1. Phase structure of graphene from Hybrid Monte-Carlo simulations
[ArXiv: , , ]
Pavel Buividovich1, L. von Smekal2,
D. Smith2, M. Ulybyshev1,
(Uni Regensburg1 and Uni Giessen2)

2. Semimetal-insulator phase transition
Good gap in graphene +
High carrier mobility =
Graphene-based semiconductors
Interesting for theorists:
Gap due to interactions?
Particle-hole bound states
Spontaneous breaking of
chiral symmetry
Attracted a lot of people from
HEP and Lattice QCD!
Different occupation
of А and В sublattices
ΔN = NA – NB

3. HMC Suspended graphene is a semimetal
Experiments by Manchester group [Elias et al. 2011,2012]:
Gap < 1 meV
HMC simulations (ITEP, Regensburg and Giessen)
[ , ]
Unphysical αc ~ 3 > αeff = 2.2
Schwinger-Dyson equations
[talk by M. Bischoff, ]
Unphysical αc ~ 5 > αeff = 2.2
In the meanwhile:
Graphene Gets a Good Gap
on SiC [M. Nevius et al ] – interactions are not
so important…
HMC
Schwinger-Dyson
Insulator
in HMC

4. Phase diagram in the V00 – V01 space
Tunable interactions and spontaneous symmetry breaking can be still realized:
In artificial graphene
In strained graphene
In graphene “superlattices” made with adatoms
Novel phases from tunable interactions:
Charge density wave
Quantum Spin Hall state (TI)
Spin liquid
Kekule distortion…
Mostly mean-field and RG studies so far ...
Vxy not
positive-definite
Difficult for HMC
[I. Herbut, cond-mat/ ]
[Raghu, Qi, Honerkamp,
Zhang ]

5. Hybrid Monte-Carlo simulations
Graphene tight-binding model with interactions
Particles = spin-up, Holes = spin-down
(bipartite lattice allows that)
Hubbard-Stratonovich + Suzuki-Trotter for partition function
Fermionic operator
Particle-hole symmetry:
No sign problem!!!

6. = Molecular Dynamics + Metropolis
Hybrid Monte-Carlo
= Molecular Dynamics + Metropolis
Molecular Dynamics trajectories as Metropolis proposals
Numerical error is corrected by accept/reject
Exact algorithm within the tight-binding model
Ψ-algorithm [Technical]:
Represent determinants
as Gaussian integrals
Molecular Dynamics
Trajectories
Molecular dynamics
Classical motion with
𝑯= 𝒙 𝛑 𝟐 𝒙 𝟐 +𝑺 𝝋 𝒙
If ergodic:
𝑷 𝝋 𝒙 ~𝒆𝒙𝒑(−𝑺[𝝋(𝒙)])
π(x) – conjugate to φ(x)

7. Detecting the phase transition
No spontaneous symmetry breaking in finite volume!
Phase transition = Large fluctuations of order parameter
Practical solutions:
Small symmetry breaking parameter δ, extrapolate ΔN to zero δ (also simplifies HMC, but bias for specific channel)
Calculate susceptibility dΔN/d δ
Volume scaling of squared order parameter
(in principle no bias)
[ ]
[ ]
[Talk by
M. Ulybyshev]

8. On-site interactions (Hubbard model)
Previous results at T ~ 0.01 eV [ ]: Uc ~ 10 eV
But: lattices up to 18 x 18 only due to different algorithm…

9. On-site interactions (Hubbard model)
Runs at T = eV: Uc likely > 13 eV
High sensitivity to temperature

10. Effect of V01 – first glimpse Shift of phase transition to higher V00

11. [8x8 lattice] [12x12 lattice] “Geometric” mass gap
Lattice energy spectrum has no zero energy levels if Lx ≠ 3 n, Ly ≠ 2 m
This ensures invertibility of fermionic operators in HMC simulations
All symmetries are preserved!!!
[8x8 lattice]
[12x12 lattice]

12. Results with geometric mass gap

13. Conclusions
Hybrid Monte-Carlo for graphene: lattices up to 48x48, electron gas temperature 103 K
Semimetal behavior for suspended monolayer graphene with screened Coulomb potential [ ], confirmed by Schwinger-Dyson with dynamical screening [Talk by M. Bischoff]
Critical Uc ~ 5 κ for Hubbard model on hex lattice
V01 shifts Uc up
Geometric energy gap: unbiased scan of phase diagram