quantum spin liquids in simple model systems - mesoscopic and
TRANSCRIPT
Quantum Spin Liquids in Simple Model Systemsp q p y
Hong YaoHong YaoStanford University & Tsinghua University
Hong‐Chen Jiang, Hong Yao, and L. Balents, arXiv: 1112:2241.Hong Yao and Steve Kivelson, Phys. Rev. Lett. 108, 247206 (2012).g y ( )
Collaborators: Hong‐Chen Jiang (KITP, UCSB), Leon Balents (KITP, UCSB), and Steve Kivelson (Stanford University)
06/20/2012Landau Institute for Theoretical Physics, Russia
OutlineOutline
• Introduction and Motivations• DMRG study of the S = 1/2 AFM square J1‐J2 model
‐ Neither magnetic order nor VBS orderf l l l‐ finite topological entanglement entropy
• Possible experimental realization of the J1‐J2 model• Generalized quantum dimer models
‐ Exact spin liquid ground states‐ Low energy effective theory and correlations
• Conclusions
OutlineOutline
• Introduction and Motivations• DMRG study of the S = 1/2 AFM square J1‐J2 model
‐ Neither magnetic order nor VBS orderf l l l‐ finite topological entanglement entropy
• Possible experimental realization of the J1‐J2 model• Generalized quantum dimer models
‐ Exact spin liquid ground states‐ Low energy effective theory and correlations
• Conclusions
States of States of MatterMatter• Condensed matter physics:
‐ Discover, characterize, and classify states of ʺmatterʺ., , y• Conventional states of matter:
Broken symmetry principle (thanks to Landau):Broken symmetry principle (thanks to Landau):‐ Local order parameters‐ Ginzburg‐Landau description of phase transitionsGinzburg Landau description of phase transitions
Exotic states of Exotic states of ʺmatterʺmatterʺ̋• Exotic phases beyond broken symmetry principle:
‐ Fractional quantum Hall states (Laughlin, 1983)Fractional quantum Hall states (Laughlin, 1983)
Quasi‐particles carry fractional charge (and statistics)‐ Quantum spin liquids (elusive one) (Anderson, 1973)Quantum spin liquids (elusive one) (Anderson, 1973)
Spin‐charge separations
• New principles for exotic phases:New principles for exotic phases: ‐ Topological order: e.g. ground state degeneracy depending on topology of manifold (Wen, 1989)epe i g o opo ogy o a i o ( , )
‐ Long‐range entanglement for gapped states: universal correction to the usual area law of entanglement entropy. g py(Levin & Wen, 2006; Preskill & Kitaev, 2006)
Conventional ordersConventional orders in magnetsin magnets
Magnetic Long Range Order Non‐Magnetic Long Range Order
Conventional orders Conventional orders in magnetsin magnets
g g g g g g
Anti‐ferromagnetic order Valence bond solid order
Spin liquid ground states of magnetsSpin liquid ground states of magnets
• The first 2D spin liquid wave function: resonating
Spin liquid ground states of magnetsSpin liquid ground states of magnets
• The first 2D spin liquid wave function: resonating valence bond (RVB) (Anderson 1973)
YRVB= + + ▪▪▪
‐ No magnetic order no translational symmetry breakingNo magnetic order, no translational symmetry breaking.For one electron per site, band theory would predict a metal instead of an insulator.
‐ fractionalized excitations (spinon = charge 0 but spin‐1/2)
Important motivations to study Important motivations to study spin liquids in 2D?spin liquids in 2D?• New states of matter:
‐ fractionalized excitations e g spin‐charge separation
p yp y p qp q
‐ fractionalized excitations, e.g. spin‐charge separation in two and higher dimensionsspinon = charge‐0, spin‐1/2; holon = charge‐e, spin‐0p g p g p
• Possible routes to high temperature superconductivity‐ RVB picture for high Tc superconductivity in cuprates.
(Anderson, 1987; Kivelson, Rokhsar, and Sethna, 1987; Lee, Nagaosa, and Wen, RMP 2003)Nagaosa, and Wen, RMP 2003)
RVB theory for highRVB theory for high‐‐TTcc superconductivity?superconductivity?
• Reality: the undoped cuprates are AFM Mott insulators instead of quantum spin liquidsinsulators, instead of quantum spin liquids.
• Possible phase diagram: f
QSL
AFM SCWhat is f? Frustration or fantasy
doping
W i it thi bl b l ki f ibl i li idWe revisit this problem by looking for possible spin liquids on the square lattice as a function of frustration.
One candidate material for spin liquids: dmit‐131
• Organic insulator: EtMe3Sb[Pd(dmit)2]2‐ No magnetic order down to sub‐Kelvin even though J ~ 250K.
‐ Thermal conductivity kxx~T : infinite violation of Wiedmann‐y xxFranz law: spinon Fermi surface? (Yamashita et al, Science 2010)
Theoretical search for Theoretical search for quantum spin quantum spin liquidsliquids
• DMRG study of Kagome anti‐ferromagnet:(H‐C. Jiang et al, PRL 2009; Yan, Huse & White, Science 2011)(H C. Jiang et al, PRL 2009; Yan, Huse & White, Science 2011)A short‐range RVB‐like spin liquid ground state.
• Examples of Exactly solvable models with spin liquid ground states: Proof of Principle‐ Quantum dimer model and generalizations (Rokhsar and Kivelson, 1988; Moessner and Sondhi, 2001; Yao and Kivelson, 2011)‐ Toric code model (Kitaev, 2003)( )‐ Kitaev model and generalizations (Kitaev, 2006; H. Yao and Kivelson, 2007; V. Chua, H. Yao, & G. Fiete, 2011; Tikhonov, Feigelman, and Kitaev, 2011; many others)i ae , 0 ; a y o e s)
OutlineOutline
• Introduction and Motivations• DMRG study of the S = 1/2 AFM square J1‐J2 model
‐ Neither magnetic order nor VBS orderf l l l‐ finite topological entanglement entropy
• Possible experimental realization of the J1‐J2 model• Generalized quantum dimer models
‐ Exact spin liquid ground states‐ Low energy effective theory and correlations
• Conclusions
SpinSpin‐‐1/2 AF Heisenberg J1/2 AF Heisenberg J11‐‐JJ22 ModelModel
jiji SSJSSJH 21
pp / g J/ g J11 JJ22
ij
jij
j
Strong Frustrations J2/J1 ~ 1/2
Relevance to Cuprate, Fe‐based superconductors
J2 << J1 : Neel AFM order J2 >> J1 > 0: Stripe AFM order
k=(,0) k=(0,)e.g. Cuprate e.g. Fe‐based superconductor
k=(,)
Phase diagram: Classical levelPhase diagram: Classical level
jiji SSJSSJH 21
a e iag a a i a e ea e iag a a i a e e
ij
jiij
ji SSJSSJH 21
0 ∞
Neel AFM: k=( ) Stripe AFM: k=(0 )/( 0)
J2/J1J2/J1=0.5
First order phase transition Neel AFM: k () Stripe AFM: k (0,)/(,0)in classical level
Phase diagram: Quantum level (spin 1/2)Phase diagram: Quantum level (spin 1/2)
jiji SSJSSJH 21
a e iag a Qua u e e ( pi / )a e iag a Qua u e e ( pi / )
ij
jiij
ji SSJSSJH 21
0 ∞
J /JJ /J 0 5 J /J 0 5 J2/J1J2/J1<0.5
Ph t iti i
J2/J1>0.5
Neel AFM: k=( ) Stripe AFM: k=(0 )/( 0)Phase transition inquantum level
Neel AFM: k () Stripe AFM: k (0,)/(,0)
Earlier Studies of S=1/2 Square Earlier Studies of S=1/2 Square JJ11‐‐JJ22 ModelModel
Projected BCS mean‐field state
Ea ie u ie o / qua eEa ie u ie o / qua e JJ11 JJ22 o eo e
Spin‐wave calculationZ2 Spin liquid
X.‐G. Wen, PRL 1991 Likely Spin Liquid at [0.38, 0.51]
P. Chandra, and B. Doucot, PRB 1988
Large‐N calculation Valence bond solid(Dimer order)
ED in a subset of short‐range VB
singlet l l b d l d d Valence bond solid(Dimer order)
N. Read, and S. Sachdev, PRL 1989
Series expansion
Plaquette valence bond solid orderM. Mambrini, A. Lauchli et. al., PRB 2006
Columnar dimer order [0.34,0.61]M. P. Gelfand, R. R. P. Singh, D. A. Huse, PRB 1989
PEPS (D=3) simulation Columnar dimer order [0.5,0.7]
V M F V t t d J I Ci PRB 2009
Exact diagonalization (ED) on4x4 Spin liquid (Short‐range RVB)
V. Murg, F. Verstraete, and J. I. Cirac, PRB 2009
ED on larger cluster N=20,32,36,40p q ( g )F. Figueirido, S. Kivelson, et al., PRB 1989 Gapped quantum paramagnetic
phase J. Richter, J. Schulenburg, PRB 2010
Earlier Studies of S=1/2 Square Earlier Studies of S=1/2 Square JJ11‐‐JJ22 ModelModel
ij
jiij
ji SSJSSJH 21
Ea ie u ie o / qua eEa ie u ie o / qua e JJ11 JJ22 o eo e
ijij
0 J2/J10 J2/J1
N l AFM k ( ) St i AFM k Neel AFM: k=() Stripe AFM: k=
There exists an intermediate magnetically disordered phase
Two candidate ground states for the intermediate phase:
There exists an intermediate magnetically disordered phase
‐ Quantum spin liquid (Spin‐wave calculation, ED with N=4x4, Projected BCS)
‐ VBS (Large‐N, Series expansion, PEPS, ED in subset of SR‐VB)
To get reliable ground states, we perform DMRG simulations on ladders with large circumference.
DMRG Study of S=1/2 Square JDMRG Study of S=1/2 Square J11‐‐JJ22 ModelModelu y o / qua e Ju y o / qua e J11 JJ22 o eo e
jiji SSJSSJH 21
Cylinder BCs up to L =12
ij
jiij
ji SSJSSJH 21
Cylinder BCs, up to Ly=12
Keep m>6000, error<10‐7
Fi d ti L /L 2 ith i i Fixed ratio Lx/Ly=2, with minimum
finite‐size effect,White et al., PRL (2007)
d Measurement restricted
to central‐half of the systemLL4
xL4
xL
Lyyy
x LLLL
2
Detect magnetic orders in the Detect magnetic orders in the S=1/2 Square JS=1/2 Square J11‐‐JJ22 ModelModele e ag e i o e i ee e ag e i o e i e / qua e J/ qua e J11 JJ22 o eo e
Neel AFM Stripe AFM
J2/J1k= k=
Static structure factor Staggered magnetization
Fitting function
Magnetic Order ParameterMagnetic Order Parameterag e i O e a a e eag e i O e a a e eNeel AFM order
Non‐zero for J2<0.41
At J2=0, ms(k0,∞)=0.304
Close to QMC result ms=0.307,Sandvik, PRB, 1997
Stripe AFM order1/L
p
Non‐zero for J2>0.62
1/L
Spin Singlet and Triplet GapSpin Singlet and Triplet Gappi i g e a ip e appi i g e a ip e ap
Spin triplet gap
Zero for magnetic order phase
Non‐zero in the intermediate phasep
At J2=0.5, ΔT ≈ 0.1
Spin singlet gap1/L
p g g p
Zero for magnetic ordered phase
Non‐zero in the intermediate phase Non zero in the intermediate phase
At J2=0.5, ΔS ≈ 0.05
1/L
Phase diagram of S=1/2 Square Phase diagram of S=1/2 Square JJ11‐‐JJ22 ModelModela e iag a o / qua ea e iag a o / qua e JJ11 JJ22 o eo e
N l AFM S i AFMN i h
J /J
Neel AFM Stripe AFM
0 41 0 62
Non‐magnetic phase
J2/J10.41 0.62
Neel AFM: k= Stripe AFM: k=( Neel AFM: k= Stripe AFM: k (
VBS oVBS order rder pparameterarameteroo ee ppa a e ea a e e
Dimer operator on bond (i, i+) (=x/y): Dimer structure factor:
Dimer order parameter: ( ,0), (0, )x y k k
Dimer order parameter vanishes in all region gof 0 < J2 < 1
1/L
Phase diagram of S=1/2 Square JPhase diagram of S=1/2 Square J11‐‐JJ22 ModelModela e iag a o / qua e Ja e iag a o / qua e J11 JJ22 o eo e
J /JJ2/J1Neel AFM Stripe AFM
0.41 0.62
Neel AFM: k= Stripe AFM: k=
Intermediate phase
1)No magnetic order
2)Fully gapped to all excitations
3)No dimer VBS order
Most likely it is a gapped quantum spin liquidMost likely, it is a gapped quantum spin liquid.
Any positive evidence?
Hallmark of topological order: topological entanglement entropyHallmark of topological order: topological entanglement entropy
1) Partitions A and B with smooth boundary A
B
2) Obtain reduced density matrix:
trA B B
3) Compute Von Neumann entropy:
[ log( )]A AS tr
Gapped phase: ( ) +S l a l
[ g( )]A A
pp p ( )S l a l 1) :topologically trivial state = 0
Levin and Wen, PRL 2006;
2) :topologically ordered state 0
Kitaev and Preskill, PRL 2006
NonNon‐‐zero topological entanglement entropyzero topological entanglement entropyoo e o opo ogi a e a g e e e opye o opo ogi a e a g e e e opy
Von Neumann entropy
A B( )y yS L a L
l=Ly
± γ ±
For J2/J1=0.5, γ ±, close to –ln(2)= 0.69, clearly J2/J1 , γ , o e o ( ) , y
suggesting that such a phase is a Z2 quantum spin liquid
Phase diagram of S=1/2 Square Phase diagram of S=1/2 Square JJ11‐‐JJ22 AFM ModelAFM ModelPhase diagram of S 1/2 SquarePhase diagram of S 1/2 Square JJ11 JJ22 AFMModelAFMModel
J /JJ2/J1Neel AFM Stripe AFM
0.41 0.62
Neel AFM: k=() Stripe AFM: k=()
OutlineOutline
• Introduction and Motivations• DMRG study of the S = 1/2 AFM square J1‐J2 model
‐ Neither magnetic order nor VBS orderf l l l‐ finite topological entanglement entropy
• Possible experimental realization of the J1‐J2 model• Generalized quantum dimer models
‐ Exact spin liquid ground states‐ Low energy effective theory and correlations
• Conclusions
SpinSpin‐‐1/2 Quasi1/2 Quasi‐‐2D materials2D materials
• Many spin‐1/2 Mott insulator materials with effe ti e J J i te a tio
pp
effective J1‐J2 interactions: ‐ Cuprates, J2/J1 ≈ 0.1: J2 is too weakLi VOSiO Li VOSiO J /J ≈ 1 0 10 0: J dominates‐ Li2VOSiO4, Li2VOSiO4, J2/J1 ≈ 1.0 ~ 10.0: J2 dominates ‐ VOMoO4, J2/J1 ≈ 1.0: J2 dominates
Li2VOSiO4 VOMoO4
PPressure effect in spinressure effect in spin‐‐1/2 1/2 QuasiQuasi‐‐2D materials2D materialspp
First principle study of First principle study of pressure effect in Lipressure effect in Li22VOSiOVOSiO44
P(GPa) J2/J10 7 10410 ‐7.1041 50 12.9575 100 2.1647 150 1.5521 200 1.4624 210 1.4662 220 1.4735 230 1.4849 250 1 5051250 1.5051
1) It’s still Mott insulator with ~2eV band gap when the1) It s still Mott insulator with ~2eV band gap when the hydrostatic pressure is applied.2) When the pressure reaches 150 GPa, the J2/J1 ratio stops decreasing, while J1 and J2 still increase.
P. C. Chen, P. Z. Tang, H. Yao, W. H. Duan, unpublished
OutlineOutline
• Introduction and Motivations• DMRG study of the S = 1/2 AFM square J1‐J2 model
‐ Neither magnetic order nor VBS orderf l l l‐ finite topological entanglement entropy
• Possible experimental realization of the J1‐J2 model• Caricature of spin liquids from quantum dimermodels‐ Exact spin liquid ground states‐ Low energy effective theory and correlations
• Conclusions
RVBRVB Quantum Quantum dimerdimer modelmodel
• RVB states with short‐range valence bonds
YRVB= + + ▪▪▪
• But, no known microscopic models for gapped RVB , p g ppground states on the square lattice.
• To have a caricature of RVB states: (1) valence bond dimer(2) dimer configurations are orthogonalg g(3) all possible dimer coverings consist complete Hilbert space. (4) write down a simple dimer Hamiltonian: quantum dimer
modelRohksar and Kivelson, PRL 1988
Quantum Quantum dimerdimer models on the square latticemodels on the square lattice
• Simplest RVB states: only NN valence bonds
YRVB= + + ▪▪▪
• Quantum dimer model:Q
t: kinetic energy, V: potential energy
t=V is the RK point, an exactly solvable point: 0 =c
c C l VBS Pl tt VBS St d VBSColumnar VBS Plaquette VBS Staggered VBS
V/t
1=RK: a critical pointRohksar and Kivelson, 1988
Generalized quantum Generalized quantum dimerdimer models models
• RVB states with both NN and NNN valence bonds:
YRVB= + + ▪▪▪
• Corresponding quantum dimer model:p g q
RK points: t = V tʹ = Vʹ tʹʹ = Vʹʹ any ( )
0 = dn c
c
c Exact ground state at RK points:RK points: t = V, t = V , t = V , any
H. Yao and Kivelson, 2012
Correlations of RK wave functionsCorrelations of RK wave functions( )
0 = dn c
c
c RK ground state wave functions:
Ground state correlators are equivalent to classical dimer model Classical dimer model is solvable by the ʺPfaffianʺ method
tij=1, for <ij> tij=, for <<ij>>
0 0| [ ]exp( )( )( )jk j k jk j k lm l m
Z da SS it a a it a a it a a
jk jklm
† † †[ ]x y zS d d i i d d In the continuum limit:
† † †0 1 1 2 2[ ]x y z
y xS dxdy i i m g dxdy m0 = 2, g = 42
i i l t di d l ith l NN di original quantum dimer model with only NN dimersFree massless Dirac fermionsdimer correlations <D(0) D(r)> ~ 1/r2 a critical pointdimer correlations <D(0) D(r)> 1/r , a critical point.
≠ 0: interacting fermions in 2+0 or 1+1 dimensions.
Correlations of RK wave functionsCorrelations of RK wave functions
† † †[ ]x y zS d d d d In the continuum limit, y ‐> t: massive Thirring model
† † †0 1 1 2 2[ ]x y z
t xS dxdt i i m g dxdt Bosonization:
2 20
1 1[ ( ) ( ) ] cos( 4 )2 F x t
F
S dxdt v mK v
1 /(2 ) is the Luttinger parameter1 /(2 )
gKg
The mass term is still relevant for finite g, indicating a gap.
Exact Bethe Ansatz solution of the mass gap in interacting Exact Bethe‐Ansatz solution of the mass gap in interacting massive Thirring model: m = m0 exp(‐g/)
ll l l l ti h t d→ all local correlations are short‐ranged.
Phase diagram of generalized quantum Phase diagram of generalized quantum dimerdimer modelmodelg g qg g q
RK points: a gapped quantum spin liquid
Gapped quantum spin liquid is impressible, i.e. stable against small perturbation in the Hamiltonian.
Columnar VBS Plaquette VBS Staggered VBS
V/tRVB liquid
1=RK
Stable gapped Z2 spin liquids on the square lattice is proved to
Yao and Kivelson, 2012
g pp p q q pexist in models.
,
ConclusionsConclusions
• DMRG study of the J1‐J2 model:h d l b d l d d‐ Neither magnetic order nor valence bond solid order
‐ Positives evidence of the spin liquid ground state: fi i l i l lfinite topological entanglement entropy
• Possible experimental realizationsA l l h l l l /‐ Apply pressure to materials with relatively large J2/J1
• Caricature from generalized quantum dimer models‐ Exact Z2 quantum spin liquid on the square lattice
Thank you!