Spin Liquid in the Heisenberg model on the Kagome and ... · Spectroscopy of SU(2) broken states F. Kolley, S. Depenbrock, IPM, U. Schollwöck, V. Alba, Phys. Rev. B 88, 144426 (2013)

Spin Liquid in the Heisenberg model on the Kagomeand Triangular lattices

Workshop and Symposium on DMRG Technique for Strongly CorrelatedSystems in Physics and Chemistry

Ian McCulloch

University of QueenslandCentre for Engineered Quantum Systems (EQuS)

26/6/2015

Ian McCulloch (UQ) iDMRG 26/6/2015 1 / 26

Outline

1 Kagome Lattice

2 Triangular Lattice – 3-leg cylinder

3 Triangular lattice – 2D


Kagome Lattice – Spin-1/2 Heisenberg Model

Classical phase diagram

J2 < 0 J2 = 0 – extensive degeneracy J2 > 0


J2 = 0 S. Depenbrock, IPM, U. Schollwöck, Phys. Rev. Lett. 109, 067201 (2012)

SU(2)-preserving DMRG8,000 states – equivalent to ∼ 16,000 states with U(1)allows increase in cylinder size compared with previous calculations


Triplet gap

SU(2) symmetry gives much better convergence with 1/W scalingGap ∼ 0.13(1)


Topological Entanglement Entropy

Reyni entropy Sα = 11−α log2 tr ρα

Renyi entropies for small α do not converge well - the tail isn’t wellrepresented in DMRGRenyi entropies for larger α are consistent with γ = 1γ = log2(D) – quantum dimension D = 2


Phase diagram for J1 − J2 F. Kolley, S. Depenbrock, IPM, U. Scholloöck,V. Alba, Phys. Rev. B 91, 104418 (2015)

Triplet gap as a function of J2


Spin structure factor

Clear signals of magnetic ordering√(3)×

√3 structure at J2 = −0.2

q = 0 structure at J2 = 0.4


SU(2) symmetry breaking

In 1D calculations, it is almost always beneficial to use any availablecontinuous symmetry (definitely for finite MPS, more subtle for infiniteMPS)in 2D it isn’t clear – continuous symmetries can spontaneously breakIs it a good idea to use SU(2) symmetry, even if it is broken?

In 1D the answer would almost certainly be no

compare Néel state with entropy S = 0 compared with the projection ontoa singlet state with entropy S ∼ ln N

In 2D, the answer appears to be yesadditional entropy is still only ∼ ln Nsmall correction verus S ∝ width, especially for small finite systems


Spectroscopy of SU(2) broken states F. Kolley, S. Depenbrock,IPM, U. Schollwöck, V. Alba, Phys. Rev. B 88, 144426 (2013)

The appearance of Goldstone modes kills the gap in the entanglementspectrumThe low-lying levels have a Tower of States structure


Entanglement Spectrum Scaling

Goldstone modes – linear dispersion, � ' 1/√(V)

Implies that the TOS gap should vanish in 1/WFinite gap to higher levels


Triangular lattice3-lag cylinder – S. Saadatmand, B. Powell, IPM, Phys. Rev. B 91, 245119 (2015)

Map 3-leg cylinder onto 1D in the YC configuration


Phase diagram

120ᵒ gapless

FM saturated

Majumdar–Ghosh phase

Columnar

J2/J

J1/J

165ᵒ

θ = 0

θ = π/2

θ = ±π

θ = -π/2

6.5ᵒ

70.0ᵒ

152.0ᵒ


120° State

............

(a)

Columnar State

............

(b)

A

B

C

C

A

B

A

B

C

C

A

B

A

B

C

C

A

B

A

B

C

C

A

B

............

Majumdar-Ghosh (θ=90°)

(c)

Majumdar-Ghosh State

............

(d)


iDMRG – correlation length scaling

100 1000m

1

10

100η

120° state, θ=-45°quasi-long-range

Majumdar-Ghosh state, θ=115°short-range

Columnar state, θ=38°quasi-long-range


Columar – Majumdar-Ghosh transition

Phase transition from gapless columnar state to gappedMajumdar-Ghosh stateHow to locate?

30 40 50 60 70 80 90θ (degrees)

0

0.1

0.2

0.3

0.4

0.5Sp

in G

ap

Majumdar-GhoshState

ColumnarState

θc

(a)


Better approach – Binder cumulant


2D triangular lattice S. Saadatmand (work in progress)

In the 2D model, possible spin liquid phase

Mishmash et al., Phys. Rev. Lett. 111, 157203 (2013)


More recently, preprints from S. R. White’s group (1502.04831) andSheng group (1504.00654)Two candidate groundstates, suggest Z2 spin liquidbut TEE difficult to fitMany puzzles – chiral? What are the excitiations?

Most calculations done at J2 = 0.1

0 0.2 0.4 0.6 0.8 1α=J

2/J

1

0

0.2

0.4

0.6

0.8

1

χ/χ c

lass

ical

Sublattice Magnetization(120° order parameter)Staggered Magnetization(columnar order parameter)

α1=0.109(4) α

classical α2=0.222(5)

Magnetization order parameters,finite DMRG results for THM,extrapolated to thermodynamic limitusing all or some of: YC30×3, XC36×4,YC36×4, 5leg-Cwrap=(-4,5), andYC48×6, m

max=1250.

%25(8)


iDMRG calculations

Two groundstates, in odd and even spin sectorswith SU(2) iDMRG this is a simple manipulataion of the boundaryquantum numbers – the system remains translationally invariantPhase boundaries are a bit different from previous calculationsNo chiral symmetry breakingstructure factor in the spin liquid is a bit different to previous calculationsAverage nearest-neighbor spin correlation is somewhat different (we getS.S ' −0.25 to −0.3, versus −0.18 to 0.22)


Correlation length


Structure factor120◦ state

"KxKyColor.SSF.alpha_-0.025.UnitCell18.iTHC-YC6.m_1250.grid200x200.EfficientNumbering.dat"

-6 -4 -2 0 2 4 6

Kx

-6

-4

-2

0

2

4

6

Ky

1

2

3

4

5

6

7


Structure factorSpin Liquid – even sector

"KxKyColor.SSF.alpha_0.125.UnitCell12.iTHC-YC12.m_5000.grid200x200.EfficientNumbering.dat"

-6 -4 -2 0 2 4 6

Kx

-6

-4

-2

0

2

4

6

Ky

1.5

2

2.5

3

3.5

4

4.5

5


Structure factorSpin Liquid – odd sector

yColor.SSF.alpha_0.125.SpinHalfBoundary.UnitCell24.iTHC-YC12.m_3500.grid200x200.EfficientNumbering.dat"

-6 -4 -2 0 2 4 6

Kx

-6

-4

-2

0

2

4

6

Ky

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5


Structure factorColumnar phase

"KxKyColor.SSF.alpha_0.465.UnitCell18.iTHC-YC6.m_2000.grid200x200.dat"

-6 -4 -2 0 2 4 6

Kx

-6

-4

-2

0

2

4

6

Ky

0

2

4

6

8

10

12

14

16

18


Summary

Kagome Lattice

Strong evidence for a Z2 spin liquidTriplet spin gap ∼ 0.13(1)Cannot (yet) distinguish between Z2 (toric code) and double-semionliquidsSU(2) level spectroscopy is a useful tool for symmetry-broken phases

Triangular lattice3-leg ladder shows 120◦ to columnar transition, but no spin liquidProbable spin liquid for wider cylindersPhase boundaries are probably narrower than previous calculations– but depends strongly on details, eg even width cylinders frustrate 120◦

ordercolumnar phase is diagonal in iDMRG (2-fold degenerate)


Kagome LatticeTriangular Lattice – 3-leg cylinderTriangular lattice – 2D

Documents

Spin Liquid in the Heisenberg model on the Kagome and ... · Spectroscopy of SU(2) broken states F. Kolley, S. Depenbrock, IPM, U. Schollwöck, V. Alba, Phys. Rev. B 88, 144426 (2013)