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## Quantum elasticity of graphene: Thermal expansion coefficient and specific heat

We explore thermodynamics of a quantum membrane, with a particular application to suspended graphene membrane and with a particular focus on the thermal expansion coefficient. We show that an interplay between quantum and classical anharmonicity-controlled fluctuations leads to unusual elastic properties of the membrane.

The effect of quantum fluctuations is governed by the dimensionless coupling constant, g0 1, which vanishes

in the classical limit ( → 0) and is equal to 0.05 for graphene. We demonstrate that the thermal expansion

coefficient αT of the membrane is negative and remains nearly constant down to extremely low temperatures,

T0 ∝ exp(−2/g0).We also find that αT diverges in the classical limit: αT ∝ −ln(1/g0) for g0 → 0. For graphene

parameters, we estimate the value of the thermal expansion coefficient as αT −0.23 eV−1, which applies below

the temperature Tuv ∼ g00 ∼ 500K(where 0 ∼ 1 eVis the bending rigidity) down to T0 ∼ 10−14 K. ForT <T0,

the thermal expansion coefficient slowly (logarithmically) approaches zero with decreasing temperature. This

behavior is surprising since typically the thermal expansion coefficient goes to zero as a power-law function.We

discuss possible experimental consequences of this anomaly.We also evaluate classical and quantum contributions

to the specific heat of the membrane and investigate the behavior of the Gr¨uneisen parameter.