Oleg Starykh University of Utah arXiv:1708.02980 E d i t o r - - PowerPoint PPT Presentation

Andrey Chubukov (U Minnesota) Zhentao Wang (U Tennessee) Cristian Batista (U Tennessee) Andrian Feigun (Northeastern U) Wei Zhu (LANL)

YITP workshop “Novel quantum states in condensed matter”, November 9, 2017

Oleg Starykh University of Utah

Chiral liquid phase of simple quantum antiferromagnets

arXiv:1708.02980

E d i t

• r

s ’ S u g g e s t i

• n

✴

Vector chirality

✴

1/3 magnetization plateau and its instabilities:

• spin-current phase

✴

Minimal s=1 XXZ model of spin-current phase

✴

Conclusions Outline

Talks by Togawa, Nagaosa, Tokura

Exotic ordered phases, emergent (Ising) orders spin nematic composite order parameter, bilinear in spins

• rdered

phases spin nematics composite order quantum spin liquids

S1 S2 S1 x S2

Annals of the Israel Physical Society, vol.2, p. 565 (1978)

Brief history TIsing = 0.5125J > TKT = 0.5046J TIsing = 0.513J > TKT = 0.502J

PHYSICAL REVIEW B 85, 174404 (2012)

Chiral spin liquid in two-dimensional XY helimagnets

• A. O. Sorokin1,* and A. V. Syromyatnikov1,2,†

H =

• x

(J1 cos(ϕx − ϕx+a) + J2 cos(ϕx − ϕx+2a) − Jb cos(ϕx − ϕx+b)),

Emergent Ising order parameters (finite T)

Low-Temperature Broken-Symmetry Phases of Spiral Antiferromagnets

Luca Capriotti1,2 and Subir Sachdev2,3

1

PRL 93, 257206 (2004) P H Y S I C A L R E V I E W L E T T E R S

week ending 17 DECEMBER 2004

• ^

H J1 X

hi;ji

^ Si ^ Sj J3 X

hhi;jii

^ Si ^ Sj;

J3 T

J1 / 4

Spiral LRO

Tc

ξspin~ S / T 1/2 ξspin~ ec'S2 / T

Neel LRO Lifshitz point

ξspin~ ecS2 / T

Ising nematic order

at T=0 only

1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 0.3 0.25 0.2 0.15 0.1 0.05 0.25 0.3 0.35 0.4 Tc J /J

a ^ S1 ^ S3 ^ S2 ^ S4a;

4 3 2 1

(Q,Q)

1 2 4 3

(Q,−Q)

• FIG. 2.

The two different minimum energy configurations with magnetic wave vectors ~ Q Q; Q and ~ Q? Q; Q with Q 2=3, corresponding to J3=J1 0:5.

Tising

TKT

Vector spin chiral phase is present, but the temperature interval is tiny. Can be enhanced by DM interaction + phonons, Onoda, Nagaosa PRL 2007

Vector chirality in 1d (T=0)

Today: Search for vector chirality without magnetic order in quantum 2d models

Cheshire Cat’s smile

✴ Spin-current phase

✴

Vector chirality

✴

1/3 magnetization plateau and its instabilities:

• spin-current phase

✴

Minimal s=1 XXZ model of spin-current phase

✴

Conclusions Outline

Phase diagram of the Heisenberg (XXX) model in the field

Seabra, Momoi, Sindzingre, Shannon 2011 Gvozdikova, Melchy, Zhitomirsky 2010

Z2 vortex (chirality ordering) transition

Quantum fluctuations, S >> 1, T=0.

J’ = J: Quantum fluctuations select co-planar and collinear phases hc2 - hc1 = (0.6/2S) hsat

UUD plateau is due to interactions between spin waves

up-up-down collinear state

PRB 2013 Need to understand end-points

J0

J

Spatially anisotropic model

h

sat

h h

sat

h

1/3-plateau

Spatially anisotropic model: classical vs quantum

∑ ∑

− ⋅ =

〉 〈 i z i ij j i ij

h J H S S S

Umbrella state: favored classically; energy gain (J-J’)2/J Planar states: favored by quantum fluctuations; energy gain J/S

J′ ̸= J

δ = S(J − J′)2/J2

The competition is controlled by dimensionless parameter

S = ∞ S = 1 2

J0

J

Alicea, Chubukov, OS PRL 2009

Emergent Ising order near the end-point of the 1/3 magnetization plateau

H = X

hi,ji

Jij ~ Si · ~ Sj

O hc2 hsat δ 1 3 4 A B C D E F G H

δcr

a b

C1 C2

Z3

U(1)*Z3*Z2 hc1 Z3*Z2 U(1)*Z3 U(1)*U(1) U(1)*U(1)*Z2 U(1)*Z2

U(1)*Z3

U(1)*U(1)

U(1)*U(1)*Z2 U(1)*Z2

δ = 40 3 S ⇣J − J0 J ⌘2 OAS, Reports on Progress in Physics 78, 052502 (2015), OAS, Wen Jin, Chubukov, Phys. Rev. Lett. 113, 087204 (2014)

1/3 plateau Cone

UUD-to-cone phase transition

H = X

hi,ji

Jij ~ Si · ~ Sj

O hc2 hsat δ 1 3 4 A B C D E F G H

δcr

a b

C1 C2

Z3 (UUD)

hc1 Z3*Z2 U(1)*Z3 U(1)*U(1) U(1)*U(1)*Z2 U(1)*Z2 (cone)

U(1)*Z3

U(1)*U(1)

U(1)*U(1)*Z2 U(1)*Z2

δ = 40 3 S ⇣J − J0 J ⌘2

Z3 → U(1) × Z2 or Z3 → smth else → U(1) × Z2?

• k2

d2

Low-energy excitation spectra

δ = 40 S 3 (1 − J0/J)2

✏d2 = hc2 − h + 9Jk2 4

✏d1 = h − hc1 + 3Jk2 4

for δ < 1 for δ < 3 hc2 − hc1 = 0.6 2S hsat = 0.6 2S (9JS) d1 Bose-Einstein condensation

• f d1 (d2) mode at k =0 leads to

lower (upper) co-planar phase Magnetization plateau is collinear phase: preserves O(2) rotations about magnetic field -- no gapless spin waves. Breaks only discrete Z3. Hence, very stable. vacuum of d1,2 Alicea, Chubukov, OS PRL 2009

S>>1

+k1

• k1

+k2

• k2

+k0

• k0

• k2

d2

d1 δ=4 k1 = k2 = k0

Low-energy excitation spectra near the plateau’s end-point

k0 = r 3 10S

Alicea, Chubukov, OS PRL 2009

δ = 40 S 3 (1 − J0/J)2

extended symmetry: 4 gapless modes at the plateau’s end-point vacuum of d1,2 Magnetization plateau is collinear phase: preserves O(2) rotations about magnetic field -- no gapless spin waves. Breaks only discrete Z3.

δ = 40 S 3 (1 − J0/J)2 parameterizes anisotropy J’/J

Ψ†

1,p؆ 2,q

Ψ†

1,pΨ2,q

H(4)

d1d2 = 3

N X

p,q

Φ(p, q) ⇣ d†

1,k0+pd† 2,−k0−pd1,−k0+qd2,k0−q − d† 1,k0+pd† 2,−k0−pd† 1,−k0+qd† 2,k0−q

⌘ + h.c.

magnon pair

• perators

Chubukov, OS PRL 2013

Ψ1,p = d1,k0+pd2,−k0−p

Ψ2,p = d1,−k0+pd2,k0−p

1 1 2 2

Φ(p, q) ∼ (−3J)k2 |p||q|

}

}

}

Interaction between low-energy magnons

singular magnon interaction

[Ψ1,p, Ψ2,q] = δ1,2δp,q ⇣ 1 + d†

1,k0+pd1,k0+p + d† 2,k0+pd2,k0+p

⌘ → δ1,2δp,q

Obey canonical Bose commutation relations in the UUD ground state

hd†

1d1iuud = hd† 2d2iuud = 0

In the UUD ground state

★ Interacting magnon Hamiltonian in terms of d1,2 bosons =

non-interacting Hamiltonian in terms of Ψ1,2 magnon pairs

Chubukov, OS PRL 2013

Two-magnon instability Magnon pairs Ψ1,2 condense before single magnons d1,2

`Superconducting’ solution with imaginary order parameter

1 = 1 S 1 N X

p

k0 p |p|2 + (1 − δ/4)k2

Instability = softening of two- magnon mode @ δcr = 4 - O(1/S2) no single particle condensate

hΨ1,pi = hΨ2,pi ⇠ i Υ p2

hd1i = hd2i = 0

Equations of motion for Ψ - Hamiltonian

h؆

1,p Ψ1,pi = 6Jf 2 p

Ωp 3 N X

q

f 2

q hΨ† 2,q Ψ2,qi

h؆

2,p Ψ2,pi = 6Jf 2 p

Ωp 3 N X

q

f 2

q hΨ† 1,q Ψ1,qi

Spin-current nematic state near the end-point of the 1/3 magnetization plateau (large-S analysis)

Chubukov, OS PRL 2013 J J’ J’

> 0 < 0 hˆ z · SA ⇥ SCi = hˆ z · SC ⇥ SBi = hˆ z · SB ⇥ SAi / Υ hSr · Sr0i is not affected

no transverse magnetic order

hSx,y

r

i = 0

Spontaneously broken Z2 -- spatial inversion [in addition to broken Z3 inherited from the UUD state] uud

distorted umbrella distorted umbrella

4 δ δcr hc2 hc1

spin- current domain wall

Finite vector chirality Z3*Z2

Spin current visualization

SA = (cos[ωt], sin[ωt], MA), SB = (cos[ωt ± 4π 3 ], sin[ωt ± 4π 3 ], MB), SC = (cos[ωt ± 2π 3 ], sin[ωt ± 2π 3 ], MC)

Precessing spins on sub lattices A, B, C are phase shifted by 2π/3: hSx,y

r

i = 0

hSA ⇥ SCi = hSC ⇥ SBi = hSB ⇥ SAi = ± sin[2π 3 ]

Then no dipolar transverse order: But finite chirality, determined by the sign of 2π/3 shift between the sublattices:

hSA · SCi = hSC · SBi = hSB · SAi = cos[2π 3 ]

and A C B

End-point of the plateau on kagome lattice

Kagome geometry 1/3 plateau

Spin-current pattern

PRB 2017

✴

Vector chirality

✴

1/3 magnetization plateau and its instabilities:

• spin-current phase

✴

Minimal s=1 XXZ model of spin-current phase

✴

Conclusions Outline

The minimal 2d quantum spin model

• Spin-1 model with featureless Mott ground state at large D > 0 [ ]
• Triangular lattice: two-fold degenerate spectrum, at +Q and -Q

H = X

hr,r0i

J(Sx

r Sx r0 + Sy r Sy r0 + ∆Sz rSz r0) + D

X

r

(Sz

r)2

Sz

r = 0

⟨Sr⟩ = 0 ⟨Sr × Sr+eν⟩ ̸= 0 ⟨Sr⟩ = 0 ⟨Sr × Sr+eν⟩ = 0 ⟨Sr⟩ ̸= 0 ⟨Sr × Sr+eν⟩ ̸= 0

CL

PM XY e1 e2 e3

U(1) × Z2 × U(1) Z2 × U(1) Z2

g gc gb

c

+Q

• Q

The minimal 2d quantum spin model

• Spin-1 model with featureless Mott ground state at large D > 0 [ ]
• Triangular lattice: two-fold degenerate spectrum, at +Q and -Q
• 1. Toy problem of two-spin exciton. Derive Schrodinger eqn for the pair wave function ψ

Solution which is odd under inversion is the first instability when approaching from large-D limit. Indicates chiral Mott phase. [Single-particle condensation occurs at D=3J.]

H = X

hr,r0i

J(Sx

r Sx r0 + Sy r Sy r0 + ∆Sz rSz r0) + D

X

r

(Sz

r)2

Sz

r = 0

+ charge, — charge

Schwinger boson representation of S=1

hbr0i = s

Large-D limit: b0 is condensed, are excitations about the vacuum.

, br↑,↓

ζ = Jz/J

Magnon interaction comes from Ising part of the exchange This accounts for quantum fluctuations

Interaction between magnons

Number conserving 2 -> 2 Non-conserving 3 -> 1, 1 -> 3 4 -> 0, 0 -> 4

Chiral order parameter

Vector chirality q-space Low-energy approximation Total spin Sz=0, Odd under Q -> - Q , Odd under Boson pair operators Convenient parameterization

Integral equation for pair vertices

=

¯ Q+k Q−k ¯ Q+k ¯ Q+p Q−p Q−k

+

Q−p ¯ Q+p ¯ Q+k Q−k

+

¯ Q+k Q−k ¯ Q+p Q−p ¯ Q+k Q−k ¯ Q+p Q−p

+ φL(k) =

¯ Q+k Q−k ¯ Q+k ¯ Q+p Q−p Q−k

+

Q−p ¯ Q+p ¯ Q+k Q−k

+

¯ Q+k Q−k ¯ Q+p Q−p ¯ Q+k Q−k ¯ Q+p Q−p

+ φR(k)

Shaded rectangles denote fully dressed Irreducible interactions between Low-energy magnons

First order in Jz = ζJ

Interaction is given by bare vertices Obtain for the 2-magnon instability No weak-coupling instability ! Interaction vertices are of order 1 (in units of Jz ) and are not singular :-(

k1,2

near ±Q Need to renormalize it! LW = Long wavelength, SW = short wavelength

To find the dressed interaction, we have to go to the 2nd order in Jz …

¯ Q+p Q−p ¯ Q+k Q−k ¯ Q+p Q−p ¯ Q+k Q−k

(a) (b)

¯ Q+p ¯ Q+k Q−p Q−k

(c) (d)

¯ Q+p ¯ Q+k Q−p Q−k

(e)

¯ Q+p Q−p Q−k ¯ Q+k

(f)

¯ Q+p ¯ Q+k Q−p Q−k 6.95 6.95 5.00 5.00 0.98 0.98

¯ Q+p Q−p ¯ Q+k Q−k ¯ Q+p Q−p ¯ Q+k Q−k

(a) (b) (c) (d) (e)

¯ Q+p Q−p Q−k ¯ Q+k

(f)

¯ Q+p ¯ Q+k Q−p Q−k Q−p ¯ Q+k Q−k ¯ Q+p Q−p ¯ Q+k Q−k ¯ Q+p 6.95 6.95 0.98 0.98 5.00 5.00

Kohn-Luttinger like mechanism but for bosons

F 22 F 04

The result

Pair vertex is real, renormalized interaction is singular Two magnon condensation takes place before the single magnon one:

α = 2.49

F 22o

k,p = −4F 04 k,p =

−αζ2J3 ωQ−pωQ−k

ζ = Jz J = ∆

More checks: Bethe-Salpeter equation (low-density approximation)

k1 ↑ k2 ↓ k2 − q ↓ k1 + q ↑ = q q + q − q′ q′ k1 ↑ k1 + q ↑ k2 ↓ k2 − q ↓ k1 ↑ k2 ↓ k1 + q ↑ k2 − q ↓ k1 + q′ ↑ k2 − q′ ↓ + k1 ↑ k2 ↓ −k2 + q′ ↓ −k1 − q′ ↑ k1 + q ↑ k2 − q ↓ q − q′ q′ k1 ↑ k2 ↓ −k1 − q ↑ −k2 + q ↓ q q = k1 ↑ k2 ↓ −k1 − q ↑ −k2 + q ↓ k1 ↑ k2 ↓ k1 + q′ ↑ k2 − q′ ↓ −k1 − q ↑ −k2 + q ↓ q − q′ q′ + k1 ↑ k2 ↓ −k1 − q′ ↑ −k2 + q′ ↓ −k1 − q ↑ −k2 + q ↓ q − q′ q′ ΓN

q (k1, k2, ω)

ΓA

q (k1, k2, ω)

1 2 3 2 2.5 3 3.5 (a) Ising XY PM SS 2 2.5 3 3.5 (b) Ising XY PM SS 0.01 0.1 1 10 10−5 10−4 10−3 10−2 (c) ζb

c ≈ 8.22√D − Dc

10−5 10−4 10−3 10−2 (d) ζb

c ≈ 4.30√D − Dc

ζ D 1 2 3 2 2.5 3 3.5 CL D 2 2.5 3 3.5 CL ζb

c

D − Dc 0.01 0.1 1 10 10−5 10−4 10−3 10−2 D − Dc 10−5 10−4 10−3 10−2

Full problem Only normal (2 -> 2) vertices

DMRG on 6x6 triangular lattice

1 2 3 4 5 (a) ζ = 0.5 (b) ζ = 1.0 1 2 3 4 5 2 3 4 5 (c) ζ = 1.5 2 3 4 5 (d) ζ = 2.0 Gaps ∆b ∆s Gaps D D

Two-magnon gap Single magnon gap

νIsing = 0.63, νXY = 0.67

Gap crossing: VC order via Ising transition before U(1) order via XY transition

Summary

⟨Sr⟩ = 0 ⟨Sr × Sr+eν⟩ ̸= 0 ⟨Sr⟩ = 0 ⟨Sr × Sr+eν⟩ = 0 ⟨Sr⟩ ̸= 0 ⟨Sr × Sr+eν⟩ ̸= 0

CL

PM XY e1 e2 e3

U(1) × Z2 × U(1) Z2 × U(1) Z2

g gc gb

c

Δ

1 2 3 CM = Chiral Mott = Chiral Liquid = CL

+Q

• Q

Chiral spin liquid appears naturally in the vicinity of magnetic quantum critical point! Broken inversion, Spontaneous vector chirality, Gapped single particle excitations. Magneto-electric effect!

Conclusions

Thank you! Mott -> superfluid transition on a frustrated lattice requires U(1) x Z2 breaking. This proceeds via intermediate spin-current (chiral Mott) phase (breaking Z2 only). Spontaneously breaks spatial inversion. But preserves time-reversal All single particle excitations are gapped.

φ(k) = −φ(−k)

u ∈ R

Paramagnet XY ordered

Summary

⟨Sr⟩ = 0 ⟨Sr × Sr+eν⟩ ̸= 0 ⟨Sr⟩ = 0 ⟨Sr × Sr+eν⟩ = 0 ⟨Sr⟩ ̸= 0 ⟨Sr × Sr+eν⟩ ̸= 0

CL

PM XY e1 e2 e3

U(1) × Z2 × U(1) Z2 × U(1) Z2

g gc gb

c

Chiral liquid can be detected via inverse Dzyaloshinskii-Moriya effect: Leads to charge density wave of O2- anions

δr

Djl

j

• +

+ + +

• l

Magneto-electric phenomena