spin-orbit-induced spin-density wave in quantum wires and spin
TRANSCRIPT
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?
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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)
Saturday, February 12, 2011
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)
Saturday, February 12, 2011
Spintronics
* No inversion symmetry => 2DEG heterostructures (e.g. GaAs)
Surface states (e.g. Au[111])
* Rashba Hamiltonian (1984)
Free electrons + SO :
Saturday, February 12, 2011
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)
Saturday, February 12, 2011
Topological Insulators,M. Z. Hasan and C. L. Kane,arxiv 1002.3895
Strong spin-orbit+surface states
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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:
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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!
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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)
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, 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
Saturday, February 12, 2011
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]
Saturday, February 12, 2011
Quantum wire in magnetic field
: BS withspin-flip
: BS withoutspin-flip
marginal+oscillating =irrelevant
without the field
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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)
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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]
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011
Close parallels with helical liquids and topological insulators
Topological Insulators,M. Z. Hasan and C. L. Kane,arxiv 1002.3895
Saturday, February 12, 2011
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
Saturday, February 12, 2011
•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
Saturday, February 12, 2011
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).
Saturday, February 12, 2011
(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)
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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?!
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011
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
Saturday, February 12, 2011