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We study temperature effect on anomalous viscosity of Graphene Hall fluid within quantum many-vortex hydrodynamics. The commonly observed filling fraction, in the range is considered. An expression for anomalous viscosity dependent on a geometric parameter-Hall expansion coefficient is obtained at finite temperatures. It arises from strained induced pseudo-magnetic field in addition to an anomalous term in vortex velocity, which is responsible for renormalization of vortex-vortex interactions. We observed that both terms greatly modify the anomalous viscosity as well as an enhancement of weakly observed v fractions. Finite values of the expansion coefficient produce constant and infinite viscosities at varying temperatures. The infinities are identified as energy gaps and suggest temperatures at which new stable quantum hall filling fractions could be seen. This phenomenon is used to estimate energy gaps of already measured fractional Quantum Hall States in Graphene.

Graphene is a monolayer of carbon atoms tightly packed into a two-dimensional (2D) honeycomb lattice. Since its discovery in 2004, graphene has attracted a great deal of attention mainly due to its exceptionally high crystal and electronic quality. Shear viscosity has been studied in graphene [

Recently, there has been a great deal of interest and renewed focus on the anomalous viscosity,

In the following, we consider flows of quantized Hall vortices in the graphene Hall fluid as elementary objects forming themselves a highly correlated quantum fluid. In the regime of long-wave slow motion, an accurate hydrodynamic description becomes possible and does, in its own validity, not depend on the microscopic behavior of the electronic fluid. Vortex-vortex interactions will then be responsible for appearance of vortex FQH Effect. The hydrodynamics of vortex matter presented here differs from Euler hydrodynamics [

The remaining of the paper is organized as follows. In Section 2 we obtained vorticity equation from Euler hydrodynamic equation for graphene. From the Euler equation, we obtained a quantized Helmholtz-Kirchoff vortex solution. Vortex flux and momentum conservation laws are subsequently derived. The stress tensor is deduced from which the anomalous viscosity is read off. In Section 3, we analyzed behavior of the anomalous viscosity under density and temperature profiles for different filling fractions of the quantum hall vortex fluid. We made comparisons of our results to some recent experimental findings of FQHE energy gaps and concluded in Section 4 highlighting possible applications of the results.

Two dimensional Euler hydrodynamics can be straightforwardly derived following Boltzmann transport equation at local equilibrium [

respectively. Where

where the quantity,

where

mean density,

circulation

Equation (4) is the Helmholtz-Kirchoff equation for vortices. We will specialize in the zero energy state of the system where

In the quantum Hall regime of dissipationless flow, fluid particles do not carry heat flux, but vortices do. Vortices move in response to temperature gradient,

Using the identity

Equation (6) is crucial in these studies. In particular, our results are based on the second term which dictates discussions that follows. The term is responsible for anomalous behavior of the fluid when approaching a vortex. It is a quantum or micro-scale phenomenon which manifests itself at classical regime due to possible broken translation symmetry associated with lattice scale deformations. Its presents renormalize the vortex-vortex interactions in Equation (4). It also creates stresses perpendicular to the fluid flow with no work or dissipation. The associated transport coefficient (viscosity) is expected to be dissipationless. The momentum conservation following from broken translation invariance in the presence of external forces yields

where the stress tensor is

In order to have a complete description of our system, one need to quantize Equation (6). The process physically leads to interpretation of the quantized circulation as the filling factor of the fractional vortex Hall states [

Graphene,

same form as Equation (6). Taking the curl and utilizing the relation

As we have pointed out in the introduction, geometric deformations can lead to induction of giant synthetic magnetic fields. Vortices feel this contribution in addition to external magnetic field as an effective field,

where

where

Finally, to obtain

The term in brackets is written in such a way that the force stays constant in order that fluctuations in the zero energy states are bounded. Replacing v in Equation (12) and using

We now discuss the behavior of anomalous viscosity. Here, we observed how small temperature gradients affect viscosity of vortex fluid quantized on Hall states having filling fraction within

In

In

In

The intuitive meaning of the infinities

Generally, important aspects of the anomalous viscosity are the anomalous,

The anomalous and Hall expansion terms shift

In conclusion, we have computed dissipationless (anomalous) viscosity of quantum vortex Hall states in graphene within hydrodynamics using quantum many-vortex picture of Euler hydrodynamics. The hydrodynamics formalism allowed a great deal of simplifications as the microscopic theory is completely unnecessary and only few variables,

We constructed a general expression to compute viscosity of a fractional quantum Hall fluid. The temperature dependence is also analyzed. Using a Gaussian temperature profile, we demonstrated strongly that anomalous viscosity can be used as tool to measure strength of fractional fillings of Hall fluid. Relying on this principle, we actually showed that experimental observable fractions correspond to infinities in viscosity at some critical

temperatures. We associate the temperatures to energy gaps within

Finally, our results could be applied to strained-engineered devices to control viscosity. The studies should be able to guide future experiments towards observing new fractions. In particular, the temperature continuums may be probed, though away from the transition zones, for new fractions by controlling the Hall expansion coefficient parameter. Moreover, our work may clarify issues or resolve conflicts of different reported energy gaps, specifically, the

M. Rabiu thanks International Center for Theoretical Physics (ICTP) for hospitality and for provision of travel grants to conduct part of this research at its center in Trieste, Italy. M. Rabiu also thanks the Condense Matter Physics (CMP) section of ICTP for indebt discussions.

RabiuMusah,Samuel Y.Mensah,Ibrahim Y.Seini,Sulemana S.Abukari, (2015) Anomalous Viscosity of Vortex Hall States in Graphene. Journal of Applied Mathematics and Physics,03,1654-1661. doi: 10.4236/jamp.2015.312190