Carbon Nanorings, Lattice Gross-Neveu models of Polyacetylene

and the Stability of Quantum Information

Michael McGuiganBrookhaven National Laboratory

August 20, 2012

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.

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.

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).

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.

Tight Binding Models of Carbon Nanorings and Quantum Field Theory

Carbon Nanoring

Tight Binding Models of Carbon Nanorings and Quantum Field Theory

Carbon Nanotorus

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.

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.

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.

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).

Large N approximation

Introduce

The leading order effective potential at finite temperature for

Follow the treatment of PRD 83,065001 Daniel Fernandez-Fraile

Thermal mass gap

The thermal mass gap is defined by:

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.

Gross-Neveu Model Equation of State

Pressure Entropy

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.

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.

Dictionary for Gravity/CMT Correspondence

2+1 Dimensional Black Hole

Gravity/CMT Correspondence

Gravity Quantum Field Theory

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:

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:

2+1 Black Hole with Dilaton

Equation of State

For Usual Black hole For Dilaton Black hole

Entropy of 2+1 Dilaton Black Hole

S/T

T

Mass of 2+1 Dilaton Black Hole

M/T2

T

Trace Anomaly of 2+1 Dilaton Black Hole

Δ

T

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.

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

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.