Phase structure of graphene from Hybrid Monte-Carlo simulations [ArXiv:1206.0619, 1304.3660, 1403.3620] Pavel Buividovich1, L. von Smekal2, D. Smith2, M. Ulybyshev1, (Uni Regensburg1 and Uni Giessen2)

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

HMC Suspended graphene is a semimetal Experiments by Manchester group [Elias et al. 2011,2012]: Gap < 1 meV HMC simulations (ITEP, Regensburg and Giessen) [1304.3660,1403.3620] Unphysical αc ~ 3 > αeff = 2.2 Schwinger-Dyson equations [talk by M. Bischoff, 1308.6199] Unphysical αc ~ 5 > αeff = 2.2 In the meanwhile: Graphene Gets a Good Gap on SiC [M. Nevius et al. 1505.00435] – interactions are not so important… HMC Schwinger-Dyson Insulator in HMC

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/0606195] [Raghu, Qi, Honerkamp, Zhang 0710.0030]

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!!!

= 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)

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) [1304.3660] [1206.0619] [Talk by M. Ulybyshev]

On-site interactions (Hubbard model) Previous results at T ~ 0.01 eV [1304.6340]: Uc ~ 10 eV But: lattices up to 18 x 18 only due to different algorithm…

On-site interactions (Hubbard model) Runs at T = 0.125 eV: Uc likely > 13 eV High sensitivity to temperature

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

[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]

Results with geometric mass gap

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 [1304.3660], 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