Spin & doped ladders

8/3/2019 Spin & doped ladders

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Spin & doped laddersSpin & doped ladders

Spin liquid

& superconducting properties

Spin liquid

& superconducting properties



OutlineOutline

• Ladder geometry: some materials &

physical properties of spin ladders

• Doping: microscopic models & fieldtheory (bosonisation), numerical results,

phase diagram, magnetic properties …

• Pairing mechanism & beyond



Ladder geometryLadder geometry

Two-leg ladders Three-leg ladders

Spin ladders: each site carries a S=1/2 (typically Cu-II atom)

Even & odd-leg ladders have different magnetic properties …



A typical spin ladder:A typical spin ladder:

Derived from 2D high-Tc cuprate structure



Copper oxide Mott insulator

Heisenberg AF super-exchange

Copper oxide Mott insulator

Heisenberg AF super-exchange

On-site repulsion UAF exchange J

between copper spins



A ladder with 3D couplingsA ladder with 3D couplings



Chaboussant et al., 1998



A few simple theoretical

considerations

From the strong & weak coupling limits…



A RVB-like spin liquidA RVB-like spin liquid

Resonating Valence Bond= linear superposition of short-range VB configurations

Anderson 87



Strong coupling limitStrong coupling limit

• For isolated rungs: cost to excite a triplet• At finite magnon can propagate to gain

kinetic enegy:

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

π

E

k

2

0

continuum of 2 magnons states

one magnon branch

Exact diag. (G. Roux et al.)



The spin gap is robust !The spin gap is robust !• For even # of chains

→ finite spin gap

• 2-leg ladder & strong

coupling:

• 2-leg ladder & weak

coupling:

QMC determination of the spin gap

Greven et al., PRL 96



Spin correlation length vs T

Scaling behavior

Spin correlation length vs T

Scaling behavior

Greven et al. (QMC)

Weak coupling result



Even-odd effects !!Even-odd effects !!• GS (& low-T low-

energy) propertiesdepend on the # of

legs

• For nc even: finite

gap

• For nc odd: gapless

Scaling of spin gap (DMRG)

White et al. (1994)



Simple qualitative argumentSimple qualitative argument

(b)

(a)

“spin pairing”

Similar to 1D chain



Spin gap revealed by

experiments

Spin gap revealed by

experiments



Susceptibility: 2-leg vs 3-leg laddersSusceptibility: 2-leg vs 3-leg ladders

Azuma et al., PRL 94

Spin gap signature

No spin gap

I m p u r i t y P a u

l i c o n t r i b

u t i o n

Single magnon branchcontribution (2-leg ladder)

Troyer et al. 96



Thermodynamic propertiesThermodynamic propertiesExemple of Cu(HpN)Cl (2-leg)

Specific heat vs TComparison with full diagonalizationMagnetization curve

Comparison with LanczosHayward et al., 1996

Calemczuk et al., 1998

Clear signatures of spin gap !





The “telephone number” compound:

A superconducting ladder

The “telephone number” compound:

A superconducting ladder

P

l a n e s o f

c h a i n s

P l a n

e s o f 2 - l e g

l a d d e r s



Doping the ladders:

Zhang-Rice singlets

Doping the ladders:

Zhang-Rice singlets

Basic model for lightly doped

system:

t-J Hamiltonian(or large-U Hubbard model)



Conformal Field theoryConformal Field theory

Prediction for long wavelength (low-energy) physics

CnSm phases (Balents & Fisher)

- n =0,1 or 2 charge modes

- m=0,1 or 2 spin modes

Insulating (Mott) spin gap phase: C0S0 phase

If spin gap robust under doping: C1S0 phase:

- Spin correlations:

- Charge correlations:

- Superconducting correlations:



Weak coupling approachWeak coupling approach

• Anisotropic Hubbardladder & small U

• Weak coupling RG:

C1S0 phase stable

Balents & Fisher 96

H a l f -

f i l l i n g

( I n s u l a t i n g

s p i n

l i q

u i d )

Single chain physics



Strong coupling argumentStrong coupling argument

Separated holes

unpaired spins

Rung hole pair

Spin gap survises

Balance between kinetic & magnetic energies

at large rung coupling:



Numerical results (t-J)Numerical results (t-J)Correlations vs distance (DMRG) Phase diagram (ED)

Hayward PRL 95

Dominant pairing correlations

C1S1

C1S0 superconductor

Phase separation



Tunneling density of statesTunneling density of states

-6 -4 -2 0 20

0.1

0.2

0.3

-0.4 -0.2 0 0.2 0.4

0

0.1

0.2

0.3

0.4

0.5

ΔQP

ω

ω

N( )ω

nh=1/8

J=0.4

hole weight nh

elec. weight (1-nh)/2

Ω

Ω

Lower Hubbard band

Superconducting gap

Magnon resonance peak

Poilblanc et al. 2004

(Lanczos)

Low energy triplet excitation of energy



Getting more insight into thenature of the (spin-fluctuation

driven) pairing interaction ?



Unconventional d-wave pairingUnconventional d-wave pairing

have opposite signs

Typical of d-wave





Superconducting Green function(Lanczos)

Superconducting Green function(Lanczos)

Generalize calculation of dynamical correlation

to grand-canonical ensemble





Dynamics of d-wave pairingDynamics of d-wave pairing

-1 -0,5 0 0,5 1 -1 -0,5 0 0,5 1

2π/3

qy=0

5π/6

π

ω

I m

F ( q ,

)

2π/3

π/2

π/3

π/2

ω

qy= π(a) (b)

π/3

π/6

0

ω

Poilblanc, Scalapino, Capponi, PRL 2003

Change of sign ! => d-wave symmetry

Deviations from BCS

due to retardation

and

opposite signs !





Summary/conclusionSummary/conclusion• Ladders exhibit fascinating behaviors

observable experimentally (except fewexperiments on doped ladders, yet)

• Simple system to investigate non-

conventional superconductivity

• Not addressed here: role of disorder,

magnetic field induced QCP, role of spinanisotropy, …

Documents

Spin & doped ladders