasing the randomness.
* The randomness-relevant state is gapless (nearly gapless), with a T –linear low-T specific heat .
* The spin-liquid state realized here is a “random-singlet” state or a “valence bond glass (VBG) ” state, with no direct relevance to quantum criticality.
* The random-singlet phase picture seems to explain various features of experimental results on herbertsmithite.
Valence-bond glass picture [R.R.P. Singh, ’10] Site-random model
![Page 38: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/38.jpg)
Related systems Is frustration essential ? S =1/2 random square-lattice Heisenberg AF Corresponding classical system ? Classical random triangular-lattice Heisenberg AF
![Page 39: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/39.jpg)
unfrustrated system --- square lattice
sublattice mag. ms2 spin freezing parameter q
Random-bond S=1/2 AF Heisenberg model on the square lattice
AF LRO persists up to the maximal randomness
Frustration plays a role
![Page 40: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/40.jpg)
Classical system ー random triangular AF Monte Carlo simulations
![Page 41: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/41.jpg)
Classical system
・L×L triangular lattice (L= 12 ~ 96) ・prdiodic B.C. ・Δ ≧0.7: temperature-exchange method ・sample # 128
uniform distribution
0
no randomness maximal randomness
Δ : randomness parameter (0 ≦ Δ ≦ 1 )
Antiferromagnetic classical Heisenberg model on the triangualr lattice
![Page 42: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/42.jpg)
3-‐subla)ce AF order qy
/ 2π
qx / 2π
(×105)
:wavevector :position vector
Spin structure factor ( Δ = 1, L=96,T =0.038)
3-subaltice AF SRO is kept even under randomness
peak at K-point
![Page 43: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/43.jpg)
Coplanar or noncoplanar?
< Δ=0~0.6 >
< Δ=0.7 >
< Δ=0.8 >
< Δ=0.9 >
< Δ=1 >
Scalar chirality = 0
≠ 0
planar
noncoplanar
T
|Χl
ocal|
0 ≦ Δ ≦ Δc1 ⇒ planar Δc1 ≦ Δ ≦ 1 ⇒ noncoplanar
Δc1 = 0.6~0.7
Local chirality amplitude|Χlocal| < L=96 >
![Page 44: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/44.jpg)
3-‐subla)ce AF : correla/on length <3-sublattice correlation length ξs >
< L=96 >
ξs
T
ξs
< Δ=0.8 >
T
ξs
T
< Δ= 1 >
ξs
T
< Δ=0.9 >
AF LRO No AF LRO
0 ≦ Δ ≦ Δc2 ⇒ AF LRO
Δc1 ≦ Δ ≦ 1 ⇒ AF SRO
Δc2 = 0.8~0.9
![Page 45: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/45.jpg)
Spin-glass order
T
ξ SG
< Δ=1 >
T
ξ CG
< Δ=1>
T
ξ SG
< L=96 >
T
ξ CG
< L=96 >
Δc1 ≦ Δ ≦ 1 ⇒ No Af LRO SG LRO with chirality
Δc2 = 0.8~0.9
spin-glass(SG)correlation length ξSG
chiral-glass(CG)correlation length ξCG
![Page 46: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/46.jpg)
Critical properties (Δ=1)
T <SG correlation-length ratio ξSG / L> <CG correlation-length ratio ξCG / L>
< spin >
< chirality >
slope = -1.47
slope = -3.01
νSG ~ 1.5, νCG ~ 3.0 ⇒spin-chirality decoupling [ H.Kawamura, ’92 ] (2D Heisenberg SG νSG ~ 0.9, νCG ~ 2.0 [ H.Kawamura, H.Yonehara , ’03 ] )
νCG ~3.0 > νSG ~1.5
![Page 47: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/47.jpg)
Phase diagram of a classical system
Randomness induces a noncoplanar spin structures.
For sufficiently strong randomness, AF LRO gives way to the spin-glass LRO.
<T = 0 >
planar noncoplanar
3-‐subla)ce AF LRO SG
Δ 0 0.6 0.8 1.0 0.7 0.9 Δc1
( Δ = 0.6 ~ 0.7 ) Δc2 ( Δ = 0.8~0.9 )
Deduced from finite-T Monte Carlo
![Page 48: Quantum Spin Liquid Behaviors in the Random Spin-1/2 ...nqs2014.ws/archive/Presen...Quantum Spin Liquid Behaviors in the Random Spin-1/2 Heisenberg Antiferromagnets on the Triangular](https://reader035.fdocuments.in/reader035/viewer/2022062508/607e0f38755f6464d7121573/html5/thumbnails/48.jpg)
Summary * S=1/2 random Heisenberg AFs on the triangular and the kagome lattices exhibit a quantum spin-liquid-like behavior, if the randomness exceeds a certain critical value. * The randomness-relevant spin-liquid state is gapless (or nearly gapless), with a T –linear low-T specific heat .
* The spin-liquid state realized here is a “random-singlet” state or a “valence bond glass (VBG) ” state, with no direct relevance to quantum criticality.
* The random-singlet phase picture seems to explain various features of available experimental data on organic salts (triangular) and herbertsmithite (kagome).
* Quantum effect, randomness and frustration are all essential to stabilize the random singlet state.