Self-consistent Calculations for Inhomogeneous Correlated Systems

Amit GhosalIISER, Kolkata

HRI, 12 Nov, 2010

Mean-Field decoupling of four-fermionic interaction term

Diagonalize full H using guess for these expectation values, calculate them

in terms of the eigenfunctions (and eigenvalues) of H. Modify guess andkeep iterating until guess and calculated values are equal.

Then calculate expectation values self-consistently:

Works well for disordered/inhomogeneous situations

(disorder breaks all symmetries)

Scheme for self-consistent calculation for interacting systemScheme for self-consistent calculation for interacting system::

Justification comes in comparison with other methods, e.g. QMC, ED etc.

See, for example, Chen et al. J. Phys.: Cond. Matter, 20, 345211

Outline:

Results from Anderson-Hubbard model

→ metallic phase in a 2D interacting disordered system?

Results from Attractive Hubbard model with disorder

→ Superconductor-Insulator transition driven by impurities

Structure of a d-wave vortex lattice

→ Charged vortices?

Outlook and Conclusions

Part I:

Anderson-Hubbard model in 2D

Anderson-Hubbard Model

MF decoupling:

with:

Effective Hamiltonian contains expectation valuesof operators with respect to its own eigenstates.

→ Must be calculated in a self-consistent manner

For a system of size N = L X L, there are (3N+1) SC parameters:

2N values of <n > for σ = ↑ and ↓ iσ

N values of h

One value of μ to fix <n>=1 (half-filling)i

(Heiderian & Trivedi, PRL, 2004)

Model parameters: U = 4t, N = 28 X 28, <n>=1

P(V )

-V V V i

i

Motivation: 2D Metal-Insulator Transition (2D MIT):

real metals, 2D or not 2D??

For dimension d ≤ 2, ALL non-interacting (single particle) states are localized for arbitrarily small disorder!!

Scaling theory of localization (1979)

Kravchenko, Sarachik, Rep. Prog. Phys. 67, 1 (2004)

• What’s “NEW” ??

strong correlation

disorder?

Si-MOSFET

Clean system (V=0) @ <n>=1 is a Mott insulator (U >> t)

Hallmark of Mott insulator:(a) Mott gap

(b) AFM spin correlation

U = 4t, N = 24 X 24, <n>=1

Anderson-Hubbard Model (contd.)

How do Mott-gap and AFM order behave as a function of V?

Energy scale for charge fluctuation (gap) ⁓ U

Energy scale for AFM coupling ⁓ t²/U

Expect AFM order tovanish with V before

the gap does!

Surprise!!

Insight comes from spatial correlations ...

Look at the spatial structure / distributions:Staggeredmagnetization

Correlation of AFM sites (m > 0.3)paramagnetic (PM) sites (m < 0.1)with site disorder

AFM in WD regions, singly occupied sitesand large Mott gap

SD regions have low lyingexcitations, 0/2 occupations,AFM vanishes locally.

SD regions grow with V

V=3 t

Why does AFM order survives up to large V?

Percolation-based model is at work for U >> t

sites with Vⁱ < -U/2 → doubly occupied, and with Vⁱ > U/2 → empty

sites with |Vⁱ| < U/2 are singly occupied, and have free spinswith t → 0

Turning on t up to 2nd order leads to AFM order bystandard mechanism of exchange (J modified by V)

→ Fraction of singly occupied (magnetic) sites x = U/2V

AFM vanishes at the critical V when doubly and unoccupied sites percolate!

V = 3.4t consistent with classical percolation of vacancies on a 2D square latticeC2

Condition for AFLRO: x > x ⁓ 0.59 (percolation threshold)c

Localization properties of the Wave-function at ɛ F

Metallic phase in 2D in Anderson-Hubbard modelat intermediate disorder!!

What is the phase between V and V ? C1 C2

Screening of strong disorder by interaction:

Why metal in 2D for effective non-interacting disorderd system?

PM sites (SD region) suffer significant screening

Screening of AFM sites (WD region) negligible

Screened potential correlated

Inhomogeneous magnetic field (hⁱ) correlated with disorder

Herbut, PRB, 63, 113102

Note: direct evidence for a metallic phase from conductivity calculation

Kobayashi et al. arXiv:08073372

ProposedPhase Diagram:

0 < V < U/2 → Mott Insulator (brown)

U/2 < V < 5U/6 → Insulator A (pink)

V > 5U/6 → Insulator B (blue)

metal in between (gray)

Part II:

Disordered s-wave superconductors

Dirorder driven SIT: Motivation

(1) Cooper Pairing:

(2) Phase Coherence:

Pair size ξ

phasePairing amplitude

externallyapplied phasetwist Φ

Clean system

What happens when we add another axis -- disorder?

clear separation of SCand I phase for T → 0

V

Haviland et al. PRL, 62, 2180

Model and Method: Attractive Hubbard model with impurities

minimal model to study the interplay of SC and disorder

Mean-Field decomposition in HF-Bogoliubov channel

attractiondisorder

Self-consistently determine:

Local pairing amplitude

Local density

μ that fixes average density

Standard way to diagonalize effective H using Bogoliubov-de Gennes transformation:

with:

→ so that:

Recalculate local pairing amplitude and local density using u's and v's (T = 0)

Distribution of pairing amplitude:

ξ(V = 0) ⁓ 10a

V ⁓ 1.75 tC

Δ(V=0)

How is inhomogeneous Δdistributed spatially?

AG, Randeria & Trivedi,PRL, 81, 3940; PRB, 65, 14501

Spatial profile of Δ:

SC “islands”(of size ξ)

Where in space do SC-islands form?

SC-islands support low-lying excitations!

“Islands” form where |Vⁱ – μ| ⁓ 0

μ

Δ ⁓ 0Δ finite

Evolution of Energy Gap and

superfluid stiffness with V

Gap robust to V

Stiffness decreases

obtained from Kubo Formula

Once the SC-islands are formed, system susceptible to phase fluctuations

Quantum Phase Fluctuations:

→ Renormalization of D due to quantum phase fluctuationsS

In clean system anyapplied phase twistdistributed uniformly

Inhomogeneous systemgain energy by distributingphase twist non-uniformly

Most of the twist lives inthe sea, keeping the phaseon SC-islands almost uniform

Evolution of D renormalized by phase fluctuations:S

Good agreement with QMC

QMC by Trivedi et al, PRB, 54, R3756

Proposed phase diagram on U-V plane based on BdG results:

Of the 3 possibilities for U → 0,(a) represents the correct result

Part II:

d-wave vortex lattice withcompeting AFM order

d-wave vortex lattice:

Model for a d-wave SC (t-J type)

Orbital magnetic field introduced through the Peierl's factor:

→ results into Abrikosov vortex lattice

Study the interplay of d-SC and AFM at the vortex cores

Mean-Field decoupling:

Self-consistent variables

Working parameters: J = 1.15 t, <n> = 0.875

Minimum energy configuration has m = 0 at all sites, in the absence of orbital field.

Allows AFM + d-SC ordering

In HTSC cuprates, AFM stabilized at <n> = 1,and suppressed quickly away from half-filling.

AG, Kallin & Berlinsky, PRB, 66, 214502

where,

Spatial distributions on a cell containing one vortex:

d-SC pairingamplitude

AFM orderparameter

Chargedensity

AFM develops only @ vortex cores, where d-SC vanishes due to orbital field

In the absence of AFM, spatial charge density structureless

When AFM is allowed to develop self-consistently,<n> reorganizes itself to accommodate AFM.

<n> → 1 locally near core, a filling favorablefor AFM order to stabilize!

Coulomb repulsion, at the HF level, does not wash out such charge accumulationin AFM vortex core Knapp et al. PRB,71, 64504

AG, Kallin & Berlinsky, PRB, 66, 214502

Other self-consistent calculations oninhomogeneous correlated system:

(1) Superfluid-Bose glass transition in disordered Bose-Hubbard model

Sheshadri et al. PRL, 75, 4075

(2) Electrodynamics of s-wave vortex lattice

Atkinson & MacDonald, PRB, 60, 9295

(3) Impurity effects on d-wave superconductors

AG, Randeria & Trivedi, PRB, 63, R20505; Hirschfeld Group

(4) Metal Insulator transition in 3D Anderson-Hubbard model

Chen & Gooding, PRB, 80, 115125

And many others ...

Self-consistent calculations are relatively simple, and are capable ofexploring interesting physics in inhomogeneous correlated systems

Conclusions:

Quantum Phase Fluctuations:

Phenomenology: Quantum XY-Model

Variational Method: self-consistent harmonic approximationWood & Stroud, PRB, 25, 1600

Estimate D by finding out the best harmonic H that describes H S θ 0

→ Renormalization of D due to quantum phase fluctuationsS

Note: disorder enters only through

TVR, Phys. Scripta, T27, 24