gapless spin liquids in frustrated heisenberg models · 1 ∼ 0.5: disordered phase (rvb liquid,...
TRANSCRIPT
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Gapless spin liquids in frustrated Heisenberg models
Federico Becca
CNR IOM-DEMOCRITOS and International School for Advanced Studies (SISSA)
New Horizon of Strongly Correlated Physics, June 2014
Y. Iqbal (ICTP, Trieste), W.-J. Hu (now CSUN)
A. Parola (Como), D. Poilblanc (CNRS, Toulouse), and S. Sorella (SISSA)
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1 Introduction
2 The Heisenberg model on the frustrated square lattice
3 The Heisenberg model on the Kagome lattice
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Ultimate frustration?
Looking for a magnetically disordered ground state?
• Many theoretical suggestions since P.W. Anderson (1973)Anderson, Mater. Res. Bull. 8, 153 (1973)
Fazekas and Anderson, Phil. Mag. 30, 423 (1974)
“Resonating valence-bond” (quantum spin liquid) states
Idea: the best state for two spin-1/2 spins is a valence bond (a spin singlet):
|VB〉R,R′ =1√2(| ↑〉R| ↓〉R′ − | ↓〉R| ↑〉R′)
Every spin of the lattice is coupled to a partnerThen, take a superposition of different valence bond configurations
Ψ = + + + ...
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Valence-bond states: liquids and solids
Valence-bond solidbreaks translational/rotationalsymmetries
Short-range RVB + + ...
Long-range RVB + + ...
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The Heisenberg model on the frustrated square lattice
H = J1∑
〈ij〉
Si · Sj + J2∑
〈〈ij〉〉
Si · Sj
Neel order Collinear order
Spin singlet
0 0.5 1
For J2/J1 ≪ 1: Antiferromagnetic order at Q = (π, π)
For J2/J1 ≫ 1: Antiferromagnetic order at Q = (π, 0) and Q = (0, π)
For J2/J1 ∼ 0.5: Disordered phase (RVB liquid, dimer order or more exotic?)
Experimental realization in Li2VOSiO4 (J2 & J1) and VOMoO4 (J2 < 0.5J1)
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Recent DMRG calculations
Jiang, Yao, and Balents, PRB 86, 024424 (2012)
• It has been criticized by Sandvik (possible dimer order)Sandvik, PRB 85, 134407 (2012)
• More recent calculations with both spin-liquid and dimer phases
Gong, Zhu, Sheng, Motrunich, Fisher, arXiv:1311.5962
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Describing magnetically disordered phases
Sµi =
1
2c†i,ασ
µα,βci,β
H = −1
2
∑
i,j,α,β
Jij
(
c†i,αcj,αc
†j,βci,β +
1
2c†i,αci,αc
†j,βcj,β
)
c†i,αci,α = 1 ci,αci,βǫαβ = 0
• At the mean-field level:
HMF =∑
i,j,α
(χij + µδij)c†i,αcj,α +
∑
i,j
(ηij + ζδij)(c†i,↑c
†j,↓ + c
†j,↑c
†i,↓) + h.c.
〈c†i,αci,α〉 = 1 〈ci,αci,β〉ǫαβ = 0
• Then, we reintroduce the constraint of one-fermion per site:
|Φ(χij , ηij , µ, ζ)〉 = PG |ΦMF(χij , ηij , µ, ζ)〉
PG =∏
i (1− ni,↑ni,↓)
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Symmetry of the mean-field ansatz
In 2001, we guessed:χk = 2t(cos kx + cos ky )ηk = ∆x2−y2(cos kx − cos ky ) + ∆xy (sin 2kx sin 2ky )After Wen, called Z2Azz13
〈s〉 =∑
x
|〈x |Φ〉|2sign {〈x |Φ〉〈x |Ψex〉}
After the Wen’s classification, it is the best projected fermionic state
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Towards the exact ground state
How can we improve the variational state?By the application of a few Lanczos steps!
|Ψp−LS〉 =(
1 +∑
m=1,...,p
αmHm
)
|ΨVMC 〉
For p → ∞, |Ψp−LS〉 converges to the exact ground state, provided 〈Ψ0|ΨVMC 〉 6= 0
On large systems, only FEW Lanczos steps are affordable: We can do up to p = 2
In addition, a fixed-node (FN) projection is possibleten Haaf et al., PRB 51, 13039 (1995)
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The variance extrapolation
A zero-variance extrapolation can be done
Whenever |ΨVMC 〉 is sufficiently close to the ground state:
E ≃ E0 + const× σ2 E = 〈H〉/Nσ2 = (〈H2〉 − E 2)/N
How does it work?Example: the t−J model
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Going to the excitation spectrum
If a variational approach works also low-energy excitations must be described
HMF =∑
i,j,α
(χij + µδij)c†i,αcj,α +
∑
i,j
(ηij + ζδij)(c†i,↑c
†j,↓ + c
†j,↑c
†i,↓) + h.c.
After a Bogoliubov transformation:
HMF =∑
k
(Ekψ†kψk − Ekφ
†kφk)
The ground state is:
|Φ0MF〉 =
∏
k
φ†k |0〉
Excited states are obtained by:
φq1. . . φqnψ
†p1. . . ψ†
pm |Φ0MF〉
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Spin excitations
Considering excited states that can be described by a SINGLE determinant, we have:
• The S=0 ground state• The S=2 with momentum k = (0, 0)• The S=1 with momentum k = (π, 0) or k = (0, π)
Test case: J2/J1 = 0.5 and 6× 6 cluster
0.00 0.01 0.02
–0.495
–0.490
–0.485
–0.480
–0.475
0 0.01 0.02
–0.495
–0.485
–0.475
–0.465
Variance Variance0 3.10–3 6.10–3 9.10–3
–0.504
–0.502
–0.500
–0.498
–0.496
–0.494
Variance
E(S
) pe
r si
te
6x68x810x1014x1418x18
(a) (b) (c)
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Spin excitations
• The S=2 gap vanishes in the Neel phase• The S=1 gap at k = (π, 0) is instead finite in the Neel phase
The Lanczos extrapolation is performed on each state separately
Calculations on the 6× 6 cluster vs exact results
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.550.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
J 2/J 1
∆ 1
S=1 gap
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.550.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
J 2/J 1
∆ 2
exactp=0p=1p=2variance extrapolation
S=2 gap
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The S=2 spin excitation for large sizes
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.180.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1/L
∆ 2J2/J1=0.5, p=0
J2/J1=0.5, variance extrapolation
J2/J1=0.5, triplet gap from DMRG
J2/J1=0.55, p=0
J2/J1=0.55, variance extrapolation
J2/J1=0.55, triplet gap from DMRG
• J2/J1 = 0.5 −→ ∆2 = −0.04(5)• J2/J1 = 0.55 −→ ∆2 = −0.07(7)
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The S=1 spin excitation for large sizes
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.180.0
0.2
0.4
0.6
0.8
1.0
1/L
∆ 1
J 2/J1=0.45, p=0
J 2/J1=0.45, variance extrapolation
J 2/J1=0.5, p=0
J 2/J1=0.5, variance extrapolation
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.180.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1/L
J 2/J1=0.3
J 2/J1=0.4
J 2/J1=0.45
J 2/J1=0.5
J 2/J1=0.55
∆1
0.30 0.35 0.40 0.45 0.50 0.550.0
0.1
0.2
0.3
0.4
0.5
0.6
J 2/J 1
∆ 1
0.48(2)
The spin gap is FINITE for J2/J1 < 0.48
Instead, it vanishes for J2/J1 > 0.48
NON-trivial aspect of the SL phase!
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Conclusions (first part)
Very good energiesWith few (TWO-THREE) variational parameters: Educated guessTo be compared with about 16000 parameters in DMRG: Brute-force calculation
Direct calculation of the spin gap for S=2 and S=1 excitationsIn both cases, we find evidence for a GAPLESS spin liquid
Our calculations are done on L× L clusters with PBC in both directions
DMRG calculations are done on 2L× L cylinders with OBC along x and PBC along y
(but the gap is computed only in the central part of the cluster)
Our approach may capture both gapless and gapped states
DMRG algorithm favors low-entangled (gapped) states
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The Heisenberg model on the Kagome lattice
H = J∑
〈ij〉
Si · Sj + . . .
• No magnetic order down to 50mK (despite TCW ≃ 200K)
• Spin susceptibility rises with T → 0 but then saturates below 0.5K
• Specific heat Cv ∝ T below 0.5K
• No sign of spin gap in dynamical Neutron scattering measurements
Mendels et al., PRL 98, 077204 (2007)
Helton et al., PRL 98, 107204 (2007)
Bert et al., PRB 76, 132411 (2007)
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Some of the previous results
Nearest-neighbor Heisenberg model on the Kagome lattice
Author GS proposed Energy/site Method used
P.A. Lee U(1) gapless SL −0.42866(1)J Fermionic VMC
Singh 36-site HVBC −0.433(1)J Series expansion
Vidal 36-site HVBC −0.43221 J MERA
Poilblanc 12- or 36-site VBC QDM
Lhuillier Chiral gapped SL SBMF
White Z2 gapped SL −0.4379(3)J DMRG
Schollwoeck Z2 gapped SL −0.4386(5)J DMRG
Xie et al. gapped SL −0.4364(1)J PESS
Ran, Hermele, Lee, and Wen, PRL 98, 117205 (2007)
Yan, Huse, and White, Science 332, 1173 (2011)
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Results with projected wave functions
A variational ansatz with ONLY hopping but non-trivial fluxes has been proposed with
• π fluxes through hexagons and 0 fluxes through triangles• Dirac points in the mean-field spinon spectrumRan, Hermele, Lee, and Wen, PRL 98, 117205 (2007)
a
b c
d
1 5
6
a
b c
d
dd
1
3
5
6
(a)
(b)
(d) (e)
0 a1
2a
0
1 2
3
4 5
6
(c)
0
0 0
• 0 fluxes everywhere• Fermi surface in the mean-field spinon spectrum
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Can we have a Z2 gapped spin liquid (as in DMRG)?
Projective symmetry-group analysis
Lu, Ran, and Lee, PRB 83, 224413 (2011)
uij =
(
χ∗ij ∆ij
∆∗ij −χij
)
a
b c
d
1 5
6
a
b c
d
dd
1
3
5
6
(a)
(b)
(d) (e)
0 a1
2a
0
1 2
3
4 5
6
(c)
0
0 0
Only ONE gapped SL connected with the U(1) Dirac SL:
FOUR gapped SL connected with the Uniform U(1) SL:
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The Dirac U(1) SL is stable against opening a gap
• By optimizing the variational state, the breaking terms go to zero
• The best variational energy is obtained by the U(1) Dirac state
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Calculations on the 48-site cluster
Our zero-variance extrapolation gives: E/N ≃ −0.4378
0 0.005 0.01 0.015 0.02 0.025
Variance of energy (σ2)
-0.435
-0.43
-0.425
-0.42
-0.415
-0.41E
nerg
y pe
r si
te
Uniform RVB ([0,0]) SL {VMC}Z
2[0,π]β SL {VMC}
U(1) Dirac ([0,π]) SL {VMC}U(1) Dirac ([0,π]) SL {FN}DMRG (Torus) [Jiang et al.]
0 0.001 0.002-0.438
-0.437
-0.436
E/N ≃ −0.4387 by ED, A. Lauchli (seen at APS in Boston)E/N ≃ −0.4381 by DMRG, S. White (private communication)
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Calculations on larger clusters
0 0.005 0.01 0.015
Variance of energy (σ2)
-0.438
-0.436
-0.434
-0.432
-0.43
-0.428
Ene
rgy
per
site
48 sites {VMC}108 sites {VMC}144 sites {VMC}192 sites {VMC}DMRG (Torus 108) [Depenbrock et al.]
0 0.001 0.002
-0.438
-0.437
0 0.005 0.01 0.015
Variance of energy (σ2)
-0.438
-0.436
-0.434
-0.432
-0.43
-0.428
Ene
rgy
per
site
48 sites {FN}108 sites {FN}144 sites {FN}192 sites {FN}DMRG (Torus 108) [Depenbrock et al.]
0 0.001 0.002
-0.438
-0.437
• NO substraction techniques to get the energy• The state has ALL symmetries of the lattice• The extrapolated values are essentially size consistent
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Extrapolations for Dirac and Z2 states
Starting from a Z2 state the extrapolation is worse than for the Dirac state
0 0.005 0.01 0.015 0.02 0.025
Variance of energy (σ2)
-0.436
-0.432
-0.428
-0.424
Ene
rgy
per
site
192 sites - U(1) Dirac SL {VMC}192 sites - Z
2[0,π]β SL {VMC}
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The thermodynamic limit
0 0.01 0.02 0.031/N
-0.4385
-0.438
-0.4375
-0.437
-0.4365G
roun
d st
ate
ener
gy p
er s
ite 2d estimate from Lanczos+VMC (current work)2d estimate from DMRG (Depenbrock et al.)2d estimate from DMRG (Yan et al.)2d estimate from CCM (Götze et al.)
• OUR thermodynamic energy is: E/J = −0.4365(2)• DMRG thermodynamic energy is: E/J = −0.4386(5)
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The variational S = 2 gap
0 0.05 0.1 0.15 0.2 0.25Inverse geometrical diameter (1/L)
0
0.2
0.4
0.6
0.8
1
1.2
S=2
Spi
n G
ap (∆
2)
Projected U(1) Dirac SLMean field U(1) Dirac SL
• The Gutzwiller projected state is still gapless• Remarkable stability of the U(1) Dirac spin liquid
Federico Becca (CNR and SISSA) Gapless Spin Liquids NHSCP at ISSP 26 / 32
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The Lanczos step extrapolations
0 0.005 0.01 0.015
Variance of energy (σ2)
-0.438
-0.436
-0.434
-0.432
-0.43
-0.428
Ene
rgy
per
site
(E/J
)
48-sites108-sites192-sites432-sites
0 0.005 0.01 0.015 0.02
Variance of energy (σ2)
-0.436
-0.432
-0.428
-0.424
-0.42
-0.416
Ene
rgy
per
site
(E/J
)
48-sites108-sites192-sites432-sites
• We separately extrapolate both S = 0 and S = 2 energies• Then the gap (zero-variance) gap is computed
Federico Becca (CNR and SISSA) Gapless Spin Liquids NHSCP at ISSP 27 / 32
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The S = 2 gap of the kagome Heisenberg model
0 0.05 0.1 0.15 0.2 0.25Inverse geometrical diameter (1/L)
0
0.1
0.2
0.3
0.4
0.5
S=2
Spi
n G
ap (∆
2)
p = 0 Lanczos stepsp = 2 Lanczos steps extrapolated
• The final result is ∆2 = −0.04± 0.06• The “upper” bound is given by ∆2 ≃ 0.02• The S = 1 gap should be ∆1 . 0.01
Much smaller than previous DMRG estimationsMore similar to recent calculations by Nishimoto et al. ∆1 = 0.05± 0.02Nishimoto, Shibata, and Hotta, Nat. Commun. 4, 2287 (2013)
Federico Becca (CNR and SISSA) Gapless Spin Liquids NHSCP at ISSP 28 / 32
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Adding the next-nearest-neighbor super-exchange
0 0.1 0.2 0.3 0.4 0.5J
2
-0.5
-0.48
-0.46
-0.44
-0.42
-0.4
-0.38
Ene
rgy
per
site
U(1) Dirac SLZ
2[0,π]β SL
q=0 magnetically ordered state2 Lanczos steps + ZVE {Z
2[0,π]β SL}
48-sites
0 0.1 0.2 0.3 0.4 0.5J
2
-0.5
-0.48
-0.46
-0.44
-0.42
-0.4
-0.38
Ene
rgy
per
site
U(1) Dirac SLZ
2[0,π]β SL
q=0 magnetically ordered state
192-sites
• The gapped Z2 state overcomes the U(1) Dirac one for J2/J1 > 0.1
• The magnetic state with q = 0 (Jastrow) is stabilized for J2/J1 > 0.3
Federico Becca (CNR and SISSA) Gapless Spin Liquids NHSCP at ISSP 29 / 32
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Adding the next-nearest-neighbor super-exchange
Applying Lanczos steps on 48 sites
0 0.005 0.01 0.015 0.02 0.025 0.03
Variance of energy-0.46
-0.456
-0.452
-0.448
-0.444
-0.44
-0.436
Ene
rgy
per
site
Z2[0,π]β SL
U(1) Dirac SL
48-site torus, J2=0.20
• For finite J2/J1 the gapped Z2 state has a “better” extrapolation with Lanczos steps
• Calculations on larger clusters are in progress...
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Conclusions
Results up to now:
Very good energiesWith TWO variational parameters: Educated guessTo be compared with about 16000 parameters in DMRG: Brute-force calculation
Direct calculation of the S = 2 gapNo evidence for a finite gap in thermodynamic limit
Pros• Very flexible approach that may describe several different phases
(gapped and gapless, not only low-entanglement states)
• Natural way of constructing and understanding low-energy excitations
• Applying few Lanczos steps allows for a sizable improvement
Cons• Still, it is a biased approach and more work must be done
Federico Becca (CNR and SISSA) Gapless Spin Liquids NHSCP at ISSP 31 / 32
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Different kinds of spin liquids
The folklore is that
• DMRG always produces excellent spin liquids
• Other numerical methods produce no good liquids
We are trying to improve the quality of our liquids
Federico Becca (CNR and SISSA) Gapless Spin Liquids NHSCP at ISSP 32 / 32