Frustration-driven multi magnon condensates and their excitations - - PowerPoint PPT Presentation

Frustration-driven multi magnon condensates and their excitations

Oleg Starykh, University of Utah, USA

Current trends in frustrated magnetism, ICTP and Jawaharlal Nehru University, New Delhi, India, Feb 9-13, 2015

Collaborators

Andrey Chubukov, Univ of Minnesota Leon Balents, KITP, UCSB

not today but closely related findings: spin-current state at the tip of 1/3 magnetization plateau, spontaneous generation of

• rbiting spin currents

(ask me for details after the talk :))

Condensed Matter Physics at the University of Utah

Scanning Probe Microscopy Nano-optics Low Temperature Transport Exotic Matter and High Pressure Spin electronics Organic Semiconductors NMR and MRI

• Strongly Correlated Electron Physics

Topological insulators Frustrated magnetism Superconductivity

Life at (and near) University of Utah

Outline

• Frustrated magnetism (brief intro)
• emergence of composite orders from competing

interactions

• Nematic vs SDW in LiCuVO4

✓ spin nematic: “magnon superconductor” ✓ collinear SDW: “magnon charge density wave”

• Volborthite kagome antiferromagnet
• experimental status - magnetization plateau
• Nematic, SDW and more
• Field theory of the Lifshitz point
• Conclusions

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

Low-Temperature Broken-Symmetry Phases of Spiral Antiferromagnets

Luca Capriotti1,2 and Subir Sachdev2,3

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

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

= X

triangle

~ Si × ~ Sj

Ising order: spin chirality

Ising nematic in collinear spin system = ~ N1 · ~ N2 = ±1

Outline

• Frustrated magnetism (brief intro)
• emergence of composite orders from competing

interactions

• Nematic vs SDW in LiCuVO4

✓ spin nematic: “magnon superconductor” ✓ collinear SDW: “magnon charge density wave”

• Volborthite kagome antiferromagnet
• experimental status - magnetization plateau
• Nematic, SDW and more
• Field theory of the Lifshitz point
• Conclusions

LiCuVO4 : magnon superconductor?

estimates: J1 = - 1.6 meV J2 = 3.9 meV J5 = -0.4 meV (subject of active debates)

High-field analysis: condensate of bound magnon pairs

hS+i = 0 hS+S+i 6= 0

Ferromagnetic J1 < 0 produces attraction in real space

Chubukov 1991 Kecke et al 2007 Kuzian and Drechsler 2007 Hikihara et al 2008 Sudan et al 2009 Zhitomirsky and Tsunetsugu 2010

Magnon binding

E-EFM = ε1 + h E-EFM = ε2 + 2h

E h

Sz=-1 Sz=-2

ε2 < 2 ε1 : “molecular” bound state Formation of molecular fluid

For d>1 at T=0 this is a molecular BEC = true spin nematic

1-magnon 2-magnon bound state

Hidden order

No dipolar order

hS+

i i = 0

hS+

i S− j i ⇠ e−|i−j|/ξ Sz=1 gap

Nematic order

hS+

i S+ i+ai 6= 0

Magnetic quadrupole moment Symmetry breaking U(1) → Z2 can think of a fluctuating fan state nematic director

LiCuVO4: NMR lineshape - collinear SDW along B

Buttgen et al 2012 Hagiwara, Svistov et al, 2011

LiCuVO4

No spin-flip scattering above ~ 9 Tesla: longitudinal SDW state SF = spin flip, ΔS = 1 NSF = no spin flip, ΔS = 0

Geometry (motivated by LiCuVO4)

• J1< 0 (ferro)

J2 >0, J’ > 0 (afm) in magnetic field

• No true condensation [ U(1) breaking] in d=1.
• Inter-chain interaction is crucial for establishing

symmetry breaking in d=2.

• Need to study weakly coupled “superconducting” chains

Sato et al 2013 Starykh and Balents 2014

Inter-chain interaction H?

inter =

X

y

Z dx J0hS+

y (x)S y+1(x + 1)inematic ground state ! 0

Superconducting analogy: single-particle (magnon) tunneling between magnon superconductors is strongly suppressed at low energy (below the single-particle gap)

Hnem ∼ (J02/J1) X

y

Z dx [T +

y (x)T y+1(x) + h.c.]

Superconducting analogy: fluctuations generate two-magnon (Josephson coupling) tunneling between chains. They are generically weak, ~ J1(J’/J1)2 << J’ , but responsible for a true two-dimensional nematic order At the same time, density-density inter-chain interaction does not experience any

• suppression. It drives the system toward a two-dimensional collinear SDW order.

Away from the saturation, SDW is more relevant [and stronger, via J’ >> (J’)2/J1 ] than the nematic interaction: coupled 1d nematic chains order in a 2d SDW state.

Hinter−chain = X

y

Z dx ~ Sy · ~ Sy+1 ∼ X

y

Z dx S+

y S− y+1 + Sz ySz y+1

Hz

interchain = Hsdw ∼ J0 X y

Sz

ySz y+1 ∼ J0 X y

Z dx cos[ √ 2π β (ϕ+

y − ϕ+ y+1)]

Sz

y = M − 2npair = M − ˜

A1ei

√ 2π β

ϕ+

y (x)eiksdwx

T +

y (x) ∼ S− y (x)S− y (x + 1)

Simple scaling

Hnem ∼ (J02/J1) X

y

Z dx [T +

y (x)T y+1(x) + h.c.]

• describes kinetic energy of magnon pairs, linear in magnon pair density

Hz

interchain = Hsdw ∼ J0 X y

Sz

ySz y+1 ∼ J0 X y

Z dx cos[ √ 2π β (ϕ+

y − ϕ+ y+1)]

• describes potential energy of interaction between magnon pairs on

neighboring chains, quadratic in magnon pair density

• Competition
• Hence:
• Spin Nematic near saturation, for npair < n*pair
• SDW for npair > n*pair

npair npair

(J0)2 J1 npair ∼ J0n2

pair, hence npair,c ∼ J0/J1

n⇤

pair ∼ J0/J1

T=0 schematic phase diagram of weakly coupled nematic spin chains

M

1/2 - O(J’/J)

SDW

Spin Nematic

Fully Polarized

BEC physics

1/2

cf: Sato, Hikihara, Momoi 2013

Cautionary remark: maybe impurity effect

Excitations (via spin-spin correlation functions)

• 2d SDW
• preserves U(1) [with respect to magnetic field] ->

hence NO transverse spin waves

• breaks translational symmetry -> longitudinal

phason mode at ksdw = π(1-2Μ) and k=0 hSz(r)i = M + Re ⇣ Φeiksdw·r⌘

phason

OS, Balents PRB 2014 (solitons (kinks) of massive sine-Gordon model which describes 2d ordered state)

Excitations (via spin-spin correlation functions)

• 2d Spin Nematic
• breaks U(1) but ΔS=1 excitations are gapped

(magnon superconductor)

• gapless density fluctuations at k=0

hS+(r)i = 0

OS, Balents PRB 2014 (+ sector: solitons of massive sine-Gordon model which describes 2d ordered state.) Energy scale (- sector: solitons of massive sine-Gordon model describing 1d zig-zag chain.) Energy scale

(J0)2/J1

J1

hS+(r)S+(r0)i ⇠ Ψ 6= 0

Intermediate Summary

• Interesting magnetically ordered states: SDW and Spin

Nematic

• Gapped ΔS=1 excitations (no usual spin waves!)
• Linearly-dispersing phason mode with ΔS=0 in 2d

SDW

• SDW naturally sensitive to structural disorder
• Linearly-dispersing magnon density waves in 2d

Spin Nematic

• analogy with superconductor/charge density wave

competition

Outline

• Frustrated magnetism (brief intro)
• emergence of composite orders from competing

interactions

• Nematic vs SDW in LiCuVO4

✓ spin nematic: “magnon superconductor” ✓ collinear SDW: “magnon charge density wave”

• Volborthite kagome antiferromagnet
• experimental status - magnetization plateau
• Nematic, SDW and more
• Field theory of the Lifshitz point
• Conclusions

• n

e t-

• n.

l in i- s l del

• e

t y s f en s, e

• en
• s

e is e

• a

b c

Cu1 Cu2 O5 H

V

b a c Cu1 Cu2 a c b

Volborthite

Volborthite’s timeline

2001

quantum spin liquid?!

r

C/T (J/K2 mol Cu)

T (K)

20 40 60 1 2 3

Cmag T -1 / mJ K-2 Cu-mol-1

T* *

• T / K
• 0 T

• 2009

impurity ordering at low T? magnetization steps?

• 150

2012 2014

magnetic order ! magnetization plateau

time = material quality

High-field magnetization

high-field mag. meas. @ Tokunaga & Kindo labs

more different MH curves in a pile of 50 large “thick” arrowhead-shaped crystals

0.5 0.4 0.3 0.2 0.1 0.0

M (

B / Cu)

70 60 50 40 30 20 10

B (T)

~2/5 polycrystals

a pile of thin crystals B ⊥ ab a pile of ~50 thick crystals B // ab B ⊥ ab

Huge 1/3 plateau! further optical meas. @ Takeyama lab It survives over 120 T!

Van Vleck

Kagome plateau or ferrimagnetic state? coupled to lattice, but already distorted

T = 1.4 K

30 days growth

2014: huge plateau!

• H. Ishikawa…M.Takigawa…Z.Hiroi, unpublished, 2014

• Phase diagram

SDW

1/3 plateau

B ?

?

T 1K 1T 26T

• ur interpretation
• H. Ishikawa et al,

unpublished

Frustrated ferromagnetism

Jic J1 J2

a c b a b

Coupled frustrated quantum spin-1

2 chains with orbital order in volborthite Cu3V2O7(OH)2·2H2O

• O. Janson,1,* J. Richter,2 P. Sindzingre,3 and H. Rosner1,†

Received 9 August 2010; published 30 September 2010

• PHYSICAL REVIEW B 82, 104434 2010

50 150 200 h (T) 0.2 0.4 0.6 0.8 1 m/ms

Jic/|J1| = 2

kagome J2/|J1| = 1.1 J2/|J1| = 1.4 J2/|J1| = 1.6 T (K)

DFT gets it right! FM

J1 < 0, J2 > 0, J0 > 0

Ferrimagnetic state

Coupled frustrated quantum spin-1

2 chains with orbital order in volborthite Cu3V2O7(OH)2·2H2O

• O. Janson,1,* J. Richter,2 P. Sindzingre,3 and H. Rosner1,†

Received 9 August 2010; published 30 September 2010

• PHYSICAL REVIEW B 82, 104434 2010

J1 FM, J2 AF J’ AF

polarized chains?! J1 < 0, J2 > 0, J0 > 0

• Phase diagram

SDW

1/3 plateau

B

spin nematic?

?

T 1K 1T 26T may be a spin nematic??

• H. Ishikawa et al,

unpublished

Frustrated ferromagnetic chain

J1 FM

H = J1

i

Si · Si+1 + J2

i

Si · Si+2 − h

i

Si

z,

J2/(|J1|+J2) H/(|J1|+J2) 1/5 1 FM quasi-spin-nematic

J2 AF

Spin chain redux

Quasi-1d nematic

1d J1-J2 chain is only quasi-spin-nematic

power-law correlations

Ψ ~ (S+)2 : spin-nematic ɸ ~ Sz ei q x : SDW

• 4
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1

J1/J2

0.2 0.4 0.6 0.8 1

m/msat

• 0.25
• 0.3
• 0.275
• 0.35
• 0.4
• 0.5
• 0.75
• 1

J2/J1

0.2 0.4 0.6 0.8 1

h/hsat Vector Chiral Order

p=2 p=3

p=4

quadrupolar SDW (p=2)

• ctupolar

SDW (p=3)

hexadecupolar ?

Ψ “dominant” ɸ “dominant”

Hikihara et al, 2008 Sudan et al, 2009

Multipolar phases

Frustrated ferromagnetic chain

J1 FM

H = J1

i

Si · Si+1 + J2

i

Si · Si+2 − h

i

Si

z,

−4 −3 −2 −1 0.2 0.4 0.6 0.8

SDW2 VC F SDW3 N T Q IN h / J2 J1 / J2 (a)

J2 AF

Is it an infinite progression?

Hikihara et al, 2008

Frustrated ferromagnetic chain

J1 FM

H = J1

i

Si · Si+1 + J2

i

Si · Si+2 − h

i

Si

z,

J2/(|J1|+J2) H/(|J1|+J2) 1/5 1 FM quasi-spin-nematic

J2 AF

“Lifshitz” QCP

A QCP parent?

Lifshitz Point

S = Z dxdτ

• isAB[ ˆ

m] + δ|∂x ˆ m|2 + K|∂2

x ˆ

m|2 + u|∂x ˆ m|4 − h ˆ mz

Berry phase term tunes QCP

AB = ˆ m1∂τ ˆ m2 ˆ m2∂τ ˆ m1 1 + ˆ m3 .

two symmetry allowed interactions at O(q4)

All properties near Lifshitz point obey “one parameter universality” dependent upon u/K ratio

• Unusual QCP: order-to-order transition
• Effective action - NLσM

Lifshitz Point

S = Z dxdτ

• isAB[ ˆ

m] + δ|∂x ˆ m|2 + K|∂2

x ˆ

m|2 + u|∂x ˆ m|4 − h ˆ mz

τ → K δ2 τ

x → s K |δ|x

S = r K δ Z dxdτ

• isAB[ ˆ

m] + sgn(δ)|∂x ˆ m|2 + |∂2

x ˆ

m|2 + v|∂x ˆ m|4 − h ˆ mz

Large parameter: saddle point!

v = u K

h = hK δ2

• Intuition: behavior near the Lifshitz

point should be semi-classical, since “close” to FM state which is classical

Saddle point

S = r K δ Z dxdτ

• isAB[ ˆ

m] + sgn(δ)|∂x ˆ m|2 + |∂2

x ˆ

m|2 + v|∂x ˆ m|4 − h ˆ mz

v derives from quantum fluctuations

By a spin wave analysis, one finds v ~ -3/(2S) < 0

h

FM

first order

local instability of FM state (1-magnon condensation)

IC cone

spiral

−1 < v < − 1

4

−δ

hc = δ2 8K p |v|(1 − p |v|)

Phase diagram

Frustrated ferromagnetic chain

−4 −3 −2 −1 0.2 0.4 0.6 0.8

SDW2 VC F SDW3 N T Q IN h / J2 J1 / J2 (a)

First order metamagnetic transition near Lifshitz point

Higher dimensions?

Hikihara et al, 2008

d>1

τ → K δ2 τ

x → s K |δ|x S = Z dxdd−1ydτ

• isAB[ ˆ

m] + δ|∂x ˆ m|2 + c|∂y ˆ m|2 + K|∂2

x ˆ

m|2 + u|∂x ˆ m|4 − h ˆ mz

y → √ cK δ y

S = √ Kdcd−1 δd−1/2 Z dxdd−1ydτ

• isAB[ ˆ

m] + sgn(δ)|∂x ˆ m|2 + |∂2

x ˆ

m|2 + |∂y ˆ m|2 + v|∂x ˆ m|4 − h ˆ mz

∴ Similar theory applies in d>1, and very similar conclusions apply

• Rescaling:

Phase diagram

−4 −3 −2 −1 0.2 0.4 0.6 0.8

SDW2 VC F SDW3 N T Q IN h / J2 J1 / J2 (a)

multipolar phases from QCP?

Origin of multipolar phases

−4 −3 −2 −1 0.2 0.4 0.6 0.8

SDW2 VC F SDW3 N T Q IN h / J2 J1 / J2 (a)

First order transition: partially polarized state coexists with plateau one

With enough quantum fluctuations, “bubbles” of partially polarized phase may become many-magnon bound states and form multipolar phases

Origin of multipolar phases

−4 −3 −2 −1 0.2 0.4 0.6 0.8

SDW2 VC F SDW3 N T Q IN h / J2 J1 / J2 (a)

First order transition: partially polarized state coexists with plateau one

With enough quantum fluctuations, “bubbles” of partially polarized phase may become many-magnon bound states and form multipolar phases

Summary

• Spin chains keep showing up in

unexpected places

✓Nematic physics of frustrated

ferromagnets

✓Explored Lifshitz point as a “parent”

for multipolar states and metamagnetism

Quasi-1d nematic

J’ij 1d J1-J2 chain J’/J H FM

“dominant” SN

“dominant” SDW SDW2

SN

c.f. J’/J ~ 0.1 in LiCuVO4

• O. Starykh + LB, 2013

True nematic occurs in narrow range near FM state

Quasi-1d nematic

J’ij 1d J1-J2 chain J’/J H FM

“dominant” SN

“dominant” SDW SDW2

SN

q = 1/4 - M/2

• M. Mourigal et al, 2012

LiCuVO4

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

χ (10-3 cm3/mol Cu)

400 300 200 100

T (K)

Volborthite Cu3V2O7(OH)2·2H2O

High-temperature series expansion J/kB = 84.1(2) K g = 2.205(1)

H = 7 T

3.5 3.0 χ (10-3 cm3/mol Cu) 60 40 20 T (K)

H = 0.01 T

FC ZFC

(a)

r

0.8 0.6 0.4 0.2 0.0

C/T (J/K2 mol Cu)

70 60 50 40 30 20 10

T (K)

CD /T Cm /T

T (K)

1.14 1.12 1.10 T = 200 K f = 12.5 MHz

(b)

7.96 7.94 7.92 T = 230 K f = 89 MHz

(a)

K 1.4 1.2 1.0

Magnetic Field (T)

T = 1.7 K f = 12.8 MHz

(c)

Spin Echo Intensity

2001: a QSL?

smooth thermodynamics

no spontaneous fields

20 40 60 1 2 3

Cmag T -1 / mJ K-2 Cu-mol-1

T* *

• T / K
• 0 T

○ 1 T △ 3 T ▼ 5 T □ 7 T

• fermionic

spinons?

c.f. herbertsmithite: ximp ~ 5%

2009: Impurity ordering at 1K? Fermionic QSL?

20 40 60 1 2 3

Cmag T -1 / mJ K-2 Cu-mol-1

T* *

• T / K
• 0 T

○ 1 T △ 3 T ▼ 5 T □ 7 T

• fermionic

spinons?

c.f. herbertsmithite: ximp ~ 5%

2009: Impurity ordering at 1K? Fermionic QSL?

• c
• 150

100 50 4 3 2 1 C/T (mJ K– 2 mol-Cu–1) C/T (mJ K– 2 mol-Cu–1) T 2 (K2) T (K) T * 120 80 40 1.5 1.0 2.0 0.5 Polycrystal Single crystal

Amazing how much sample quality matters!

2012: Ordering transitions! Not a QSL

Frustrated ferromagnetism

Jic J1 J2

a c b a b

Coupled frustrated quantum spin-1

2 chains with orbital order in volborthite Cu3V2O7(OH)2·2H2O

• O. Janson,1,* J. Richter,2 P. Sindzingre,3 and H. Rosner1,†

Received 9 August 2010; published 30 September 2010

• PHYSICAL REVIEW B 82, 104434 2010

50 150 200 h (T) 0.2 0.4 0.6 0.8 1 m/ms

Jic/|J1| = 2

kagome J2/|J1| = 1.1 J2/|J1| = 1.4 J2/|J1| = 1.6 T (K)

Prospects of observing novel quantum liquids FM

• Phase diagram

SDW

1/3 plateau

B ?

?

T 1K 1T 26T

• ur interpretation
• H. Ishikawa et al,

unpublished

Quasi-1d nematic

J’ij Interchain coupling ~ J’ ɸy ɸy+1 + (J’)2/J Ψy Ψy+1

extra suppression of spin- nematic order in quasi-1d limit

1d J1-J2 chain

Nematic chain

Sz-Sz (SDW) channel: in-chain J1< 0 gaps out relative mode

• J1-J2 chain

ϕ−

y = (ϕy,odd − ϕy,even)/

√ 2

y, odd y, even

S+

y (x) ∼ (−1)xA3ei β

√ 2 θ+ y (x)ei(−1)x β √ 2 θ− y (x)

quantum-disordered, decays exponentially: Sz = 1 excitations are gapped

T +

y = S+ y (x)S+ y (x + 1) ∼ ei √ 2βθ+

y (x)

Standard (in 1d) power-law decay: critical nematic spin correlations, but U(1) is preserved Kolezhuk, Vekua (2005); Hikihara et al. (2008); Sato, Hikihara, Momoi (2013)

Physical picture: 1d magnon “superconductor”

hS+i = 0

Hintra−chain = X

y

Z dxJ1 sin[πM] cos[ √ 8π β ϕ−

y ]

hT +i = hS+S+i ! 0

local pair formation

2d commensurate SDW (such as 1/3 magnetization plateau)

gapped phason

soliton breather kx

π (1 - 2 M) π (1 + 2 M) π

B1 B1 s, s s, s

continuum ω 2ms ms + m1

s, s

~ ~

2 π M

s, s

phason

gapped transverse excitations

∆plat ∆plat

2d Spin Nematic

vanishing spectral weight as k -> 0 (isotropic) soliton soliton

(+ sector: solitons of massive sine-Gordon model which describes 2d ordered state.) Energy scale (J’)2/J1 (- sector: solitons of massive sine-Gordon model describing 1d zig-zag chain.) Energy scale J1

2d SDW

vanishing spectral weight as kx -> 0 (anisotropic)

phason (longitudinal)

soliton breather bosonization, sine- Gordon + inter- chain RPA/LG analysis kx

π (1 - 2 M) π (1 + 2 M) π

B1 B1 s, s s, s

continuum ω 2ms ms + m1

s, s

~ ~

2 π M

s, s

phason

gapped transverse excitations

Single crystals of volborthite

0.2 mm

• H. Yoshida’s crystal
• 1. natural leaf crystals, long time ago
• 2. low-quality polycrystalline samples by

precipitation, 2001

• 3. high-quality polycrystalline samples by

hydrothermal annealing, 2009

• 4. small single crystals, 2012
• 5. large arrowhead-shaped crystals, 2013

Ishikawa’s crystal

1 mm

Spin nematic redux

Frustrated ferromagnetic chain

J1 FM

H = J1

i

Si · Si+1 + J2

i

Si · Si+2 − h

i

Si

z,

J2/(|J1|+J2)

H/(|J1|+J2)

1/5 1 FM

quasi-spin-nematic

J2 AF