[PPT] - Competing order in the fermionic Hubbard model on the hexagonal PowerPoint Presentation

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Fachbereich 7 | Institut für Theoretische Physik | Lorenz von Smekal | 01

Southampton, 27 July 2016

Pavel Buividovich, Maksim Ulybyshev

(Regensburg)

Dominik Smith, Lorenz von Smekal

(Giessen)

Competing order in the fermionic Hubbard model

n the hexagonal graphene lattice

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[courtesy L. Holicki]

27 July 2016 | Lorenz von Smekal | p.

Introduction

2 Lattice 2016

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Honeycomb Lattice

3 Lattice 2016

graphene

triangular lattice

(2 atoms per unit cell)

hexagonal Brillouin zone
single-particle energy bands

K K' M

E±(k) = ± |Φ(k)|

Φ(k) = t X

i

eik·δi

structure factor:

E(p) = ±~vf|p| , vf = 3ta/2 ' 1 ⇥ 106m/s ' c/300

massless dispersion around Dirac points K±

[Wallace, 1947]

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Honeycomb Lattice

4 Lattice 2016

mass terms (gaps)

Hm = 1 N 2 X

k,σ

mσ

a†

k,σak,σ − b† k,σbk,σ

(pseudo-spin) staggered on-site potential

Graphene Gets a Good Gap on SiC Nevis et al., PRL 115 (2015) 136802

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Honeycomb Lattice

4 Lattice 2016

mass terms (gaps)

Hm = 1 N 2 X

k,σ

mσ

a†

k,σak,σ − b† k,σbk,σ

(pseudo-spin) staggered on-site potential
mcdw = 1

2 (mu + md) msdw = 1 2 (mu md)

spin (flavor) dependence

Graphene Gets a Good Gap on SiC Nevis et al., PRL 115 (2015) 136802

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Honeycomb Lattice

4 Lattice 2016

mass terms (gaps)

Hm = 1 N 2 X

k,σ

mσ

a†

k,σak,σ − b† k,σbk,σ

(pseudo-spin) staggered on-site potential
mcdw = 1

2 (mu + md) msdw = 1 2 (mu md)

spin (flavor) dependence

Graphene Gets a Good Gap on SiC Nevis et al., PRL 115 (2015) 136802

Coulomb interaction

αg = e2 4πε ~vf

effective coupling

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Honeycomb Lattice

4 Lattice 2016

mass terms (gaps)

Hm = 1 N 2 X

k,σ

mσ

a†

k,σak,σ − b† k,σbk,σ

(pseudo-spin) staggered on-site potential
mcdw = 1

2 (mu + md) msdw = 1 2 (mu md)

spin (flavor) dependence

Graphene Gets a Good Gap on SiC Nevis et al., PRL 115 (2015) 136802

− → − →

m → 0

with strong interactions: Mott-insulator transition charge-density wave (CDW) AF spin-density wave (SDW)

Coulomb interaction

αg = e2 4πε ~vf

effective coupling

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Honeycomb Lattice

4 Lattice 2016

mass terms (gaps)

Hm = 1 N 2 X

k,σ

mσ

a†

k,σak,σ − b† k,σbk,σ

(pseudo-spin) staggered on-site potential
mcdw = 1

2 (mu + md) msdw = 1 2 (mu md)

spin (flavor) dependence

Graphene Gets a Good Gap on SiC Nevis et al., PRL 115 (2015) 136802

− → − →

m → 0

with strong interactions: Mott-insulator transition charge-density wave (CDW) AF spin-density wave (SDW)

2 4 1 2 2 4 CDW U QSH CDW SM V1 QSH SDW SDW V2

Raghu et al., PRL 100 (2008) 156401

Coulomb interaction

αg = e2 4πε ~vf

effective coupling

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Honeycomb Lattice

4 Lattice 2016

mass terms (gaps)

Hm = 1 N 2 X

k,σ

mσ

a†

k,σak,σ − b† k,σbk,σ

(pseudo-spin) staggered on-site potential
mcdw = 1

2 (mu + md) msdw = 1 2 (mu md)

spin (flavor) dependence

Graphene Gets a Good Gap on SiC Nevis et al., PRL 115 (2015) 136802

− → − →

m → 0

with strong interactions: Mott-insulator transition charge-density wave (CDW) AF spin-density wave (SDW)

2 4 1 2 2 4 CDW U QSH CDW SM V1 QSH SDW SDW V2

Raghu et al., PRL 100 (2008) 156401

Coulomb interaction

αg = e2 4πε ~vf

effective coupling

sign-problem in HMC with mcdw > 0

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Potentially Strong Interactions

5 Lattice 2016

suspended graphene

ε → 1

αg = e2 4π ~vf ≈ 300 137 ≈ 2.19

remains conducting, semimetal Elias et al., Nature Phys. 2049 (2011)

puzzle

predictions at the time

αcrit ∼ 1

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Potentially Strong Interactions

5 Lattice 2016

suspended graphene

ε → 1

αg = e2 4π ~vf ≈ 300 137 ≈ 2.19

remains conducting, semimetal Elias et al., Nature Phys. 2049 (2011)

puzzle

predictions at the time

αcrit ∼ 1

screening at short distances

Wehling et al., PRL 106 (2011) 236805

:

1 10 0.2 0.4 0.6 0.8 1 V(r) [eV] r [nm]

part. screened

CRPA ITEP screened

std. Coulomb

from σ-band electrons and localised higher energy states

U V1 V2

(1 = 2.4 and d = 2.8 )

−1(⃗

k) = 1

1 1 + 1 + (1 − 1)e−kd 1 + 1 − (1 − 1)e−kd

interpolate at intermediate distances

with dielectric thin-film model

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HMC on Hexagonal Lattice

6 Lattice 2016

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 <Δn> m [eV] α = 2.73 α = 3.36 α = 4.37 α = 4.86 0.1 0.2 0.3 0.4 2 2.5 3 3.5 4 4.5 5 <ΔN > αeff Nx = 36 Nx = 18

chiral extrapolation

msdw → 0

semimetal-insulator transition in unphysical regime

αcrit ≈ 3 > 2.19

Ulybyshev, Buividovich, Katsnelson, Polikarpov, PRL 111 (2013) 056801 Smith, LvS, PRB 89 (2014) 195429

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Dyson-Schwinger Equations

7 Lattice 2016

hexagonal lattice, screened Coulomb

Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015)

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Dyson-Schwinger Equations

8 Lattice 2016

hexagonal lattice, screened Coulomb

Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015)

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Dyson-Schwinger Equations

9 Lattice 2016

hexagonal lattice, screened Coulomb

Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015)

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Dyson-Schwinger Equations

10 Lattice 2016

hexagonal lattice, screened Coulomb

Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015)

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Dyson-Schwinger Equations

11 Lattice 2016

hexagonal lattice, screened Coulomb

Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015)

graphene’s single-particle band structure

no Lindhard screening

αcrit ≈ 1.5

Π(!, ~ q) =

+ +

what about CDW and the other insulating phases?

from π-band electrons

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Dyson-Schwinger Equations

12 Lattice 2016

hexagonal Hubbard model, Hartree-Fock

iΣ(~ p) =

−1 − −1

= −

fermion self-energy

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Dyson-Schwinger Equations

12 Lattice 2016

hexagonal Hubbard model, Hartree-Fock

iΣ(~ p) =

−1 − −1

= −

fermion self-energy

Katja Kleeberg et al., in preparation

Araki and Semenoff, PRB 86 (2012) 121402(R) with on-site U and nearest-neighbor V first order

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HMC on Hexagonal Lattice

13 Lattice 2016

chiral extrapolation, SDW
nly on-site U first

2 4 6 8 4 4.4 4.8 5.2 5.6 <χdis> U00 / κ m = 0.0259259 κ Nx = 6 Nx = 12 Nx = 18

Nt = 80

msdw → 0

6 7 8 9 10 11 12 0.01 0.02 0.03 0.04 χcon

max

m / κ Nx = 6 Nx = 12

2 3 4 5 6 0.01 0.02 0.03 0.04 Upeak / κ m / κ Nx = 6 Nx = 12

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HMC on Hexagonal Lattice

13 Lattice 2016

chiral extrapolation, SDW
nly on-site U first

2 4 6 8 4 4.4 4.8 5.2 5.6 <χdis> U00 / κ m = 0.0259259 κ Nx = 6 Nx = 12 Nx = 18

Nt = 80

msdw → 0

6 7 8 9 10 11 12 0.01 0.02 0.03 0.04 χcon

max

m / κ Nx = 6 Nx = 12

2 3 4 5 6 0.01 0.02 0.03 0.04 Upeak / κ m / κ Nx = 6 Nx = 12

Sorella, Tosatti, EPL 19 (1992) 699: Uc ≈ 4.5κ Assaad, Herbut, PRX 3 (2013) 031010: Uc ≈ 3.8κ

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HMC with Geometric Mass

14 Lattice 2016

8 × 8 lattice 12 × 12 lattice

hexagonal Brillouin zone
removes Dirac points
preserves symmetries
improves invertibility

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Suitable Order Parameters

15 Lattice 2016

O = 1 L2 s⌦ X

i∈A

Oi 2↵ + ⌦ X

i∈B

Oi 2↵

for zero(geometric)-mass simulations, use with

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Suitable Order Parameters

15 Lattice 2016

O = 1 L2 s⌦ X

i∈A

Oi 2↵ + ⌦ X

i∈B

Oi 2↵

spin-density wave:

Oi → ~ Si = X

σ,σ0

c†

i,σ

~ σσ0 2 ci,σ0 ci = ⇢ ai , i ∈ A bi , i ∈ B

for zero(geometric)-mass simulations, use with

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Suitable Order Parameters

15 Lattice 2016

O = 1 L2 s⌦ X

i∈A

Oi 2↵ + ⌦ X

i∈B

Oi 2↵

charge-density wave:

Oi → Qi = X

σ

c†

i,σci,σ − 1

spin-density wave:

Oi → ~ Si = X

σ,σ0

c†

i,σ

~ σσ0 2 ci,σ0 ci = ⇢ ai , i ∈ A bi , i ∈ B

for zero(geometric)-mass simulations, use with

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HMC with Geometric Mass

16 Lattice 2016

0.05 0.1 0.15 0.2 0.25 0.02 0.04 0.06 0.08 0.1 0.12 0.14

<Sx> 1/L

T=0.046 κ, dt κ=0.27 , V01=0 Sx, V00=2.96 κ Sx, V00=3.33 κ Sx, V00=3.70 κ Sx, V00=4.07 κ Sx, V00=4.44 κ Sx, V00=4.81 κ

pure on-site U, SDW

L = 8, 14, 20, 26

U as before: Uc ≈ 3.8κ

Nt = 80

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HMC with Geometric Mass

17 Lattice 2016

violation of spin symmetry!

L = 8, 14, 20, 26

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.02 0.04 0.06 0.08 0.1 0.12 0.14

<Si> 1/L

T=0.046 κ, dt κ=0.27 Sx, V00=4.07 κ Sx, V00=3.33 κ Sz, V00=4.07 κ Sz, V00=3.33 κ

Sx, U Sz, U

Nt = 80

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0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.02 0.04 0.06 0.08 0.1 0.12 0.14

<Si> 1/L

T=0.023 κ, dt κ=0.27 Sx, V00=4.07 κ Sx, V00=3.33 κ Sz, V00=4.07 κ Sz, V00=3.33 κ

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HMC with Geometric Mass

18 Lattice 2016

violation of spin symmetry!

Sx, U Sz, U

Nt = 160

lower temperatures don’t help

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0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.02 0.04 0.06 0.08 0.1 0.12 0.14

<Si> 1/L

T=0.023 κ, dt κ=0.0675 Sx, V00=4.07 κ Sx, V00=3.33 κ Sz, V00=4.07 κ Sz, V00=3.33 κ

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HMC with Geometric Mass

19 Lattice 2016

Sx, U Sz, U continuum limit in time does

Nt = 640

violation of spin symmetry!

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Perfect Action

20 Lattice 2016

time-discretisation breaks sublattice symmetry

already in non-interacting tight-binding theory

replace in fermion matrix 1 − Htb ∆τ → e−Htb ∆τ Nt:

with exponential for continuous time-evolution in fermion matrix

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.1 0.2 0.3 0.4 0.5 0.6

<Sx,z> dt κ

T=0.046 κ <Sz>, V00=3.33 κ, lattice 8x8 <Sx>, V00=3.33 κ, lattice 8x8 <Sz>, V00=4.07 κ, lattice 8x8 <Sx>, V00=4.07 κ, lattice 8x8 <Sz>, V00=3.33 κ, ExpLattice 8x8 <Sx>, V00=3.33 κ, ExpLattice 8x8 <Sz>, V00=4.07 κ, ExpLattice 8x8 <Sx>, V00=4.07 κ, ExpLattice 8x8

U = 4.07κ U = 3.33κ

Sx = Sz

L = 8

Sx Sz

40 80 160 320

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Phase Diagram

21 Lattice 2016

hexagonal Hubbard model

with on-site U and nearest-neighbor V

0.5 1 1.5 2 2.5 3 4 5 6 7 V01/κ V00/κ T=0.046 κ AFM phase SM phase CDW phase (??)

V/κ U/κ SDW SM CDW?

preliminary Hartree-Fock phase coexistence? V = U/3

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Conclusions

22 Lattice 2016

HMC on hexagonal graphene lattice
continuous time-evolution in improved fermion matrix
geometric mass, no explicit sublattice symmetry breaking
study competing CDW/SDW order in extended Hubbard model

screened Coulomb interactions ➟ suspended graphene in semimetal phase maintain full spin and sublattice symmetries no explicit symmetry breaking ➟ study competition between various insulating phases Uc ≈ 3.8 κ confirmed for anti-ferromagnetic Mott insulator transition (SDW) extend results into U-V plane with first order transition to CDW (sign-problem)

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Conclusions

22 Lattice 2016

HMC on hexagonal graphene lattice
continuous time-evolution in improved fermion matrix
geometric mass, no explicit sublattice symmetry breaking
study competing CDW/SDW order in extended Hubbard model

screened Coulomb interactions ➟ suspended graphene in semimetal phase maintain full spin and sublattice symmetries no explicit symmetry breaking ➟ study competition between various insulating phases Uc ≈ 3.8 κ confirmed for anti-ferromagnetic Mott insulator transition (SDW) extend results into U-V plane with first order transition to CDW (sign-problem)