It's been quite some time since my last post. All of my writing efforts have gone toward producing the first draft of my Ph.D. dissertation, The Effects of Static Electric Potentials on Single Electrons and Excitons in Carbon Nanotubes: A Theoretical Study. I'm just about done --- with the first draft. I've got a 200 page monster with no figures and incomplete references that I've got to hammer into shape by the middle of May, but it's a lot better than staring at a blank page!
Although I haven't posted anything recently, I've been reading a lot. Going over background materials for my dissertation reminded me of how fascinating semiconductor physics, quantum field theory, and excitons are! I've had some time for deviations from my primary research focus as well. My pile of interesting papers continues to grow, plus I've started reading some textbooks on econophysics and nonlinear dynamics. Fascinating stuff.
Hopefully I'll get back in the habit of posting regularly. I hope to make the blog more friendly to others too. It will still primarily focus on interesting papers I've read (I'm still worried that once they go into my filing cabinet, I may never be able to find them again...), but I'd like to make the blog into more of a journal than a database. Maybe we can get a little dialog going on some topics. Maybe I can sharpen my skills at explaining complicated things in simple terms. Maybe I'll be so busy finishing my dissertation and finding a job that I won't have time for anything else. Who knows?
Check back in from time to time to see what's new. And drop me an e-mail if you like.
Jesse
Monday, February 18, 2008
Sunday, August 12, 2007
Peapods are the Same as Nanotubes
Transport Properties of Carbon Nanotube C_{60} Peapods
C.H.L. Quay, et al.
PRB 76, 073404 (2007)
URL: http://link.aps.org/abstract/PRB/v76/e073404
In this article, the authors present data that suggest room temperature transport properties of carbon nanotube peapods are basically the same as unfilled carbon nanotubes. The peapods are either semiconducting or metallic and there is a Coulomb blockade.
The authors are surprised by the similarities with unfilled nanotubes. Since the nanotubes tested were selected from an ensemble containing mostly --- but not all --- peapods, one possible explanation is that the authors happened to select 7 unfilled nanotubes from the ensemble. They present a Bayesian statistical analysis to show that the chance of that happening are small, and that the most likely number of peapods in the experiment is roughly 6 of 7.
The authors' conclusion is that the transport properties of fullerene peapods are not that different from unfilled nanotubes. Although their statistical analysis seems correct, they could bolster their claim with more samples. I share their initial surprise that a lattice of buckyballs inside a nanotube has virtually no effect on its transport properties.
If the authors are correct, there are no signatures of the buckyball lattice in the transport spectra of a nanotube. How else might one go about detecting them?
C.H.L. Quay, et al.
PRB 76, 073404 (2007)
URL: http://link.aps.org/abstract/PRB/v76/e073404
In this article, the authors present data that suggest room temperature transport properties of carbon nanotube peapods are basically the same as unfilled carbon nanotubes. The peapods are either semiconducting or metallic and there is a Coulomb blockade.
The authors are surprised by the similarities with unfilled nanotubes. Since the nanotubes tested were selected from an ensemble containing mostly --- but not all --- peapods, one possible explanation is that the authors happened to select 7 unfilled nanotubes from the ensemble. They present a Bayesian statistical analysis to show that the chance of that happening are small, and that the most likely number of peapods in the experiment is roughly 6 of 7.
The authors' conclusion is that the transport properties of fullerene peapods are not that different from unfilled nanotubes. Although their statistical analysis seems correct, they could bolster their claim with more samples. I share their initial surprise that a lattice of buckyballs inside a nanotube has virtually no effect on its transport properties.
If the authors are correct, there are no signatures of the buckyball lattice in the transport spectra of a nanotube. How else might one go about detecting them?
Tuesday, July 10, 2007
Photons and the Aharonov Bohm Effect
Bound on the Photon Charge from the Phase Coherence of Extragalactic Radiation
Brett Altschul
PRL 98, 261801 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e261801
The conclusion of this paper will come as a surprise to very few people: the photon probably doesn't have a charge. Altschul, from Indiana University, has deduced an upper bound on the photon charge that is 32 orders of magnitude less than the electron charge (46 if photons have both a positive and a negative charge). It is his analysis rather than his conclusion that I found interesting.
Altshcul's analysis starts from the observation that we can use interferometry to study astrophysical objects. Basically, one collects light from the same source at two different receivers. By studying the interference between the signals at the two receivers, one can obtain information about the source object. For this to work, the light from the source must be coherent --- i.e., the phase difference between two photons traveling along the same path must be small compared to the phase difference they acquire due to the path difference between the two receivers.
Altschul points out a source of phase difference that does not immediately come to mind: the Aharonov-Bohm Effect. If photons have a charge, then photons at the two detectors of an interferometer will acquire a phase difference that depends on the magnetic flux through the triangle made up of the two detectors and the source. (The assumption here is that a charged photon would interact with an external electromagnetic field exactly the same way an electron does.) The fact that interferometry works means the Aharonov Bohm phase small. (Conservatively, Altschul interprets "small" as "less than one".)
Making order of magnitude estimates for the interstellar magnetic field and using the baseline of the Very Long Baseline Interferometry Space Observatory Program (VSOP) with a source distance of 1 Gpc (about 3 billion light years), Altschul places an upper bound of 10^{-32} on the ratio of the photon charge to the electron charge.
The small bound is possible because of the huge distances involved in astronomical observations. It's almost like running a lab experiment designed to probe the Aharonov Bohm effect for 3 billion years --- that's a lot of data!
If the photon can have both positive and negative charges (like electrons) or positive, negative, and neutral charges (like pions), then the bounds are even tighter. This is because the Aharonov Bohm phase is proportional to the charge of the particle. If a particle with positive charge and a particle with negative charge travel along the same path, they acquire equal and opposite phases. If photons have two charges, the fact that interferometry works places an upper limit of 10^{-46} on the ratio of photon and electron charges.
The major source of uncertainty in this analysis is the current lack of understanding regarding interstellar magnetic fields. Perhaps Altschul's study will inspire new methods of studying these fields using interferometry.
As I said, the fact that the one can place a very small upper bound on the photon charge is not surprising. The fact that it can be done by analyzing the Aharonov Bohm effect is.
Altschul mentions a couple interesting facts about the theory of photons. First, the problem of the photon mass has been studied much more than that of photon charge. He mentions three theories of photon mass (Proca, Higgs, and Stuckleberg). I've never heard of the third. He also points out that not much is known about the consequences of charged photons. This is surprising, given the large number of models studied in quantum field theory --- many of which have little relevance to the physical world as revealed by experiments. It sounds like the kind of problem one might find at the end of a chapter in Peskin and Schroeder.
Brett Altschul
PRL 98, 261801 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e261801
The conclusion of this paper will come as a surprise to very few people: the photon probably doesn't have a charge. Altschul, from Indiana University, has deduced an upper bound on the photon charge that is 32 orders of magnitude less than the electron charge (46 if photons have both a positive and a negative charge). It is his analysis rather than his conclusion that I found interesting.
Altshcul's analysis starts from the observation that we can use interferometry to study astrophysical objects. Basically, one collects light from the same source at two different receivers. By studying the interference between the signals at the two receivers, one can obtain information about the source object. For this to work, the light from the source must be coherent --- i.e., the phase difference between two photons traveling along the same path must be small compared to the phase difference they acquire due to the path difference between the two receivers.
Altschul points out a source of phase difference that does not immediately come to mind: the Aharonov-Bohm Effect. If photons have a charge, then photons at the two detectors of an interferometer will acquire a phase difference that depends on the magnetic flux through the triangle made up of the two detectors and the source. (The assumption here is that a charged photon would interact with an external electromagnetic field exactly the same way an electron does.) The fact that interferometry works means the Aharonov Bohm phase small. (Conservatively, Altschul interprets "small" as "less than one".)
Making order of magnitude estimates for the interstellar magnetic field and using the baseline of the Very Long Baseline Interferometry Space Observatory Program (VSOP) with a source distance of 1 Gpc (about 3 billion light years), Altschul places an upper bound of 10^{-32} on the ratio of the photon charge to the electron charge.
The small bound is possible because of the huge distances involved in astronomical observations. It's almost like running a lab experiment designed to probe the Aharonov Bohm effect for 3 billion years --- that's a lot of data!
If the photon can have both positive and negative charges (like electrons) or positive, negative, and neutral charges (like pions), then the bounds are even tighter. This is because the Aharonov Bohm phase is proportional to the charge of the particle. If a particle with positive charge and a particle with negative charge travel along the same path, they acquire equal and opposite phases. If photons have two charges, the fact that interferometry works places an upper limit of 10^{-46} on the ratio of photon and electron charges.
The major source of uncertainty in this analysis is the current lack of understanding regarding interstellar magnetic fields. Perhaps Altschul's study will inspire new methods of studying these fields using interferometry.
As I said, the fact that the one can place a very small upper bound on the photon charge is not surprising. The fact that it can be done by analyzing the Aharonov Bohm effect is.
Altschul mentions a couple interesting facts about the theory of photons. First, the problem of the photon mass has been studied much more than that of photon charge. He mentions three theories of photon mass (Proca, Higgs, and Stuckleberg). I've never heard of the third. He also points out that not much is known about the consequences of charged photons. This is surprising, given the large number of models studied in quantum field theory --- many of which have little relevance to the physical world as revealed by experiments. It sounds like the kind of problem one might find at the end of a chapter in Peskin and Schroeder.
Wednesday, July 4, 2007
Relativity on the Table-Top
Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion
L. Lamata, J. Leon, T. Schatz, and E. Solano
PRL 98, 253005 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e253005
In this article, the authors demonstrate the experimental possibility of simulating a Dirac Hamiltonian in an atomic system. In short, they propose a table-top experiment that would probe the effects of a relativistic system.
The requirements are relatively modest. To simulate the (1+1)-D or (2+1)-D Dirac equation, one needs a two-level atomic system; for the (3+1)-D Dirac equation, one needs a 4-level system. Three types of couplings are also required:
• Carrier Interaction -- a resonant coupling between two of the internal atomic states (such as a laser tuned to the transition frequency).
• Jaynes-Cummings Interaction -- couples two internal states with a vibrational mode of the center of mass. An upward transition between internal states is accompanied by the destruction of a phonon.
• Anti-Jaynes-Cummings Interaction -- also couples two internal states with a vibrational mode, but the frequency is tuned so that an upward transition between is accompanied instead by the creation of a phonon.
When the phases of the laser fields used to generate the above interactions are appropriately tuned, the effective Hamiltonian for the two-level or four-level system is identical in form with the free Dirac equation. By varying the couplings, one can tune the effective particle mass and velocity of light.
The authors dedicate a lot of space to a discussion of Zitterbewegung, which is a rapid oscillatory motion of an electron about its mean position. It was predicted in 1930 by Schrodinger, but has not yet been observed experimentally. For electrons, the amplitude of the oscillations is on the order of 10^{-13} m, and the frequency is on the order of 10^{21} Hz. In addition, Zitterbewegung is an effect of the single-particle Dirac equation. There is some controversy over whether or not the effect persists in QED.
The authors point out that by controlling the effective electron mass and the effective speed of light, one should be able to bring the amplitude and frequency of the oscillations into an experimentally accessible range. Another point they don't mention: even if the effect does not occur for real electrons, the dynamics of their atomic system are (in theory) accurately described by a single-particle Dirac equation. Whether or not real electrons jitter around, the trapped ions on their desks should.
The authors mention a few other relativistic effects that might be simulated: by controlling the particle mass, one could observe something akin to the Higgs mechanism for mass generation; one could simulate the Klein paradox, in which particle-hole pairs are created in a strong potential; one could simulate a (1+1)-D axial anomaly, the lower-dimensional counterpart of the chiral anomaly in (3+1)-D.
The possibility of tuning the parameters of the Dirac equation is interesting. However, such simulations of relativistic systems is a bit perplexing. A similar situation exists in the theory of graphene and carbon nanotubes, where the low-energy excitations are described by a Dirac equation. On a very fundamental level, the simulation can't be right. So how far can one push the analogy?
How similar are a pseudospinor and a spinor? Electron spin has to do with angular momentum, but pseudospin has nothing to do with it. The simulated system can't have pseudospin-orbit coupling, but the Dirac equation can.
How effective is an effective speed of light? In the table-top simulation, nothing prohibits the electron or the center of mass of the ion system from exceeding the effective speed of light. What then? Would the authors claim they were simulating tachyons?
In the Klein paradox, where are the holes going to come from? No corresponding anti-cesium ion is going to materialize in the trap.
It would be quite interesting to probe these unphysical effects in a trap and see exactly how the effective Dirac physics breaks down.
A final note: This was a well-written paper, but I must criticize the authors for a particular choice of phrase. On page 2, they refer to the "notorious analogy" between the Dirac equation and the effective Hamiltonian of the trapped ion. Jesse James was notorious. The three-body problem is notoriously difficult. An analogy between Hamiltonians is not "famous or well known, typically for some bad quality or deed."
L. Lamata, J. Leon, T. Schatz, and E. Solano
PRL 98, 253005 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e253005
In this article, the authors demonstrate the experimental possibility of simulating a Dirac Hamiltonian in an atomic system. In short, they propose a table-top experiment that would probe the effects of a relativistic system.
The requirements are relatively modest. To simulate the (1+1)-D or (2+1)-D Dirac equation, one needs a two-level atomic system; for the (3+1)-D Dirac equation, one needs a 4-level system. Three types of couplings are also required:
• Carrier Interaction -- a resonant coupling between two of the internal atomic states (such as a laser tuned to the transition frequency).
• Jaynes-Cummings Interaction -- couples two internal states with a vibrational mode of the center of mass. An upward transition between internal states is accompanied by the destruction of a phonon.
• Anti-Jaynes-Cummings Interaction -- also couples two internal states with a vibrational mode, but the frequency is tuned so that an upward transition between is accompanied instead by the creation of a phonon.
When the phases of the laser fields used to generate the above interactions are appropriately tuned, the effective Hamiltonian for the two-level or four-level system is identical in form with the free Dirac equation. By varying the couplings, one can tune the effective particle mass and velocity of light.
The authors dedicate a lot of space to a discussion of Zitterbewegung, which is a rapid oscillatory motion of an electron about its mean position. It was predicted in 1930 by Schrodinger, but has not yet been observed experimentally. For electrons, the amplitude of the oscillations is on the order of 10^{-13} m, and the frequency is on the order of 10^{21} Hz. In addition, Zitterbewegung is an effect of the single-particle Dirac equation. There is some controversy over whether or not the effect persists in QED.
The authors point out that by controlling the effective electron mass and the effective speed of light, one should be able to bring the amplitude and frequency of the oscillations into an experimentally accessible range. Another point they don't mention: even if the effect does not occur for real electrons, the dynamics of their atomic system are (in theory) accurately described by a single-particle Dirac equation. Whether or not real electrons jitter around, the trapped ions on their desks should.
The authors mention a few other relativistic effects that might be simulated: by controlling the particle mass, one could observe something akin to the Higgs mechanism for mass generation; one could simulate the Klein paradox, in which particle-hole pairs are created in a strong potential; one could simulate a (1+1)-D axial anomaly, the lower-dimensional counterpart of the chiral anomaly in (3+1)-D.
The possibility of tuning the parameters of the Dirac equation is interesting. However, such simulations of relativistic systems is a bit perplexing. A similar situation exists in the theory of graphene and carbon nanotubes, where the low-energy excitations are described by a Dirac equation. On a very fundamental level, the simulation can't be right. So how far can one push the analogy?
How similar are a pseudospinor and a spinor? Electron spin has to do with angular momentum, but pseudospin has nothing to do with it. The simulated system can't have pseudospin-orbit coupling, but the Dirac equation can.
How effective is an effective speed of light? In the table-top simulation, nothing prohibits the electron or the center of mass of the ion system from exceeding the effective speed of light. What then? Would the authors claim they were simulating tachyons?
In the Klein paradox, where are the holes going to come from? No corresponding anti-cesium ion is going to materialize in the trap.
It would be quite interesting to probe these unphysical effects in a trap and see exactly how the effective Dirac physics breaks down.
A final note: This was a well-written paper, but I must criticize the authors for a particular choice of phrase. On page 2, they refer to the "notorious analogy" between the Dirac equation and the effective Hamiltonian of the trapped ion. Jesse James was notorious. The three-body problem is notoriously difficult. An analogy between Hamiltonians is not "famous or well known, typically for some bad quality or deed."
Saturday, June 16, 2007
The Geometry of Phase Space
Geometry of Hamiltonian Chaos
L. Horwitz, et al.
PRL 98, 234302 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e234302
This group from Israel has recast classical Hamiltonian dynamics in the language of differential geometry. In doing so, they have discovered a new criterion for chaotic motion: the deviation of parallel geodesics. The geodesics of the system are classical trajectories. If nearby geodesics diverge, it indicates an instability in the system. This is a different measure than the Lyapunov exponent, although it’s possible there is a deeper connection between the two. I would speculate that the Lyapunov exponents could be derived from this geometric theory.
The general idea behind the method is to rewrite the Hamiltonian using a metric in the space of conjugate momenta. From this metric, one can derive the connection and curvature. From the curvature, one can calculate geodesic deviation. However, the authors find that the geodesic equation and the Hamiltonian equations of motion do not agree. It turns out that the geodesics in the dual manifold (defined by dx^{i} = g^{ij} dx_{j}) do indeed agree with the Hamiltonian equations of motion. The authors simply state this fact without discussion. There must be a deeper significance of this result. Or maybe the authors just started their calculation on the wrong manifold. Regardless of the connection, the geodesics on the dual manifold and the Hamiltonian equations of motion are mathematically equivalent; thus, one has a geometric way of thinking about phase space trajectories.
The new connection is anti-symmetric, so there is torsion. I don’t know much about torsion. In my course on general relativity, one of the postulates we used in deriving Einstein’s equations was the “no torsion condition.” It has something to do with twist of the geodesics. It is irrelevant in the derivation of the authors because the antisymmetric terms cancel from the geodesic equation, leaving only a symmetric connection.
In the end, the authors find that the curvature derived from this connection determines the stability of an orbit. If the curvature is negative, geodesics diverge and the system is chaotic. If curvature is positive, geodesics converge, and the system is stable. This seems like a very useful tool for analyzing unstable systems. The authors point out that this approach does not require one to use approximations of the true equations of motion, unlike other methods.
The idea behind the paper is quite interesting. My one complaint with the paper is that the phrase “a complete discussion will be given elsewhere” was used at least four times, but the authors don’t indicate where. It would be interesting to see these discussions, as well as a discussion of the connection between the curvature studied by the authors and the Lyapunov exponents.
L. Horwitz, et al.
PRL 98, 234302 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e234302
This group from Israel has recast classical Hamiltonian dynamics in the language of differential geometry. In doing so, they have discovered a new criterion for chaotic motion: the deviation of parallel geodesics. The geodesics of the system are classical trajectories. If nearby geodesics diverge, it indicates an instability in the system. This is a different measure than the Lyapunov exponent, although it’s possible there is a deeper connection between the two. I would speculate that the Lyapunov exponents could be derived from this geometric theory.
The general idea behind the method is to rewrite the Hamiltonian using a metric in the space of conjugate momenta. From this metric, one can derive the connection and curvature. From the curvature, one can calculate geodesic deviation. However, the authors find that the geodesic equation and the Hamiltonian equations of motion do not agree. It turns out that the geodesics in the dual manifold (defined by dx^{i} = g^{ij} dx_{j}) do indeed agree with the Hamiltonian equations of motion. The authors simply state this fact without discussion. There must be a deeper significance of this result. Or maybe the authors just started their calculation on the wrong manifold. Regardless of the connection, the geodesics on the dual manifold and the Hamiltonian equations of motion are mathematically equivalent; thus, one has a geometric way of thinking about phase space trajectories.
The new connection is anti-symmetric, so there is torsion. I don’t know much about torsion. In my course on general relativity, one of the postulates we used in deriving Einstein’s equations was the “no torsion condition.” It has something to do with twist of the geodesics. It is irrelevant in the derivation of the authors because the antisymmetric terms cancel from the geodesic equation, leaving only a symmetric connection.
In the end, the authors find that the curvature derived from this connection determines the stability of an orbit. If the curvature is negative, geodesics diverge and the system is chaotic. If curvature is positive, geodesics converge, and the system is stable. This seems like a very useful tool for analyzing unstable systems. The authors point out that this approach does not require one to use approximations of the true equations of motion, unlike other methods.
The idea behind the paper is quite interesting. My one complaint with the paper is that the phrase “a complete discussion will be given elsewhere” was used at least four times, but the authors don’t indicate where. It would be interesting to see these discussions, as well as a discussion of the connection between the curvature studied by the authors and the Lyapunov exponents.
Wednesday, June 13, 2007
Excitons Insensitive to Environment
Screening of Excitons in Single, Suspended Carbon Nanotubes
A.G. Walsh, et al.
Nanoletters 7, 1485--1488 (2007)
URL: http://pubs.acs.org/cgi-bin/abstract.cgi/nalefd/2007/7/i06/abs/nl070193p.html
This group from Boston University has investigated the binding energy of excitons in nanotubes as a function of the dielectric constant of the environment. They repeated photoluminescence measurements on nanotubes in dry air, humid air, and water. In general, they find that the binding energy is not very sensitive to changes in the dielectric environment. The binding energy only changes by a couple tens of meV between dry air and water.
The theoretical model they use is a 1D interaction of the form V / ( |z| + Z ). Surprisingly, they find that the parameter Z scales linearly with the dielectric constant! They claim this model was solved exactly in 1959. I’m going to track down that reference. With the exact solution and scaling relation for Z, they are able to reproduce the scaling of their data as well as a scaling relation reported by Perebeinos a couple years ago.
A.G. Walsh, et al.
Nanoletters 7, 1485--1488 (2007)
URL: http://pubs.acs.org/cgi-bin/abstract.cgi/nalefd/2007/7/i06/abs/nl070193p.html
This group from Boston University has investigated the binding energy of excitons in nanotubes as a function of the dielectric constant of the environment. They repeated photoluminescence measurements on nanotubes in dry air, humid air, and water. In general, they find that the binding energy is not very sensitive to changes in the dielectric environment. The binding energy only changes by a couple tens of meV between dry air and water.
The theoretical model they use is a 1D interaction of the form V / ( |z| + Z ). Surprisingly, they find that the parameter Z scales linearly with the dielectric constant! They claim this model was solved exactly in 1959. I’m going to track down that reference. With the exact solution and scaling relation for Z, they are able to reproduce the scaling of their data as well as a scaling relation reported by Perebeinos a couple years ago.
Tuesday, June 12, 2007
Exchange and Correlation in Graphene
Chirality and Correlations in Graphene
Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari, and A.H. MacDonald
PRL 98, 236601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e236601
This article was a pleasure to read. I've been reading about the use of techniques from field theory in condensed matter physics lately, and I got to see them in action here.
This group from Texas has evaluated the effects of exchange and correlation on the charge and spin susceptibility of graphene. They start from the massless Dirac equation that describes low energy excitations in graphene and calculate the interaction energy. The technique they used was new to me.
First, they write down an expression for the interaction energy that depends on the structure factor of the interacting system. The interaction energy appears to be calculated by adiabatically turning on the coupling between particles. The structure factor is calculated from the density-density response function, a relation based on the fluctuation dissipation theorem. The response function can be calculated in the random phase approximation using methods of quantum field theory. The authors combine a lot of neat tools to obtain the interaction energy. It's all condensed into one paragraph in this paper. I'll have to check out their reference, Giuliani and Vignale's "Quantum Theory of the Electron Liquid."
Though the techniques are well-established, they lead to surprising results in graphene. The authors find that the sign of the correections to the spin and charge susceptibilities is opposite that of the usual electron gas. The similarity of the response functions is also different from an electron gas. The exchange effect decreases the susceptibilities in graphene, while it enhances them in the normal 2DEG. The authors trace the difference in behavior to the renormalized fermi velocity, which increases with the interaction strength.
Once I am more comfortable with the techniques I'm learning from Abrikosov and Mattuck, I'm going to try and reproduce these results.
Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari, and A.H. MacDonald
PRL 98, 236601 (2007)
URL: http://link.aps.org/abstract/PRL/v98/e236601
This article was a pleasure to read. I've been reading about the use of techniques from field theory in condensed matter physics lately, and I got to see them in action here.
This group from Texas has evaluated the effects of exchange and correlation on the charge and spin susceptibility of graphene. They start from the massless Dirac equation that describes low energy excitations in graphene and calculate the interaction energy. The technique they used was new to me.
First, they write down an expression for the interaction energy that depends on the structure factor of the interacting system. The interaction energy appears to be calculated by adiabatically turning on the coupling between particles. The structure factor is calculated from the density-density response function, a relation based on the fluctuation dissipation theorem. The response function can be calculated in the random phase approximation using methods of quantum field theory. The authors combine a lot of neat tools to obtain the interaction energy. It's all condensed into one paragraph in this paper. I'll have to check out their reference, Giuliani and Vignale's "Quantum Theory of the Electron Liquid."
Though the techniques are well-established, they lead to surprising results in graphene. The authors find that the sign of the correections to the spin and charge susceptibilities is opposite that of the usual electron gas. The similarity of the response functions is also different from an electron gas. The exchange effect decreases the susceptibilities in graphene, while it enhances them in the normal 2DEG. The authors trace the difference in behavior to the renormalized fermi velocity, which increases with the interaction strength.
Once I am more comfortable with the techniques I'm learning from Abrikosov and Mattuck, I'm going to try and reproduce these results.
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