Phase structure of graphene from Hybrid MonteCarlo simulations
Phase structure of graphene from Hybrid Monte-Carlo simulations [Ar. Xiv: 1206. 0619, 1304. 3660, 1403. 3620] Pavel Buividovich 1, L. von Smekal 2, D. Smith 2, M. Ulybyshev 1, (Uni Regensburg 1 and Uni Giessen 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 Δ N = N A – NB
Suspended graphene is a semimetal Experiments by Manchester group [Elias et al. 2011, 2012]: Gap < 1 me. V HMC simulations (ITEP, Regensburg and Giessen) HMC [1304. 3660, 1403. 3620] Unphysical αc ~ 3 > αeff = 2. 2 Schwinger-Dyson equations [talk by M. Bischoff, Bischoff 1308. 6199] Schwinger-Dyson Unphysical αc ~ 5 > αeff = 2. 2 In the meanwhile: Graphene Gets a Good Gap on Si. C [M. Nevius et al. 1505. 00435] – interactions are not so important… Insulator in HMC
Phase diagram in the V 00 – V 01 space Tunable interactions and spontaneous symmetry breaking can be still realized: Vxy not positive-definite Difficult for HMC • 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. . . [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!!!
Hybrid Monte-Carlo = Molecular Dynamics + Metropolis • Molecular Dynamics trajectories as Metropolis proposals Molecular Dynamics Trajectories • Numerical error is corrected by accept/reject • Exact algorithm within the tight-binding model • Ψ-algorithm [Technical]: Represent determinants as Gaussian integrals
Detecting the phase transition No spontaneous symmetry breaking in finite volume! Phase transition = Large fluctuations of order parameter [1304. 3660] Practical solutions: Small symmetry breaking parameter δ, extrapolate ΔN to zero δ (also simplifies HMC, but bias for specific channel) [1206. 0619] Calculate susceptibility dΔN/d δ Volume scaling of squared order parameter (in principle no bias) [Talk by M. Ulybyshev]
On-site interactions (Hubbard model) Previous results at T ~ 0. 01 e. V [1304. 6340]: Uc ~ 10 e. V But: lattices up to 18 x 18 only due to different algorithm…
On-site interactions (Hubbard model) Runs at T = 0. 125 e. V: Uc likely > 13 e. V High sensitivity to temperature
Effect of V 01 – first glimpse Shift of phase transition to higher V 00
“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!!! [8 x 8 lattice] [12 x 12 lattice]
Results with geometric mass gap
Conclusions • Hybrid Monte-Carlo for graphene: lattices up to 48 x 48, 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 • V 01 shifts Uc up • Geometric energy gap: unbiased scan of phase diagram
- Slides: 13