Spin-orbit-induced spin-density wave in quantum wires and spin

Oleg Starykh, University of UtahSuhas Gangadharaiah, University of Basel

Jianmin Sun, Indiana University

Spin-orbit-induced spin-density wave in quantum wires and spin chains

Dahlem Center, Freie Universitat Berlin, Sept. 29, 2010

PRL 98, 126408; PRL 100, 156402; PRB 78, 054436; PRB 78, 174420 and work in progress

also appears in quasi-1d Kagome antiferromagnet, work withAndreas Schnyder (MPI Stuttgart) and Leon Balents (KITP)

Saturday, February 12, 2011

Motivation:

Why be interested in weak relativistic interaction -- spin-orbit?


Spin - Orbital (SO) coupling

—Relativistic effect:

v

EB

spin in magnetic field

—Atoms:

—Magnetic materials: Dzyaloshinskii-Moriya interaction via exchange + SO (1957)

D ~ λ J

requires absence of inversion symmetry r -> -r

Textbook (Landau-Lifshits VIII p.286) example: MnSi pitch ~ 170 A


50 years later: MnSi - quantum phase transition under pressure

• MnSi – itinerant ferromagnet with long pitch spiral order. At ambient pressure: Tc=30K, Moment =0.3μB

‘Partial Order’ PhaseE

NERGY

Itinerant ferromagnet with long pitch spiral - non-Fermi liquid under pressure

(slide from A. Vishwanath)


Field-induced gap in 1D antiferromagnet

•Cu benzoate: specific heat in the magnetic field C ~ exp[-Δ/kBT]

δq ~ H: standard Heisenberg

Δ ~ H2/3 : staggered DM

Massive incommensurate S=1 excitations

Dender et al PRL 79, 1750 (1997)

Oshikawa, Affleck PRL 79, 2883 (1997)


Spintronics

* No inversion symmetry => 2DEG heterostructures (e.g. GaAs)

Surface states (e.g. Au[111])

* Rashba Hamiltonian (1984)

Free electrons + SO :


Spin splitting of an Au(111) surface states: ARPES

Surface obtained by cutting along (111) plane

Spin-split Fermi surface Brillouin zone

ARPES spectra dispersion fit: Δ ~ 55meV = EF/8 !

LaShell et al. PRL 77, 3419 (1996)


Topological Insulators,M. Z. Hasan and C. L. Kane,arxiv 1002.3895

Strong spin-orbit+surface states


9


1D setting: magnetized wires with SOI and in proximity with s-wave superconductors

arxiv 1003.1145 and 1006.4395Saturday, February 12, 2011

• Spin-orbit interactions show up in different physical situations— Dresselhaus, Rashba, Dzyaloshinskii-Moriya...

• Result in interesting symmetry reductions— Momentum dependent magnetic field — Symmetry reduction SU(2) => U(1)

• Are not that [ (v/c)2 ] small : can be (and, are) observed currently!

Thus

Interplay of e-e interactions and spin-orbit is very interesting


Outline• Warm-up: van der Waals like coupling between spins in quantum dots

• Brief intro to quantum wires

Key scattering processes in a single-subband wire Effect of magnetic field: Zeeman splitting

• Spin-orbital effects

Cooper channel Spin-density wave formation (orthogonal magnetic field) Transport: suppressed backscattering Connection with spin chain physics: uniform DM interaction Unusual implications for ESR experiments

Conclusions


Warm-up exercise: spin-orbit mediated coupling of spins in the absence of exchange (no tunneling!)

Idea: Spin-Orbit correlates spin and orbital motion,while Coulomb correlates orbital motion of electrons

Coupled single-electron quantum dots


Unitary transformation to “remove” Spin-Orbit

Shahbazyan, Raikh (1994)Aleiner, Fal’ko (2001)

• Spin-orbit form indeed

• Assumes that (SO length) << (confining length)

Unitary rotation:

Transforms SOI into:


Two single-electron dots coupled by Coulomb interaction

• Four harmonic oscillators: along X (Y), symmetric (anti-symmetric)

• Perturbation: spin-orbit

• 2nd order energy correction

• van der Waals-like spin-spin interaction


Generalizations

• vdW interaction is absent in strict d=1 limit, when

• Effect of magnetic field: appearance of dipolar coupling

for

• Implications for exchange interactions: expect symmetry breaking of DM form only in αR

4 order.

Hidden SU(2) symmetry (Shekhtman et al 1992, Koshibae et al 1994)

Flindt et al 2006, Trif et al 2007

but external magnetic field will again result in two non-commuting perturbations!


Serious consequences for Wigner crystals

• Electron lattice with exponentially small exchange competing multi-spin exchanges extensive spin degeneracy (e.g. Pomeranchuk effect)

• Spin vdW coupling: ferromagnetic Ising interaction Non-exchange type (no overlap of wave functions) No frustration lifts degeneracy

• Ferromagnetic ground state (GaAs rs~100; InAs rs~20)

Sun, Gangadharaiah, OS, PRL 100, 156402 (2008)

SOI + Coulomb does lead to interesting new physics







Conclusions



Vicinal Au(111) surface states: one-dimensional electrons on terraces

•Cut at small miscut angle α~3.50 : surface composed of {111} steps (terraces)

d ~ 38 A

•Terrace: one-dimensional states (d ~ λF)

Disperses along the terrace

But notperpendicularto it!

•Dispersing states are spin-split :

kF1 = 0.157 A-1

kF2 = 0.184 A-1

Mugarza et al PRL 87, 107601 (2001)PRB 66, 245419 (2002)

no magnetic field here


Quantum wire

Coulomb interaction is screened by the gate => short-ranged U(x)

Slow modes: right and left movers

-e

+e

-e

+ea/d=0.1-1


Interaction leads to two-particle scattering

• characterized by momentum transfer q

Forward q~ 0 (mostly controls charge )

Backscattering q~ 2kF (mostly spin )

Screened interaction: U(0) ~ U(2kF)

• Must conserve momentum (at T=0)

Long-range interaction: U(0) >> U(2kF)


Hydrodynamic description: bosonization

• Two independent liquids: charge and spin are decoupled

• All excitations are density waves

=>

Charge density

Charge current

PE KE

= “coordinate”

jc = “momentum”

controlled by spin-rotational[SU(2)] symmetry

charge

spin

Dual pair φ and θ


Correlation functions are determined by interaction-dependent Kc & Ks

• Charge correlations

• Spin correlations

(zz) and (xx, yy) are equivalent only if Ks = 1/Ks => Ks =1 ( SU(2) fixed point)

gz0

This happens via BKT renormalization:spin backscattering is marginally irrelevantThus initially

Butat the “end”

initial = high-energy

final = low-energySaturday, February 12, 2011

Spin backscattering is noticeable: NMR in Sr2CuO3

Spin decomposition:

Spin correlations

uniform and staggered magnetization

NMR relaxation rate

noninteracting spinons

(OS, Singh, Sandvik;Takigawa,OS,Sandvik,Singh 1997)

Sr2CuO3

N

M

free part, H0


Spin backscattering is noticeable: NMR in Sr2CuO3

Spin decomposition:

Spin correlations

uniform and staggered magnetization

NMR relaxation rate

noninteracting spinons

(OS, Singh, Sandvik;Takigawa,OS,Sandvik,Singh 1997)

Sr2CuO3

N

M

free part, H0


Transport: Ballistic conductance G=I/V

Kc=1 Kc=1Kc<1

wirespin degeneracy

Number of subbands

perfect transmissiondue to multiple scatteringof plasmon waves

• spins play no role !

• Very fragile: single impurity“cuts” the wire

[Kane,Fisher 1992]


Quantum wire in magnetic field

: BS withspin-flip

: BS withoutspin-flip

marginal+oscillating =irrelevant

without the field


Renormalization Group: BKT flow in magnetic field

• Initial values of BS constants:

=1+gz

The fixed point

• The meaning: spin-flip scattering is frozen.

• Note

SU(2) U(1) : spins are in the plane perpendicular to BKs

* > 1


Hint of a new scattering channel: Cooper scattering

• But Sz is conserved

• Consequence of U(1) symmetry - need to break it!

Sz conservation forbids Cooper scattering







Conclusions


Spin - orbit interaction

• Two dimensions: Rashba Hamiltonian

• One dimension:

Confining potential Vconf(x) = mω x2/2 Transverse momentum is quantized <px> = 0(standing wave)

SOI = momentum-dependent magnetic fieldPreferred axis - σx : spin-rotational symmetry is reduced to U(1)


Single particle problem

• Eigenvalues

• Eigenstates

spinors

k > 0:counterclock-wiserotation of spins

k < 0:clock-wise rotation of spins

µ

χ+ and χ− : orthogonal at the same kbut not at the same energy

1 2

N.B: different precessionfrequencies at k1 and k2


Cooper scattering

U(k1- k2) but small overlap

U(k1+ k2) but bigger overlap

(almost) always:

1 2 1 2

• Cooper channel: spin non-conserving inter-subband pair tunnelingpossible due to Spin-Orbit only

[relative minus sign]


SDW instability

• Easy limit: EF >> gµB >> αkF

Free charge:

Interacting spin: + Cooper process

Kc < 1

Ks > 1 relevant!

• Strong-coupling limit: minimal energy @

Thus θs is frozen, hence φs fluctuates wildly.

• 2kF component of spin operators:

but

Power-law decay is controlled by charge sector:quasi Long Range Order


SDW: transport properties• Density: suppressed Friedel oscillations

• Should we expect better conductance? Impurity = potential scatterer => preserves spin

No single particle scattering off the potential impurity in SDW phase!

• But two-particle backscattering off the impurity does get generatedCorrection to conductance

Relevant (divergent) for strong e-e interaction: Kc < 1/2

Sx ordered component

• The physics: k1/2 => -k1/2 backscattering suppressed due to opposite ordering of Sx Inter-subband backscattering k1/2 => -k2/1 suppressed by destructive interference

at 2k1 and 2k2

at 2kF=k1+k2

N.B. magnetic impurity will scatter strongly


Close parallels with helical liquids and topological insulators

Topological Insulators,M. Z. Hasan and C. L. Kane,arxiv 1002.3895


H0 =2πv

3

Z

x

�J2R+ �J2

L→ 2πv

3

Z

x

�M2R+ �M2

L

Spin chain with uniform DM term via non-abelian rotations

•Rotate right (left) current by γ (−γ) about y axis

•Backscattering interaction of spin currents is modified

Gangadharaiah, Sun, OS, PRB 78, 054436; and Schnyder, OS, Balents, PRB 78, 174420

∑j

Dx̂ · �S j×�S j+1→ D̃Z

x(Jx

R− JxL)

•This rotation leaves invariant, thanks to emergent SU(2)R x SU(2)L symmetry

x

z

γ−γ

yh-D D

odd under inversion


•Magnetic field can now be absorbed

•Transverse to total field t components Mx,y oscillate with x

So that

•The final (momentum-conserving) Hamiltonian

Cooper term

Spin chain with DM cont’d


Y

YC

γ=π/2

γ=π/4

γ=0

BKT phase diagram: always in strong coupling phase for h perp. D

(h=0)

(D=0)

• SDW for arbitrary ratio of D/h = S.O. coupling/Zeeman

LL (massless)

Massive

Moroz et al. PRB 62, 16900 (2000);Gritsev et al. PRL 94, 137207 (2005).


(D+hsin[β])JxR +hJz

R

(−D+hsin[β])JxL +hJz

L

Arbitrary angle between SO axis and magnetic field

x

zh

-D Dh cos(β)

h sin(β)

Field experienced by right-moving electrons

Field experienced by left-moving electrons

∂xϕσ and ∂xθσ

Chiral rotation angles for right/left currents are different:linear shifts in both are required.

Cooper process does not conserve momentum anymore.Backscattering is reduced to purely marginal term:Hbs→−gcos[γR− γL] M

z

RM

z

L

β

★ End result: critical Luttinger state with slightly renormalized exponents

Detailed phase diagram via numerical solution of coupled RG equations:Garate and Affleck, PRB 81, 144419 (2010)


Iesr(ω) ∝ E2r ω χ��

xx(q = 0,ω)

χ��xx(q = 0,ω) ∝ δ(ω−gµBH)

Implications for ESR experimentsMeasures absorption of linearly polarized, and perpendicular to external magnetic field, radiation

SU(2) symmetric system of spins:Oshikawa, Affleck PRB 65, 134410 (2002)

Spin chain with uniform DM (quantum wire with SO interaction):right and left movers absorb at different frequencies !χ��

xx(q = 0,ω) ∝ δ�

ω−�

(D−hsinβ)2 +(hcosβ)2�

+δ�

ω−�

(D+hsinβ)2 +(hcosβ)2�

ideal Heisenberg chainChain with uniform DM

shift due to momentum boost ~ D/J

carbon nanotubes:A. De Martino et al, PRL (2002);generation of DC currentsin quantum wires:Ar. Abanov et al, arxiv 1008.1225


Conclusions• Interplay of magnetic field, spin-orbit and interactions: novel and interesting many-body physics

• SDW driven by electron pair tunneling between Zeeman-split subbands

Possible due to SU(2) breaking by the spin-orbit interaction

• Spin-density wave instability affects (charge) conductance

• Spin chains with uniform DM interaction

✓ Chiral rotations of right- and left- spin currents

❖ ESR experiments as a chiral probe of 1d excitations

★ Consequences for Majorana fermions?!


ESR study of Cs2CuCl4Schrama et al, Physica B 256-258, 637 (1998)

Single peak at T = 4.2 Kevolves into two peaks atT < 1.1 K

This spin-1/2 quasi-1d materialis known to possess uniformDM couplings, OS, Katsura, Balents PRB 82, 014421 (2010)

Experiments in Institute for Physical Problems, Moscow:K. Povarov, A.I. Smirnov et al(unpublished) confirm orientationdependent ESR doublets


Can we really get there?

• So far: assumed fully developed SDW state• With impurities present, what happens first:

SDW instability or strong-impurity limit - detailed RG required.

Naively: impurity is washed away if V0 < ΔSDW

1/Ks

Magnetic field

Affleck, Oshikawa PRB 60, 1038 (1999)

Weak field: Ks=1+1/[2 ln(EF/gµB)]

Strong field: Ks = 2


Tilted magnetic field: pair momentum is NOT conserved

• SDW stable when SO axis and magnetic field are orthogonal.Narrow (but finite) angular stability.

h

D


Monolayer Graphene on Ni (111)Dedkov et al. PRL 2008


Spin-orbit-induced spin-density wave in quantum wires and spin

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