1. Carbon Nanorings, Lattice Gross-Neveu models of Polyacetylene and the Stability of Quantum Information Michael McGuigan Brookhaven National Laboratory August 20, 2012

2. Outline  Motivation for study of Carbon Nanorings.  Examples of Carbon Nanorings.  Tight binding Models of Polyacetylene as Gross Neveu Model.  Approaches to Gross-Neveu model.  AdS/CMT dual of Gross Neveu model.  Using the AdS dual theory to describe the stability of Quantum Information and entanglement.  Conclusions.

3. Motivation for Carbon Nanorings  As a novel superstructure, single-walled carbon nanorings exhibit interesting transport properties, such as  Aharonov-Bohm effects.  Magnetotransport.  Establishment of persistent currents.  Polyacetelene- electrically conductive polymers.  Polyacetylene undergoes a phase transition from a solitonic phase to a metallic phase. This transition is first-order, and occurs at a doping concentration y=0.06.

4. Examples of Carbon Nanorings Carbon nanoring with 22 Carbon atoms Carbon nanotorus with 1596 Carbon atoms Trans-polyacetylene ring with 40 Carbon atoms (blue) and 40 Hydrogen atoms (white). Cis-polyacetylene ring with 40 Carbon atoms (blue) and 40 Hydrogen atoms (white).

5. Tight Binding Models of carbon nanorings and Quantum Field Theory Tight binding models are effective theories describing the interaction of matter through nearest neighbor coupling. For Carbon nanorings tight binding models define quantum field theories in 1+1 (one space and one time) dimensions.

6. Tight Binding Models of Carbon Nanorings and Quantum Field Theory  Carbon Nanoring

7. Tight Binding Models of Carbon Nanorings and Quantum Field Theory  Carbon Nanotorus

8. Tight Binding Models of Carbon Nanorings and Quantum Field Theory Trans-polyacetylene Cis-polyacetylene These tight binding models can be derived from the Su, Schrieffer and Heeger dicrete model hamiltonian including phonon propagation and interaction.

9. Gauge Theories share common approaches and features with Gross- Neveu  Asymptotic Freedom.  Mass Gap.  Equation of State.  Large N.  AdS/QFT.  Monte Carlo Simulation.  Trace Anomaly.  Schwinger-Dyson Equation.  The multiple approaches are useful as they can be used to check results.

10. Gross-Neveu Model Polyacetylene, a linear chain of carbon atoms each with one hydrogen atom (shown in Fig. 1), as a function of the concentration of dopants, undergoes a finite-density phase transition which can be described by a tight binding model, the N = 2 Gross-Neveu model Structure of polyacetylene in the trans configuration.

11. Gross-Neveu Model  The Gross-Neveu model is described by the Lagrangian density, Where are N-component fermion fields and g denotes the coupling constant. The Gross-Neveu model with 1/N corrections gives an accurate description of the phase transition in polyacetylene. Takayama, Lin-liu Maki (1980), Campbell and Bishop (1982), Chodos Minakata (1997).

12. Large N approximation Introduce The leading order effective potential at finite temperature for Follow the treatment of PRD 83,065001 Daniel Fernandez-Fraile

13. Thermal mass gap  The thermal mass gap is defined by:

14. Equation of state  Once one has the thermal mass M(T) one can obtain the pressure from:  Other thermodynamic quantities follow from the pressure like entropy, energy density and trace anomaly.

15. Gross-Neveu Model Equation of State PressureEntropy

16. Gross-Neveu Model Equation of State Trace Anomaly The pressure, entropy, and trace anomaly as a function of temperature for different values of m. m can be found in the effective potential at finite temperature for the Gross-Neveu model.

17. Gravity Duals of Quantum Field Theory A gravity dual of quantum field theory is a description of the quantum field theory in terms of a theory of gravity in one higher dimension (sometimes two) in terms of a dictionary that maps quantities in the quantum field theory to quantities in the gravity theory and vice a versa. If the quantum field theory is a gauge theory the mapping is called the gravity/gauge correspondence. If the quantum field theory is used to describe a condensed matter theory (CMT) the mapping is called the gravity/CMT correspondence.

18. Dictionary for Gravity/CMT Correspondence

19. 2+1 Dimensional Black Hole

20. Gravity/CMT Correspondence Gravity Quantum Field Theory

21. Einstein-Dilaton Action  where V( ϕ ) is a potential function and C( ϕ ) and ω( ϕ ) are the coupling functions, we can derive the With a metric of the form: Yield the following equations:

22. 2+1 Black Hole Solutions When J = 0 and where f(r) is as in the following metric, For the usual BTZ black hole and the solution is:

23. 2+1 Black Hole with Dilaton

24. Equation of State For Usual Black hole For Dilaton Black hole

25. Entropy of 2+1 Dilaton Black Hole S/T T

26. Mass of 2+1 Dilaton Black Hole M/T 2 T

27. Trace Anomaly of 2+1 Dilaton Black Hole Δ T

28. Quantum Entanglement and AdS Black Holes  Mark Van Raamsdonk (2010) argues that product states with no entanglement leads to disjoint spacetime dual geometries.  Entangled states of the form Correspond to AdS black hole spacetimes with inverse Hawking Temperature β.  The more entangled the quantum state in a one parameter variation the the more connected the dual spacetime classically.

29. Stability of Quantum Information The fact that these are negative means that the black hole system and the quantum information it contains is stable. This is nontrivial as most black holes are unstable with respect to Hawking radiation. Another way of seeing the stability is to compute the specific heat. This can be shown to be positive using the fact that M is greater than J (Pidokrait 2003). This is for BTZ black holes. The stability of quantum information with the Dilaton is under investigation. To show the black hole system is stable one computes the Hessian of the entropy with respect to M and J. The Hessian matrix has negative eigenvalues given by

30. Summary  Black hole picture giving a new description of stability of quantum information and entanglement, Carbon nanorings can provide a way to study black holes using molecular systems.  2+1 dilaton black hole deviates from conformality at low T but is conformal at high T reflecting that the dual quantum field theory is asymptotically conformal or free.  Work in progress to determine dilaton potential that can reproduce the features of the Gross-Neveu equation of state at finite temperature and finite density. This will give an alternative picture of carbon nanorings with strong tight binding coupling.  Thanks to YunLi Tang (Binghampton U.) for assistance with computations and David Berenstein (UCSB) for discussions on AdS black holes.