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

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

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”

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

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]

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]

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

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]

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

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.

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)]

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

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

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

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

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

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 ?

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

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

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

Susceptibility

[Y.Shimizu, et al . PRL, 2003 ]

Susceptibility exhibits a gapless behavior.

For sufficiently strong randomness, An intrinsic Curie tail is observed.

Susceptibility

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

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]

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

?

   

 

 

 

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 !

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 　

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.

Quantum spin-liquid-like behavior in the S =1/2 random kagome-lattice Heisenberg AF

herbertsmithite

  ZnCu3(OH)6Cl2

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

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

AF Neel and SG orders (T =0)

Neither Neel or SG order at any Δ

Sublattice magnetization

ms2

Spin liquid state for any Δ !

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

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]

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.]

“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

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

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

Related systems Is frustration essential ? S =1/2 random square-lattice Heisenberg AF Corresponding classical system ? Classical random triangular-lattice Heisenberg AF

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

Classical system ー random triangular AF 　Monte Carlo simulations

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

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

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 >

　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

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

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

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

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.