Research

# In Memoriam

I never had the good fortune of meeting or personally knowing Shoucheng Zhang. Nevertheless he has had a profound influence on my academic career. As the world learned sometime last week, Zhang passed away suddenly on December 1 “after fighting a battle with depression” 1. He was one of the world’s greatest theoretical physicists and losing him at the young age of 55 is an incalculable loss for the physics community. Already in his relatively short career he had made gigantic contributions to condensed matter physics. However, his ideas permeated well beyond condensed matter and touched high energy and particle physics as well.

He deserved to be awarded the Nobel prize, not once but several times over. I have tried to make a partial list of his major works each of which independently constitutes a major leap in its respective field. A few of these are listed below.

# Scientific Accomplishments

1. Chern-Simons Landau-Ginzburg Effective Theory of the Fractional Quantum Hall Effect: It is well understood [1] that the integer quantum hall effect (IQHE) can be given an effective field theory formulation by adding the topological Chern-Simons term to the Maxwell action in $2+1$ dimensions. Zhang [2, 3] extended the effective field theory approach to cover the fractional quantum hall effect (FQHE). This approach gave results equivalent to the wave-function approach of Laughlin but in addition also allowed a mean-field understanding of the FQHE.

2. Phase Diagram of the Quantum Hall State and discovery of Quantum Hall Insulator: In collaboration with Kivelson and Lee [4, 5] Zhang uncovered a global phase diagram for the FQHE which exhibits a phenomenon known as the “law of corresponding states” according to which states at different filling fractions $\nu$ can be mapped to each other under certain transformations. This approach provided the foundation for later work by Dolan [6] who showed that this law of corresponding states followed naturally from the existence of a $SL(2, \mathbb{Z})$ symmetry in the FQHE. The existence of this symmetry was later used by Bayntun and collaborators [7, 8] to demonstrate a holographic realization of the quantum hall effect.

3. $SO(5)$ Theory of High-Tc Superconductivity: Zhang’s greatest achievement, in my opinion, was the development of a theory [9, 10] which provides a unified description of the phase diagram of high-Tc superconductivity.

The two primary regions of the phase diagram, the antiferromagnetic (AF) and the superconducting (SC) regions, are described in the long-wavelength limit by effective field theories which are governed by the gauge groups $SO(3)$ (for AF) and $U(1)$ (for SC). Zhang’s brilliant insight was that these two gauge groups could be obtained from a large group $SO(5)$ via symmetry breaking. Of course, there are many possible larger groups which contain $SO(3)$ and $U(1)$ as sub-groups. What makes $SO(5)$ unique is the fact that starting from a microscopic Hubbard model for the dynamics of electrons in the cuprate lattice, Zhang (in collaboration with Demler, Meixner, Rabello, Kohno and Hanke) [11, 12, 13] was able to show – both numerically and analytically that in the long wavelength group emergent excitations obey the $SO(5)$ symmetry.

This is where my relation with Zhang’s work comes in. While at Penn State I was exposed to the gauge formulation of general relativity [14] which lies at the foundation of Loop Quantum Gravity (LQG) [15, 16, 17]. My brownian motion like traversal through the space of ideas and papers eventually led me to the beautiful $BF$ theory formulation of general relativity [18, 19, 20, 21, 22, 23, 24] wherein one starts with a theory which has an action of the form:

\label{eqn:bf-action}
S_{BF} = \int d^4 x~ \Tr[ B \wedge F] = \int d^4 x ~ \frac{1}{2} \epsilon^{\alpha \beta \mu \nu} B_{\alpha\beta}^{IJ} F_{\mu \nu}^{KL} \delta_{IK} \delta_{JL}

where $B := B_{\mu\nu}^{IJ}$ is an anti-symmetric (in the spacetime $\mu,\nu$ indices) field (or “two-form” in more technical language) and $F := F_{\mu\nu}^{KL}$ is the field strength tensor of a gauge field $A_\mu^I$, where the $I,J,K,\ldots$ take values in the Lie-algebra of a gauge group. The precise gauge group depends on the value of the cosmological constant and whether our geometry is Lorentzian or Riemannian 2 :

$\Lambda < 0 $$\Lambda = 0$$\Lambda > 0$
$SO(3,2)$
Minkowski
$ISO(3,1)$
de Sitter (dS)
$SO(4,1)$
RiemannianHyperbolic
$SO(4,1)$
Euclidean
$ISO(4)$
Spherical
$SO(5)$
Gauge groups of Einstein-Cartan spacetimes depending on the value of the cosmological constant (positive, negative or zero) and whether the spacetime resulting from symmetry breaking is Lorentzian or Riemannian.

The action (\ref{eqn:bf-action}) is a topological action. The equation of motion for the field strength gives:

$$F = 0$$

implying that there are no local degrees of freedom. The topological symmetry is broken and local degrees of freedom are introduced by adding an interaction term to (\ref{eqn:bf-action}) of the form:

\label{eqn:bf-action-interactions}
S_{B^2 F} = \int d^4 x~ \Tr\left[B \wedge F – \theta~B \wedge B \right]

where $\theta$ measures the strength of the interaction term. After performing the Cartan decomposition of the gauge connection and the $B$ field 3, we obtain the action for general relativity with a cosmological constant $\Lambda$. The symmetry breaking term $\theta$ determines the strength of Newton’s gravitational constant $G$ and the cosmological constant $\Lambda$ in the resulting spacetime via the relation:

\label{eqn:bf-effective-lambda}
\theta = \frac{G \Lambda}{6}

Now, a priori, there is no reason to think that there should be any relation between Zhang’s $SO(5)$ theory of high-Tc superconductivity and Einstein-Cartan gravity. However, if one looks more carefully at the Lie-algebra structure of both the theories a striking similarity emerges. The Cartan decomposition can be used to write the gauge field $A_\mu^I$ in the following form:

\label{eqn:cartan-decomp}
A^I{}_J = \left( \begin{array}{cccc}
0 & \omega^0{}_1 & \omega^0{}_2 & \omega^0{}_3 & e^0/l \\
\omega^1{}_0 & 0 & \omega^1{}_2 & \omega^1{}_3 & e^1/l \\
\omega^2{}_0 & \omega^2{}_1 & 0 & \omega^2{}_3 & e^2/l \\
\omega^3{}_0 & \omega^3{}_1 & \omega^3{}_2 & 0 & e^3/l \\
\epsilon e^0/l & -\epsilon e^1/l & -\epsilon e^2/l & -\epsilon e^3/l & 0
\end{array} \right)

where $\omega^a{}_b$ represents the usual gravitational gauge connection, $e^a$ is the gravitational tetrad, $\epsilon \in { -1, 0, 1 }$ determines the sign of the cosmological constant and $l$ is a length scale related to the cosmological constant by the relation:

\label{eqn:cosmo-length}
l = \sqrt{\frac{3}{\Lambda}}

On the other hand in Zhang’s $SO(5)$ theory, one can construct an anti-symmetric five dimensional “Zhang” tensor $L_{ab}$ which obeys the commutation relations of the $\mathfrak{so}(5)$ Lie algebra:

\label{eqn:zhang-tensor}
L_{ab} = \left( \begin{array}{ccccc}
0 & & & & \\
\pi^\dagger_x + \pi_x & 0 & & & \\
\pi^\dagger_y + \pi_y & -S_z & 0 & & \\
\pi^\dagger_z + \pi_z & S_y & -S_x & 0 & \\
Q & -i(\pi^\dagger_x – \pi_x) & -i(\pi^\dagger_y – \pi_y) & -i(\pi^\dagger_z – \pi_z) & 0
\end{array} \right)

where $Q$ is charge operator for the superconducting phases, $\vect{S} = (S_x, S_y, S_z)$ is the spin-operator which measures the Néel order parameter in the anti-ferromagnetic phase and $\pi_i$ is an operator which measures the strength of the valence bond between neighboring sites in the underlying Hubbard model. Using the commutation relations of these operators one finds that the Zhang tensor satisfies the following commutators:

\label{eqn:so5-commutation}
\left[ L_{ab}, L_{cd} \right] = i \left(\delta_{ac} L_{bd} – \delta_{ad} L_{bc} – \delta_{bc} L_{ad} + \delta_{ad} L_{bc} \right)

which are precisely the defining relations of the Lie algebra of $SO(5)$. Now if we compare (\ref{eqn:cartan-decomp}) and (\ref{eqn:zhang-tensor}) we see that it possible to make the following identifications between the variables on the gravity side and those on the condensed matter side:

$L_{IJ}$$A_{IJ} RotationsS_x S_y S_z - \omega^3{}_2 \omega^3{}_1 \omega^2{}_1 Boosts \pi^\dagger_x + \pi_x \pi^\dagger_y + \pi_y \pi^\dagger_z + \pi_z \omega^1{}_0 \omega^2{}_0 \omega^3{}_0 Translations i (\pi^\dagger_x - \pi_x) i (\pi^\dagger_y - \pi_y) i (\pi^\dagger_z - \pi_z) \epsilon e^1/l \epsilon e^2/l \epsilon e^3/l Charge Q$$ \epsilon e^0/l$

This identification, if it stands scrutiny, implies that there is a direct correspondence between the different phases of high-Tc superconductors and solutions of Einstein’s equations. This work was published in 2017 in AHEP [25].

4. Four Dimensional Generalization of the Quantum Hall Effect: In [26] Zhang and Hu predicted the existence of the Quantum Hall Effect (QHE) in four spatial dimensions. In the usual 2+1D QHE, given the electric field, there is only one spatial direction orthogonal to it along which the Hall current can flow. In 4+1D, given an electric field, there are three different spatial directions along which the Hall current could flow. In order for a QHE to exist in 4+1D, therefore the charge carriers must carry an internal $SU(2)$ spin degree of freedom and the direction of the spin determines the direction of the Hall current. Now, of course, we don’t have access to four spatial dimensions, so the practical utility of this effect might be limited. However, it does point the way towards a possible realization of elementary particles as topological excitations in a quantum hall fluid.

5. Quantum Spin Hall Effect and Topological Insulators: Zhang, alongwith Hughes and Bernevig first predicted [27] the existence of the quantum spin hall state – which is a topological insulator state – in HgTe/CdTe heterostructure quantum wells. Zhang and collaborators also went on to experimentally observe this effect [28] within one year of their prediction of its existence.

This was an achievement no less remarkable than Novoselov and Geim’s discovery of a method for manufacturing graphene or Cornell and Weiman’s observation of the first Bose-Einstein condensate. Yet, whereas these two discoveries were awarded the Nobel prize within five years of the date of discovery, Zhang’s work was passed over for far longer. Of course, there might be many factors at work here, but his untimely passing without having been awarded Physics’ highest honor only highlights the absurdity of denying the Nobel to those who happen to die before the Prize Committee gets around to recognizing their phenomenal accomplishments during their lifetime.

# A Personal Note on Depression

It appears unthinkable that a man of Zhang’s talents and accomplishments would take his own life at the peak of his career. Of course, suicide among physicists is not an unknown phenomenon. Boltzmann and Ehrefest are two well-known examples. However, both faced difficult personal circumstances. Boltzmann’s work on thermodynamics and entropy attracted strong criticism and even derision from many of his peers. Ehrenfest, despite his towering talent, was not quite able to live up to his full academic potential. Zhang, on the other hand, was a celebrated physicist in his own lifetime. His accomplishments were globally recognized and he was a perennial favorite for the physics Nobel. Why then would such a person commit suicide?

Depression is a terrible disease. What amplifies its debilitating effect is its invisibility. There are no physical symptoms apparent to the external observer. In the absence of any concrete diagnostic criteria via either brain imaging or biochemical analysis, its presence must be inferred only indirectly and based primarily upon the testimony of the afflicted individual. Due to this reliance on personal testimony rather than on objective evidence, it is also easy to dismiss the existence of depression in an individual as an imagined affliction rather than a genuine pathology. To make matters worse, from time to time so-called “Gurus” and “moral leaders” setback the cause of mental illness education by decades by declaring that “most depression is self-created”!

In light of this lack of social acceptance of the reality of mental illness it becomes imperative that those who have suffered or are currently suffering from any form of mental illness to come out of the shadows and reveal their condition to the world.

I have been suffering from clinical depression since I was about fourteen years old. As someone who has battled major depression to reach some level of professional success in theoretical physics I have some understanding of the personal struggle Zhang must have faced in his own life. The story of my own struggle is the topic of a separate blog post. This post is intended to be dedicated solely to the memory of one of the greatest theoretical physicists of our generation.

2. This table is taken from Sec 2.3 of [22
3. For details of this decomposition the reader may refer to [23, 25, 21, 22
[1] J. Jain, Composite fermions, 1 ed., Cambridge University Press, 2007.
[Bibtex]
@book{Jain2007Composite,
Abstract = {When electrons are confined to two dimensions, cooled to near absolute zero temperature, and subjected to a strong magnetic field, they form an exotic new collective state of matter. Investigations into this began with the observations of integral and fractional quantum Hall effects, which are among the most important discoveries in condensed matter physics. The fractional quantum Hall effect and a stream of other unexpected findings are explained by a new class of particles: composite fermions. This textbook is a self- contained, pedagogical introduction to the physics and experimental manifestations of composite fermions. Ideal for graduate students and academic researchers, it contains numerous exercises to reinforce the concepts presented. The topics covered include the integral and fractional quantum Hall effects, the composite-fermion Fermi sea, various kinds of excitations, the role of spin, edge state transport, electron solid, bilayer physics, fractional braiding statistics and fractional local charge.},
Author = {Jain, Jainendra},
Citeulike-Article-Id = {4753044},
Date-Modified = {2018-12-13 11:34:18 +0530},
Day = {09},
Edition = {1},
Howpublished = {Hardcover},
Isbn = {0521862329},
Keywords = {composite\_fermions, condensed\_matter, hall\_effect, manybody, nonequilibrium},
Month = {April},
Posted-At = {2010-02-04 21:03:55},
Priority = {2},
Publisher = {Cambridge University Press},
Title = {Composite Fermions},
Url = {http://www.worldcat.org/isbn/0521862329},
Year = {2007},
Bdsk-Url-1 = {http://www.worldcat.org/isbn/0521862329}}
[2] S. C. Zhang, T. H. Hansson, and S. Kivelson, “Effective-field-theory model for the fractional quantum hall effect,” Physical review letters, vol. 62, iss. 1, p. 82–85, 1989.
[Bibtex]
@article{Zhang1989EffectiveFieldTheory,
Abstract = {Starting directly from the microscopic Hamiltonian; we derive a field-theory model for the fractional quantum Hall effect. By considering an approximate coarse-grained version of the same model; we construct a Landau-Ginzburg theory similar to that of Girvin. The partition function of the model exhibits cusps as a function of density and the Hall conductance is quantized at filling factors {\^I}½=(2 k -1) -1 with k an arbitrary integer. At these fractions the ground state is incompressible; and the quasiparticles and quasiholes have fractional charge and obey fractional statistics. Finally; we show that the collective density fluctuations are massive.},
Author = {Zhang, S. C. and Hansson, T. H. and Kivelson, S.},
Citeulike-Article-Id = {6444164},
Date-Modified = {2018-12-11 22:57:51 +0530},
Doi = {10.1103/PhysRevLett.62.82},
Journal = {Physical Review Letters},
Keywords = {chern\_simons, conductance, effective, field\_theory, file-import-09-12-27, fluctuations, fractional\_charge, fractional\_quantum\_hall\_effect, fractional\_statistics, incompressible, landau-ginzburg},
Month = {Jan},
Number = {1},
Pages = {82--85},
Posted-At = {2009-12-27 12:17:41},
Priority = {2},
Publisher = {American Physical Society},
Title = {Effective-Field-Theory Model for the Fractional Quantum Hall Effect},
Url = {http://dx.doi.org/10.1103/PhysRevLett.62.82},
Volume = {62},
Year = {1989},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.62.82}}
[3] S. C. Zhang, “The chern-simons-landau-ginzburg theory of the fractional quantum hall effect,” International journal of modern physics b, vol. 6, p. 25–58, 1992.
[Bibtex]
@article{Zhang1992The-Chern-Simons-Landau-Ginzburg,
Abstract = {This paper gives a systematic review of a field theoretical approach tothe fractional quantum Hall effect (FQHE) that has been developed in thepast few years. We first illustrate some simple physical ideas tomotivate such an approach and then present a systematic derivation ofthe Chern-Simons-Landau-Ginzburg (CSLG) action for the FQHE, startingfrom the microscopic Hamiltonian. It is demonstrated that all thephenomenological aspects of the FQHE can be derived from the mean fieldsolution and the small fluctuations of the CSLG action. Although thisformalism is logically independent of Laughlin's wave function approach,their physical consequences are equivalent. The CSLG theory demonstratesa deep connection between the phenomena of superfluidity and the FQHE,and can provide a simple and direct formalism to address many newmacroscopic phenomena of the FQHE.},
Author = {Zhang, S. C.},
Citeulike-Article-Id = {6925295},
Date-Modified = {2018-12-11 22:57:51 +0530},
Doi = {10.1142/S0217979292000037},
Journal = {International Journal of Modern Physics B},
Keywords = {chern\_simons, effective-theory, fqhe, fractional, hall\_effect, landau\_ginzburg},
Pages = {25--58},
Posted-At = {2010-03-28 19:29:05},
Priority = {2},
Title = {The Chern-Simons-Landau-Ginzburg Theory of the Fractional Quantum Hall Effect},
Url = {http://dx.doi.org/10.1142/S0217979292000037},
Volume = {6},
Year = {1992},
Bdsk-Url-1 = {http://dx.doi.org/10.1142/S0217979292000037}}
[4] D. H. Lee, S. Kivelson, and S. C. Zhang, “Quasiparticle charge and the activated conductance of a quantum hall liquid,” Physical review letters, vol. 68, p. 2386–2389, 1992.
[Bibtex]
@article{Lee1992Quasiparticle,
Abstract = {We provide a theoretical basis for Clark's proposal that for quantum Hall liquids at magic filling factors, where the longitudinal conductivity is exponentially activated, {{\"I}xx={\"I}xx0eB}-{{\^I}/kT}, the prefactor {\"I}xx0 is proportional to the square of the quasiparticle charge. We also propose that the same experiments uncover a remarkable law of corresponding states.},
Author = {Lee, Dung H. and Kivelson, Steven and Zhang, Shou C.},
Citeulike-Article-Id = {10783460},
Date-Modified = {2018-12-11 23:01:17 +0530},
Doi = {10.1103/PhysRevLett.68.2386},
Journal = {Physical Review Letters},
Keywords = {classic, condensed\_matter, duality, insulator, kivelson, law\_of\_corresponding\_states, manybody, phase\_diagram, quantum\_hall\_effect, sc\_zhang, selection\_rule, transport-coefficients},
Month = apr,
Pages = {2386--2389},
Posted-At = {2012-06-12 11:13:40},
Priority = {2},
Publisher = {American Physical Society},
Title = {Quasiparticle charge and the activated conductance of a quantum Hall liquid},
Url = {http://dx.doi.org/10.1103/PhysRevLett.68.2386},
Volume = {68},
Year = {1992},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.68.2386}}
[5] S. Kivelson, D. H. Lee, and S. C. Zhang, “Global phase diagram in the quantum hall effect,” Physical review b, vol. 46, p. 2223–2238, 1992.
[Bibtex]
@article{Kivelson1992Global,
Abstract = {We report recent progress in determining the global behavior of the two-dimensional electron gas in a high magnetic field. Specifically, we have: (i) derived a law of corresponding states which allows us to construct a global phase diagram and calculate many interrelations between transport coefficients; (ii) derived a ''selection rule'' governing the allowed continuous transitions between pairs of quantum Hall liquid states; and (iii) identified the ''insulating state,'' which we have named the Hall insulator, as a state in which, as the temperature T{\^a}0, {\"I}xx{\^a}{\^a}, {\"I}xx and {\"I}xy{\^a}0, but {\"I}xy tends to a constant value, roughly B/nec. Each of these results has many testable experimental consequences.},
Author = {Kivelson, Steven and Lee, Dung H. and Zhang, Shou C.},
Citeulike-Article-Id = {10783458},
Date-Modified = {2018-12-11 23:01:17 +0530},
Doi = {10.1103/PhysRevB.46.2223},
Journal = {Physical Review B},
Keywords = {classic, condensed\_matter, duality, insulator, kivelson, manybody, phase\_diagram, quantum\_hall\_effect, sc\_zhang, selection\_rule, transport-coefficients},
Month = jul,
Pages = {2223--2238},
Posted-At = {2012-06-12 11:12:25},
Priority = {2},
Publisher = {American Physical Society},
Title = {Global phase diagram in the quantum Hall effect},
Url = {http://dx.doi.org/10.1103/PhysRevB.46.2223},
Volume = {46},
Year = {1992},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevB.46.2223}}
[6] B. P. Dolan, “Duality and the modular group in the quantum hall effect,” , 1999.
[Bibtex]
@article{Dolan1999Duality,
Abstract = {We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects, including: universality; the selection rule \$|p\_1q\_2 - p\_2q\_1|=1\$ for quantum Hall transitions between filling factors \$\nu\_1=p\_1/q\_1\$ and \$\nu\_2=p\_2/q\_2\$; critical values for the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaus.},
Archiveprefix = {arXiv},
Author = {Dolan, Brian P.},
Citeulike-Article-Id = {10783046},
Date-Modified = {2018-12-13 11:49:30 +0530},
Day = {28},
Eprint = {cond-mat/9805171},
Keywords = {bp\_dolan, conductivity, duality, modular\_group, phase\_transition, quantum\_hall\_effect, renormalization-group, semicircle\_law, sl2z},
Month = jan,
Posted-At = {2012-06-12 07:01:52},
Priority = {2},
Title = {Duality and the Modular Group in the Quantum Hall Effect},
Url = {http://arxiv.org/abs/cond-mat/9805171},
Year = {1999},
Bdsk-Url-1 = {http://arxiv.org/abs/cond-mat/9805171}}
[7] A. Bayntun, C. P. Burgess, B. P. Dolan, and S. Lee, “AdS/QHE: towards a holographic description of quantum hall experiments,” , 2010.
[Bibtex]
@article{Bayntun2010AdS/QHE:,
Abstract = {Transitions among quantum Hall plateaux share a suite of remarkable experimental features, such as semi-circle laws and duality relations, whose accuracy and robustness are difficult to explain directly in terms of the detailed dynamics of the microscopic electrons. They would naturally follow if the low-energy transport properties were governed by an emergent discrete duality group relating the different plateaux, but no explicit examples of interacting systems having such a group are known. Recent progress using the {AdS}/{CFT} correspondence has identified examples with similar duality groups, but without the {DC} ohmic conductivity characteristic of quantum Hall experiments. We use this to propose a simple holographic model for low-energy quantum Hall systems, with a nonzero {DC} conductivity that automatically exhibits all of the observed consequences of duality, including the existence of the plateaux and the semi-circle transitions between them. The model can be regarded as a strongly coupled analog of the old composite boson' picture of quantum Hall systems. Non-universal features of the model can be used to test whether it describes actual materials, and we comment on some of these in our proposed model. In particular, the model indicates the value 2/5 for low-temperature scaling exponents for transitions among quantum Hall plateaux, in agreement with the measured value 0.42 \pm 0.04.},
Archiveprefix = {arXiv},
Author = {Bayntun, Allan and Burgess, C. P. and Dolan, Brian P. and Lee, Sung-Sik},
Citeulike-Article-Id = {10744245},
Date-Modified = {2018-12-13 11:49:54 +0530},
Day = {15},
Eprint = {arXiv:1008.1917},
Keywords = {adscft, composite\_boson, condensed\_matter, conductivity, duality, hall\_plateaus, holography, quantum\_gravity, quantum\_hall\_effect},
Month = aug,
Posted-At = {2012-06-05 07:54:45},
Priority = {2},
Rating = {5},
Title = {{AdS}/{QHE}: Towards a Holographic Description of Quantum Hall Experiments},
Url = {http://arxiv.org/abs/arXiv:1008.1917},
Year = {2010},
Bdsk-Url-1 = {http://arxiv.org/abs/arXiv:1008.1917}}</code></pre></div>       <div id="paperkey_7" class="papercite_entry">[8]                   A. Bayntun and C. P. Burgess, "Finite size scaling in quantum hallography," , 2011. <br/>    <a href="javascript:void(0)" id="papercite_7" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_7_block"><pre><code class="tex bibtex">@article{Bayntun2011Finite,
Abstract = {At low temperatures observations of the Hall resistance for Quantum Hall systems at the interface between two Hall plateaux reveal a power-law behaviour, \$\exd R\_{xy}/\exd B \propto T^{-p}\$ (with \$p = 0.42 \pm 0.01\$); changing at still smaller temperatures, \$T \< T\_s\$, to a temperature-independent value. Experiments also show that the transition temperature varies with sample size, \$L\$, according to \$T\_s \propto 1/L\$. These experiments pose a potential challenge to the holographic AdS/QHE model recently proposed in {\tt <a href="/abs/1008.1917">arXiv:1008.1917</a>}. This proposal, which was motivated by the natural way AdS/CFT methods capture the emergent duality symmetries exhibited by quantum Hall systems, successfully describes the scaling exponent \$p\$ by relating it to an infrared dynamical exponent \$z\$ with \$p = 2/z\$. For a broad class of models \$z\$ is robustly predicted to be \$z = 5\$ in the regime relevant to the experiments (though becoming \$z = 1\$ further in the ultraviolet). By incorporating finite-size effects into these models we show that they reproduce a transition to a temperature-independent regime, predicting a transition temperature satisfying \$T\_s \propto 1/L\$ even though \$z = 5\$. The AdS/CFT calculation predicts this is a first-order transition, suggesting new possibilities for testing the picture. Remarkably, in this interpretation the gravity dual of the transition from temperature scaling to temperature-independent resistance is related to the Chandrashekar transition from a star to a black hole with increasing mass.},
Archiveprefix = {arXiv},
Author = {Bayntun, Allan and Burgess, C. P.},
Citeulike-Article-Id = {10744263},
Date-Modified = {2018-12-13 11:49:54 +0530},
Day = {16},
Eprint = {arXiv:1112.3698},
Keywords = {adscft, condensed\_matter, duality, experiment, hall\_plateaus, phase\_transition, quantum\_gravity, quantum\_hall\_effect, scaling\_laws},
Month = dec,
Posted-At = {2012-06-05 07:56:28},
Priority = {2},
Title = {Finite Size Scaling in Quantum Hallography},
Url = {http://arxiv.org/abs/arXiv:1112.3698},
Year = {2011},
Bdsk-Url-1 = {http://arxiv.org/abs/arXiv:1112.3698}}</code></pre></div>       <div id="paperkey_8" class="papercite_entry">[9]                   E. Demler and S. Zhang, "Theory of the resonant neutron scattering of high \$T_c\$ superconductors," , 1995. <br/>    <a href="javascript:void(0)" id="papercite_8" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_8_block"><pre><code class="tex bibtex">@article{Demler1995Theory,
Abstract = {ecent polarized neutron scattering experiments on \$YBa\_2 Cu\_3 O\_7\$ have revealed a sharp spectral peak at the \$(\pi,\pi)\$ in reciprocal lattice centered around the energy transfer of \$41\ meV\$. We offer a theoretical explanation of this remarkable experiment in terms of a new collective mode in the particle particle channel of the Hubbard model. This collective mode yields valuable information about the symmetry of the superconducting gap.},
Archiveprefix = {arXiv},
Author = {Demler, Eugene and Zhang, Shou-Cheng},
Citeulike-Article-Id = {7121840},
Date-Modified = {2018-12-11 23:01:17 +0530},
Day = {15},
Eprint = {cond-mat/9502060},
Keywords = {empirical\_data, experiment, gap, high\_temperature, hubbard\_model, neutron, resonance, scattering, superconductors, demler_eugene, zhang_sc, so5, cuprate\_superconductors},
Month = {Feb},
Posted-At = {2010-05-03 21:26:07},
Priority = {2},
Title = {Theory of the Resonant Neutron Scattering of High \$T\_c\$ Superconductors},
Url = {http://arxiv.org/abs/cond-mat/9502060},
Year = {1995},
Bdsk-Url-1 = {http://arxiv.org/abs/cond-mat/9502060}}</code></pre></div>       <div id="paperkey_9" class="papercite_entry">[10]                   S. Zhang, "So(5) quantum nonlinear sigma model theory of the high tc superconductivity," , 1996. <br/>    <a href="javascript:void(0)" id="papercite_9" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_9_block"><pre><code class="tex bibtex">@article{Zhang1996SO5-Quantum,
Abstract = {We show that the complex phase diagram of high \$T\_c\$ superconductors can be deduced from a simple symmetry principle, a \$SO(5)\$ symmetry which unifies antiferromagnetism with \$d\$ wave superconductivity. We derive the approximate \$SO(5)\$ symmetry from the microscopic Hamiltonian and show furthermore that this symmetry becomes exact under the renormalization group flow towards a bicritical point. With the help of this symmetry, we construct a \$SO(5)\$ quantum nonlinear \$\sigma\$ model to describe the effective low energy degrees of freedom of the high \$T\_c\$ superconductors, and use it to deduce the phase diagram and the nature of the low lying collective excitations of the system. We argue that this model naturally explains the basic phenomenology of the high \$T\_c\$ superconductors from the insulating to the underdoped and the optimally doped region.},
Archiveprefix = {arXiv},
Author = {Zhang, Shou-Cheng},
Citeulike-Article-Id = {6925287},
Date-Modified = {2018-12-11 22:57:51 +0530},
Day = {17},
Eprint = {cond-mat/9610140},
Keywords = {antiferromagnetism, high\_temperature, nonlinear-sigma-model, phase\_diagram, renormalization-group, so5, superconductivity},
Month = {Oct},
Posted-At = {2010-03-28 19:19:34},
Priority = {2},
Title = {SO(5) Quantum Nonlinear sigma Model Theory of the High Tc Superconductivity},
Url = {http://arxiv.org/abs/cond-mat/9610140},
Year = {1996},
Bdsk-Url-1 = {http://arxiv.org/abs/cond-mat/9610140}}</code></pre></div>       <div id="paperkey_10" class="papercite_entry">[11]           <a href='http://dx.doi.org/10.1103/PhysRevLett.79.4902' class='papercite_doi' title='View document in publisher site'><img src='https://www.quantumofgravity.com/blog/wp-content/plugins/papercite/img/external.png' width='10' height='10' alt='[doi]' /></a>        S. Meixner, W. Hanke, E. Demler, and S. Zhang, "Finite-Size Studies on the SO(5) Symmetry of the Hubbard Model," <span style="font-style: italic">Physical review letters</span>, vol. 79, iss. 24, p. 4902–4905, 1997. <br/>    <a href="javascript:void(0)" id="papercite_10" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_10_block"><pre><code class="tex bibtex">@article{Meixner1997Finite-Size,
Abstract = {We present numerical evidence for the approximate SO(5) symmetry of the Hubbard model on a 10 site cluster. Various dynamic correlation functions involving the {\$}\backslashpi{\$} operators, the generators of the SO(5) algebra, are studied using exact diagonalisation, and are found to possess sharp collective peaks. Our numerical results also lend support on the interpretation of the recent resonant neutron scattering peaks in the YBCO superconductors in terms of the Goldstone modes of the spontaneously broken SO(5) symmetry.},
Archiveprefix = {arXiv},
Arxivid = {arXiv:cond-mat/9701217v1},
Author = {Meixner, Stefan and Hanke, Werner and Demler, Eugene and Zhang, Shou-Cheng},
Date-Modified = {2018-12-11 23:01:17 +0530},
Doi = {10.1103/PhysRevLett.79.4902},
Eprint = {9701217v1},
File = {:Users/deepak/mendeley/files/Meixner et al.{\_}Finite-Size Studies on the SO(5) Symmetry of the Hubbard Model{\_}1997.pdf:pdf},
Issn = {0031-9007},
Journal = {Physical Review Letters},
Keywords = {antiferromagnetism,condensed matter,demler{\_}eugene,high-tc,many body,quantum gravity,so5,superconductivity,zhang{\_}s{\_}c},
Mendeley-Groups = {SO(5) Gravity},
Mendeley-Tags = {antiferromagnetism,condensed matter,demler{\_}eugene,high-tc,many body,quantum gravity,so5,superconductivity,zhang{\_}s{\_}c},
Number = {24},
Pages = {4902--4905},
Primaryclass = {arXiv:cond-mat},
Title = {{Finite-Size Studies on the SO(5) Symmetry of the Hubbard Model}},
Url = {http://arxiv.org/abs/cond-mat/9701217 http://dx.doi.org/10.1103/PhysRevLett.79.4902 https://arxiv.org/abs/cond-mat/9701217 https://link.aps.org/doi/10.1103/PhysRevLett.79.4902},
Volume = {79},
Year = {1997},
Bdsk-Url-2 = {https://dx.doi.org/10.1103/PhysRevLett.79.4902}}</code></pre></div>       <div id="paperkey_11" class="papercite_entry">[12]           <a href='http://dx.doi.org/10.1103/physrevlett.80.3586' class='papercite_doi' title='View document in publisher site'><img src='https://www.quantumofgravity.com/blog/wp-content/plugins/papercite/img/external.png' width='10' height='10' alt='[doi]' /></a>        S. Rabello, H. Kohno, E. Demler, and S. Zhang, "Microscopic electron models with exact SO(5) symmetry," <span style="font-style: italic">Physical review letters</span>, vol. 80, iss. 16, p. 3586–3589, 1997. <br/>    <a href="javascript:void(0)" id="papercite_11" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_11_block"><pre><code class="tex bibtex">@article{Rabello1997Microscopic,
Abstract = {We construct a class of microscopic electron models with exact {SO}(5) symmetry between antiferromagnetic and d-wave superconducting ground states. There is an exact one-to-one correspondence between both single-particle and collective excitations in both phases. {SO}(5) symmetry breaking terms can be introduced and classified according to irreducible representations of the exact {SO}(5) algebra. The resulting phase diagram and collective modes are identical to that of the {SO}(5) nonlinear sigma model.},
Archiveprefix = {arXiv},
Author = {Rabello, Silvio and Kohno, Hiroshi and Demler, Eugene and Zhang, Shou-Cheng},
Citeulike-Article-Id = {12870786},
Date-Modified = {2018-12-11 23:01:17 +0530},
Day = {2},
Doi = {10.1103/physrevlett.80.3586},
Eprint = {cond-mat/9707027},
Issn = {0031-9007},
Journal = {Physical Review Letters},
Keywords = {antiferromagnetism, d-wave, demler\_eugene, exact\_solution, hamiltonian, manybody, nonlinear-sigma-model, phase\_diagram, prl, quantum\_gravity, sc\_zhang, so5, superconductivity, symmetry\_breaking},
Month = jul,
Number = {16},
Pages = {3586--3589},
Posted-At = {2013-12-26 06:04:05},
Priority = {2},
Title = {Microscopic Electron Models with Exact {SO}(5) Symmetry},
Url = {http://dx.doi.org/10.1103/physrevlett.80.3586},
Volume = {80},
Year = {1997},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/physrevlett.80.3586}}</code></pre></div>       <div id="paperkey_12" class="papercite_entry">[13]           <a href='http://dx.doi.org/10.1103/PhysRevLett.80.3586' class='papercite_doi' title='View document in publisher site'><img src='https://www.quantumofgravity.com/blog/wp-content/plugins/papercite/img/external.png' width='10' height='10' alt='[doi]' /></a>        S. Rabello, H. Kohno, E. Demler, and S. Zhang, "Microscopic Electron Models with Exact SO(5) Symmetry," <span style="font-style: italic">Physical review letters</span>, vol. 80, iss. 16, p. 3586–3589, 1998. <br/>    <a href="javascript:void(0)" id="papercite_12" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_12_block"><pre><code class="tex bibtex">@article{Rabello1998Microscopic,
Abstract = {We construct a class of microscopic electron models with exact SO(5) symmetry between antiferromagnetic and d-wave superconducting ground states. There is an exact one-to-one correspondence between both single-particle and collective excitations in both phases. SO(5) symmetry breaking terms can be introduced and classified according to irreducible representations of the exact SO(5) algebra. The resulting phase diagram and collective modes are identical to that of the SO(5) nonlinear sigma model.},
Archiveprefix = {arXiv},
Arxivid = {arXiv:cond-mat/9707027v1},
Author = {Rabello, Silvio and Kohno, Hiroshi and Demler, Eugene and Zhang, Shou-Cheng},
Date-Modified = {2018-12-11 23:01:17 +0530},
Doi = {10.1103/PhysRevLett.80.3586},
Eprint = {9707027v1},
File = {:Users/deepak/mendeley/files/Rabello et al.{\_}Microscopic Electron Models with Exact SO(5) Symmetry{\_}1998.pdf:pdf},
Issn = {0031-9007},
Journal = {Physical Review Letters},
Keywords = {condensed matter,demler{\_}eugene,duality,many body,nonlinear-sigma-model,prl,quantum gravity,so5,symmetry,zhang{\_}s{\_}c},
Mendeley-Tags = {zhang{\_}s{\_}c,demler{\_}eugene,so5,prl,symmetry,duality,nonlinear-sigma-model,condensed matter,many body,quantum gravity},
Month = {jul},
Number = {16},
Pages = {3586--3589},
Primaryclass = {arXiv:cond-mat},
Title = {{Microscopic Electron Models with Exact SO(5) Symmetry}},
Volume = {80},
Year = {1998},
Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevLett.80.3586}}</code></pre></div>       <div id="paperkey_13" class="papercite_entry">[14]           <a href='http://dx.doi.org/10.1007/BF00758384' class='papercite_doi' title='View document in publisher site'><img src='https://www.quantumofgravity.com/blog/wp-content/plugins/papercite/img/external.png' width='10' height='10' alt='[doi]' /></a>        J. D. Romano, "Geometrodynamics vs. connection dynamics," , 1993. <br/>    <a href="javascript:void(0)" id="papercite_13" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_13_block"><pre><code class="tex bibtex">@article{Romano1993Geometrodynamics,
Abstract = {The purpose of this review is to describe in some detail the mathematical relationship between geometrodynamics and connection dynamics in the context of the classical theories of 2+1 and 3+1 gravity. I analyze the standard Einstein-Hilbert theory (in any spacetime dimension), the Palatini and Chern-Simons theories in 2+1 dimensions, and the Palatini and self-dual theories in 3+1 dimensions. I also couple various matter fields to these theories and briefly describe a pure spin-connection formulation of 3+1 gravity. I derive the Euler-Lagrange equations of motion from an action principle and perform a Legendre transform to obtain a Hamiltonian formulation of each theory. Since constraints are present in all these theories, I construct constraint functions and analyze their Poisson bracket algebra. I demonstrate, whenever possible, equivalences between the theories.},
Author = {Romano, Joseph D.},
Citeulike-Article-Id = {6444191},
Date-Modified = {2018-12-13 14:48:12 +0530},
Day = {26},
Doi = {10.1007/BF00758384},
Eprint = {gr--qc/9303032},
Keywords = {adm\_formulation, ashtekar\_variables, chern\_simons, connection\_dynamics, file-import-09-12-27, geometrodynamics, loop\_quantum\_gravity, lqg, palatini},
Month = {Mar},
Posted-At = {2009-12-27 12:17:41},
Priority = {2},
Title = {Geometrodynamics vs. Connection Dynamics},
Url = {http://dx.doi.org/10.1007/BF00758384},
Year = {1993},
Bdsk-Url-1 = {http://dx.doi.org/10.1007/BF00758384}}</code></pre></div>       <div id="paperkey_14" class="papercite_entry">[15]           <a href='http://dx.doi.org/10.1088/0264-9381/21/15/R01' class='papercite_doi' title='View document in publisher site'><img src='https://www.quantumofgravity.com/blog/wp-content/plugins/papercite/img/external.png' width='10' height='10' alt='[doi]' /></a>        A. Ashtekar and J. Lewandowski, "Background independent quantum gravity: a status report," <span style="font-style: italic">Classical and quantum gravity</span>, vol. 21, iss. 15, p. R53–R152, 2004. <br/>    <a href="javascript:void(0)" id="papercite_14" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_14_block"><pre><code class="tex bibtex">@article{Ashtekar2004Background,
Abstract = {The goal of this review is to present an introduction to loop quantum gravity--a background-independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a bird's eye view of the present status of the subject, the review should also serve as a vehicle to enter the field and explore it in detail. To aid non-experts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the review is essentially self-contained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well-established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the review to a reasonable size, and to avoid overwhelming non-experts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it.},
Author = {Ashtekar, Abhay and Lewandowski, Jerzy},
Citeulike-Article-Id = {4462819},
Date-Modified = {2018-12-13 14:50:22 +0530},
Doi = {10.1088/0264-9381/21/15/R01},
Journal = {Classical and Quantum Gravity},
Keywords = {geometry, lecture-notes, lqg, quantum-gravity, review},
Number = {15},
Pages = {R53--R152},
Posted-At = {2009-06-26 02:25:51},
Priority = {0},
Title = {Background independent quantum gravity: a status report},
Url = {http://dx.doi.org/10.1088/0264-9381/21/15/R01},
Volume = {21},
Year = {2004},
Bdsk-Url-1 = {http://dx.doi.org/10.1088/0264-9381/21/15/R01}}</code></pre></div>       <div id="paperkey_15" class="papercite_entry">[16]           <a href='http://dx.doi.org/10.1017/CBO9781107706910' class='papercite_doi' title='View document in publisher site'><img src='https://www.quantumofgravity.com/blog/wp-content/plugins/papercite/img/external.png' width='10' height='10' alt='[doi]' /></a>        C. Rovelli and F. Vidotto, <span style="font-style: italic">Covariant Loop Quantum Gravity</span>, Cambridge: Cambridge University Press, 2014. <br/>    <a href="javascript:void(0)" id="papercite_15" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_15_block"><pre><code class="tex bibtex">@book{Rovelli2014Covariant,
Abstract = {Quantum gravity is among the most fascinating problems in physics. It modifies our understanding of time, space and matter. The recent development of the loop approach has allowed us to explore domains ranging from black hole thermodynamics to the early Universe. This book provides readers with a simple introduction to loop quantum gravity, centred on its covariant approach. It focuses on the physical and conceptual aspects of the problem and includes the background material needed to enter this lively domain of research, making it ideal for researchers and graduate students. Topics covered include quanta of space; classical and quantum physics without time; tetrad formalism; Holst action; lattice QCD; Regge calculus; ADM and Ashtekar variables; Ponzano-Regge and Turaev-Viro amplitudes; kinematics and dynamics of 4D Lorentzian quantum gravity; spectrum of area and volume; coherent states; classical limit; matter couplings; graviton propagator; spinfoam cosmology and black hole thermodynamics.},
Author = {Rovelli, Carlo and Vidotto, Francesca},
Date-Modified = {2018-12-13 14:50:10 +0530},
Doi = {10.1017/CBO9781107706910},
Isbn = {9781107706910},
Keywords = {area operator,arrow of time,ashtekar{\_}variables,black holes,black-hole-entropy,book,cosmology,introductionl,loop{\_}quantum{\_}gravity,pedagogical,quantum gravity,rovelli{\_}carlo,tetrad formalism,vidotto{\_}francesca,volume operator},
Mendeley-Tags = {book,pedagogical,vidotto{\_}francesca,rovelli{\_}carlo,loop{\_}quantum{\_}gravity,quantum gravity,introductionl,black holes,ashtekar{\_}variables,black-hole-entropy,area operator,volume operator,tetrad formalism,arrow of time,cosmology},
Publisher = {Cambridge University Press},
Title = {{Covariant Loop Quantum Gravity}},
Url = {http://www.cpt.univ-mrs.fr/{~}rovelli/IntroductionLQG.pdf http://ebooks.cambridge.org/ref/id/CBO9781107706910},
Year = {2014},
Bdsk-Url-1 = {http://www.cpt.univ-mrs.fr/%7B~%7Drovelli/IntroductionLQG.pdf%20http://ebooks.cambridge.org/ref/id/CBO9781107706910},
Bdsk-Url-2 = {http://dx.doi.org/10.1017/CBO9781107706910}}</code></pre></div>       <div id="paperkey_16" class="papercite_entry">[17]           <a href='http://dx.doi.org/10.1007/978-3-319-43184-0' class='papercite_doi' title='View document in publisher site'><img src='https://www.quantumofgravity.com/blog/wp-content/plugins/papercite/img/external.png' width='10' height='10' alt='[doi]' /></a>        D. Vaid and S. Bilson-Thompson, <span style="font-style: italic">Lqg for the bewildered</span>, C. Caron, Ed., Springer Nature, 2014. <br/>    <a href="javascript:void(0)" id="papercite_16" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_16_block"><pre><code class="tex bibtex">@book{Vaid2014LQG-for-the-Bewildered,
Abstract = {We present a pedagogical introduction to the notions underlying the connection formulation of General Relativity - Loop Quantum Gravity (LQG) - with an emphasis on the physical aspects of the framework. We explain, in a concise and clear manner, the steps leading from the Einstein-Hilbert action for gravity to the construction of the quantum states of geometry, known as \emph{spin-networks}, which provide the basis for the kinematical Hilbert space of quantum general relativity. Along the way we introduce the various associated concepts of \emph{tetrads}, \emph{spin-connection} and \emph{holonomies} which are a pre-requisite for understanding the LQG formalism. Having provided a minimal introduction to the LQG framework, we discuss its applications to the problems of black hole entropy and of quantum cosmology. A list of the most common criticisms of LQG is presented, which are then tackled one by one in order to convince the reader of the physical viability of the theory. An extensive set of appendices provide accessible introductions to several key notions such as the \emph{Peter-Weyl theorem}, \emph{duality} of differential forms and \emph{Regge calculus}, among others. The presentation is aimed at graduate students and researchers who have some familiarity with the tools of quantum mechanics and field theory and/or General Relativity, but are intimidated by the seeming technical prowess required to browse through the existing LQG literature. Our hope is to make the formalism appear a little less bewildering to the un-initiated and to help lower the barrier for entry into the field.},
Archiveprefix = {arXiv},
Author = {Vaid, Deepak and Bilson-Thompson, Sundance},
Citeulike-Article-Id = {13051558},
Date-Modified = {2018-12-13 14:50:38 +0530},
Day = {14},
Doi = {10.1007/978-3-319-43184-0},
Editor = {Christian Caron},
Eprint = {1402.3586},
Keywords = {adm\_formulation, area-operator, arxiv, ashtekar\_variables, bilson-thompson, connection\_dynamics, differential\_forms, introductory, lqg, lqgbewil, peter\_weyl\_theorem, quantum\_gravity, regge\_calculus, vaid\_d},
Month = feb,
Posted-At = {2014-02-17 08:20:44},
Priority = {2},
Publisher = {Springer Nature},
Title = {LQG for the Bewildered},
Url = {http://arxiv.org/abs/1402.3586},
Year = {2014},
Bdsk-Url-1 = {http://arxiv.org/abs/1402.3586},
Bdsk-Url-2 = {https://doi.org/10.1007/978-3-319-43184-0}}</code></pre></div>       <div id="paperkey_17" class="papercite_entry">[18]                   A. Randono, "Gauge gravity: a forward-looking introduction," , 2010. <br/>    <a href="javascript:void(0)" id="papercite_17" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_17_block"><pre><code class="tex bibtex">@article{Randono2010Gauge,
Abstract = {This article is a review of modern approaches to gravity that treat the gravitational interaction as a type of gauge theory. The purpose of the article is twofold. First, it is written in a colloquial style and is intended to be a pedagogical introduction to the gauge approach to gravity. I begin with a review of the {Einstein-Cartan} formulation of gravity, move on to the {Macdowell-Mansouri} approach, then show how gravity can be viewed as the symmetry broken phase of an ({A)dS}-gauge theory. This covers roughly the first half of the article. Armed with these tools, the remainder of the article is geared toward new insights and new lines of research that can be gained by viewing gravity from this perspective. Drawing from familiar concepts from the symmetry broken gauge theories of the standard model, we show how the topological structure of the gauge group allows for an infinite class of new solutions to the {Einstein-Cartan} field equations that can be thought of as degenerate ground states of the theory. We argue that quantum mechanical tunneling allows for transitions between the degenerate vacua. Generalizing the tunneling process from a topological phase of the gauge theory to an arbitrary geometry leads to a modern reformulation of the {Hartle-Hawking} "no boundary" proposal.},
Archiveprefix = {arXiv},
Author = {Randono, Andrew},
Citeulike-Article-Id = {8141655},
Date-Modified = {2018-12-13 14:46:53 +0530},
Day = {27},
Eprint = {1010.5822},
Keywords = {einstein\_cartan\_theory, gauge\_gravity\_duality, hartle\_hawking\_state, macdowell-mansouri, quantum\_tunnelling, randono, symmetry\_breaking},
Month = oct,
Posted-At = {2011-11-29 18:58:16},
Priority = {2},
Title = {Gauge Gravity: a forward-looking introduction},
Url = {http://arxiv.org/abs/1010.5822},
Year = {2010},
Bdsk-Url-1 = {http://arxiv.org/abs/1010.5822}}</code></pre></div>       <div id="paperkey_18" class="papercite_entry">[19]                   A. Randono, "Gravity from a fermionic condensate of a gauge theory," , 2010. <br/>    <a href="javascript:void(0)" id="papercite_18" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_18_block"><pre><code class="tex bibtex">@article{Randono2010Gravity,
Abstract = {The most prominent realization of gravity as a gauge theory similar to the gauge theories of the standard model comes from enlarging the gauge group from the Lorentz group to the de Sitter group. To regain ordinary Einstein-Cartan gravity the symmetry must be broken, which can be accomplished by known quasi-dynamic mechanisms. Motivated by symmetry breaking models in particle physics and condensed matter systems, we propose that the symmetry can naturally be broken by a homogenous and isotropic fermionic condensate of ordinary spinors. We demonstrate that the condensate is compatible with the Einstein-Cartan equations and can be imposed in a fully de Sitter invariant manner. This lends support, and provides a physically realistic mechanism for understanding gravity as a gauge theory with a spontaneously broken local de Sitter symmetry.},
Archiveprefix = {arXiv},
Author = {Randono, Andrew},
Citeulike-Article-Id = {7151223},
Date-Modified = {2018-12-13 14:46:53 +0530},
Day = {7},
Eprint = {1005.1294},
Keywords = {cartan-geometry, desitter, fermionic\_condensate, lorentz\_group, manybody, spontaneous\_symmetry\_breaking},
Month = {May},
Posted-At = {2010-05-19 13:44:10},
Priority = {2},
Title = {Gravity from a fermionic condensate of a gauge theory},
Url = {http://arxiv.org/abs/1005.1294},
Year = {2010},
Bdsk-Url-1 = {http://arxiv.org/abs/1005.1294}}</code></pre></div>       <div id="paperkey_19" class="papercite_entry">[20]                   H. F. Westman and T. G. Zlosnik, <span style="font-style: italic">Gravity from dynamical symmetry breaking</span>, 2013. <br/>    <a href="javascript:void(0)" id="papercite_19" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_19_block"><pre><code class="tex bibtex">@misc{Westman2013Gravity,
Abstract = {It has been known for some time that General Relativity can be regarded as a Yang-Mills-type gauge theory in a symmetry broken phase. In this picture the gravity sector is described by an SO(1,4) or SO(2,3) gauge field \$A^{a}\_{\ph{a}b\mu}\$ and Higgs field \$V^{a}\$ which acts to break the symmetry down to that of the Lorentz group SO(1,3). This symmetry breaking mirrors that of electroweak theory. However, a notable difference is that while the Higgs field \$\Phi\$ of electroweak theory is taken as a genuine dynamical field satisfying a Klein-Gordon equation, the gauge independent component \$V^2\$ of the Higgs-type field \$V^a\$ is typically regarded as non-dynamical. Instead, in many treatments \$V^a\$ does not appear explicitly in the formalism or is required to satisfy \$V^2\equiv \eta\_{ab}V^{a}V^{b}=const.\$ by means of a Lagrangian constraint. As an alternative to this we propose a class of polynomial actions that treat both the gauge connection \$A^{a}\_{\ph{a}b\mu}\$ and Higgs field \$V^a\$ as genuine dynamical fields. The resultant equations of motion consist of a set of first-order partial differential equations. We show that for certain actions these equations may be cast in a second-order form, corresponding to a scalar-tensor model of gravity. A specific choice based on the symmetry group SO(1,4) yields a positive cosmological constant and an effective mass \$M\$ of the gravitational Higgs field ensuring the constancy of \$V^2\$ at low energies and agreement with empirical data if \$M\$ is sufficiently large. More general actions are discussed corresponding to variants of Chern-Simons modified gravity and scalar-Euler form gravity.},
Archiveprefix = {arXiv},
Author = {Westman, H. F. and Zlosnik, T. G.},
Citeulike-Article-Id = {12882158},
Date-Modified = {2018-12-13 14:46:53 +0530},
Day = {5},
Eprint = {1302.1103},
Keywords = {chern-simons, cosmological\_constant, einstein\_cartan\_theory, electroweak\_model, gauge\_theory, quantum\_gravity, symmetry\_breaking, westman\_hf, zlosnik\_tg},
Month = feb,
Posted-At = {2013-12-26 13:37:26},
Priority = {2},
Title = {Gravity from dynamical symmetry breaking},
Url = {http://arxiv.org/abs/1302.1103},
Year = {2013},
Bdsk-Url-1 = {http://arxiv.org/abs/1302.1103}}</code></pre></div>       <div id="paperkey_20" class="papercite_entry">[21]           <a href='http://dx.doi.org/10.1016/j.aop.2015.06.013' class='papercite_doi' title='View document in publisher site'><img src='https://www.quantumofgravity.com/blog/wp-content/plugins/papercite/img/external.png' width='10' height='10' alt='[doi]' /></a>        H. F. Westman and T. G. Zlosnik, "An introduction to the physics of cartan gravity," <span style="font-style: italic">Annals of physics</span>, vol. 361, p. 330–376, 2015. <br/>    <a href="javascript:void(0)" id="papercite_20" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_20_block"><pre><code class="tex bibtex">@article{Westman2015An-introduction,
Author = {Westman, Hans F. and Zlosnik, Tom G.},
Date-Modified = {2018-12-13 14:46:53 +0530},
Doi = {10.1016/j.aop.2015.06.013},
File = {FULLTEXT:/home/dvaid/physicistatwork.com/librarian/library/02238.pdf:PDF},
Journal = {Annals of Physics},
Pages = {330--376},
Title = {An introduction to the physics of Cartan gravity},
Volume = {361},
Year = {2015},
Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.aop.2015.06.013}}</code></pre></div>       <div id="paperkey_21" class="papercite_entry">[22]                   D. K. Wise, "Macdowell-mansouri gravity and cartan geometry," , 2009. <br/>    <a href="javascript:void(0)" id="papercite_21" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_21_block"><pre><code class="tex bibtex">@article{Wise2009MacDowellMansouri,
Abstract = {The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous "model spacetime", including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A "Cartan connection" gives a prescription for parallel transport from one "tangent model spacetime" to another, along any path, giving a natural interpretation of the MacDowell-Mansouri connection as "rolling" the model spacetime along physical spacetime. I explain Cartan geometry, and "Cartan gauge theory", in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell-Mansouri gravity, as well as its more recent reformulation in terms of BF theory, in the context of Cartan geometry.},
Author = {Wise, Derek K.},
Citeulike-Article-Id = {6444301},
Date-Modified = {2018-12-13 14:46:53 +0530},
Day = {15},
Eprint = {gr-qc/0611154},
Keywords = {cartan-geometry, file-import-09-12-27, macdowell-mansouri},
Month = {May},
Posted-At = {2009-12-27 12:17:43},
Priority = {5},
Title = {MacDowell-Mansouri gravity and Cartan geometry},
Url = {http://arxiv.org/abs/gr-qc/0611154},
Year = {2009},
Bdsk-Url-1 = {http://arxiv.org/abs/gr-qc/0611154}}</code></pre></div>       <div id="paperkey_22" class="papercite_entry">[23]                   L. Smolin and A. Starodubtsev, "General relativity with a topological phase: an action principle," , 2003. <br/>    <a href="javascript:void(0)" id="papercite_22" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_22_block"><pre><code class="tex bibtex">@article{Smolin2003General,
Abstract = {An action principle is described which unifies general relativity and topological field theory. An additional degree of freedom is introduced and depending on the value it takes the theory has solutions that reduce it to 1) general relativity in Palatini form, 2) general relativity in the Ashtekar form, 3) \$F\wedge F\$ theory for {SO}(5) and 4) \${BF}\$ theory for {SO}(5). This theory then makes it possible to describe explicitly the dynamics of phase transition between a topological phase and a gravitational phase where the theory has local degrees of freedom. We also find that a boundary between adymnamical and topological phase resembles an horizon.},
Archiveprefix = {arXiv},
Author = {Smolin, Lee and Starodubtsev, Artem},
Citeulike-Article-Id = {10081845},
Date-Modified = {2018-12-13 14:52:47 +0530},
Day = {18},
Eprint = {hep-th/0311163},
Keywords = {antidesitter, bf-theory, desitter, phase\_transition, quantum\_gravity, smolin, starodubstev, tqft},
Month = nov,
Posted-At = {2011-11-29 17:48:29},
Priority = {2},
Title = {General relativity with a topological phase: an action principle},
Url = {http://arxiv.org/abs/hep-th/0311163},
Year = {2003},
Bdsk-Url-1 = {http://arxiv.org/abs/hep-th/0311163}}</code></pre></div>       <div id="paperkey_23" class="papercite_entry">[24]                   J. C. Baez, "An introduction to spin foam models of quantum gravity and BF theory," , 1999. <br/>    <a href="javascript:void(0)" id="papercite_23" class="papercite_toggle">[Bibtex]</a></div>          <div class="papercite_bibtex" id="papercite_23_block"><pre><code class="tex bibtex">@article{Baez1999An-Introduction,
Abstract = {In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of  foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a spin foam model' we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called {BF} theory.},
Archiveprefix = {arXiv},
Author = {Baez, John C.},
Citeulike-Article-Id = {354214},
Date-Modified = {2018-12-13 14:53:14 +0530},
Day = {21},
Eprint = {gr-qc/9905087},
Keywords = {baez, bf-theory, intertwiner, introduction, pedagogical, quantum\_gravity, simplicial\_geometry, spin-foams},
Month = may,
Posted-At = {2011-05-13 11:42:37},
Priority = {2},
Title = {An Introduction to Spin Foam Models of Quantum Gravity and {BF} Theory},
Url = {http://arxiv.org/abs/gr-qc/9905087},
Year = {1999},
Bdsk-Url-1 = {http://arxiv.org/abs/gr-qc/9905087}}
[25] D. Vaid, “Superconducting and Anti-Ferromagnetic phases of spacetime,” Advances in high energy physics, vol. 2017, iss. 1, p. 9, 2013.
[Bibtex]
@article{Vaid2013Superconducting,
Abstract = {A correspondence between the \$SO(5)\$ theory of High-T\${}\_C\$ superconductivity and antiferromagnetism, put forward by Zhang and collaborators, and a theory of gravity arising from symmetry breaking of a \$SO(5)\$ gauge field is presented. A physical correspondence between the order parameters of the unified SC/AF theory and the generators of the gravitational gauge connection is conjectured. A preliminary identification of regions of geometry, in solutions of Einstein's equations describing charged-rotating black holes embedded in deSitter spacetime, with SC and AF phases is carried out.},
Archiveprefix = {arXiv},
Author = {Vaid, Deepak},
Citeulike-Article-Id = {12883610},
Date-Modified = {2018-12-13 12:36:03 +0530},
Day = {26},
Eprint = {1312.7119},
Journal = {Advances in High Energy Physics},
Keywords = {antiferromagnetism, bf-theory, high\_temperature, manybody, quantum\_gravity, sc\_zhang, so5, superconductivity, symmetry\_breaking, vaid\_d},
Month = dec,
Number = {1},
Pages = {9},
Posted-At = {2013-12-30 05:58:11},
Priority = {2},
Title = {Superconducting and {Anti-Ferromagnetic} Phases of Spacetime},
Url = {http://arxiv.org/abs/1312.7119, https://doi.org/10.1155/2017/7935185},
Volume = {2017},
Year = {2013},
Bdsk-Url-1 = {http://arxiv.org/abs/1312.7119},
Bdsk-Url-2 = {http://arxiv.org/abs/1312.7119,%20https://doi.org/10.1155/2017/7935185}}
[26] S. Zhang and J. Hu, “A four dimensional generalization of the quantum hall effect,” Science, vol. 294, iss. 5543, p. 823–828, 2001.
[Bibtex]
@article{Zhang2001Four,
Abstract = {We construct a generalization of the quantum Hall effect, where particles move in four dimensional space under a SU(2) gauge field. This system has a macroscopic number of degenerate single particle states. At appropriate integer or fractional filling fractions the system forms an incompressible quantum liquid. Gapped elementary excitations in the bulk interior and gapless elementary excitations at the boundary are investigated.},
Archiveprefix = {arXiv},
Author = {Zhang, Shou-Cheng and Hu, Jiangping},
Citeulike-Article-Id = {5234280},
Date-Modified = {2018-12-11 23:01:17 +0530},
Day = {27},
Doi = {10.1126/science.294.5543.823},
Eprint = {cond-mat/0110572},
Journal = {Science},
Keywords = {dimensions, effect, four, geometry, hall, quantum, so5},
Month = {Oct},
Number = {5543},
Pages = {823--828},
Priority = {4},
Title = {A Four Dimensional Generalization of the Quantum Hall Effect},
Url = {http://dx.doi.org/10.1126/science.294.5543.823},
Volume = {294},
Year = {2001},
Bdsk-Url-1 = {http://dx.doi.org/10.1126/science.294.5543.823},
Bdsk-Url-2 = {http://arxiv.org/abs/cond-mat/0110572}}
[27] A. B. Bernevig, T. L. Hughes, and S. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science, vol. 314, iss. 5806, p. 1757–1761, 2006.
[Bibtex]
@article{Bernevig2006Quantum,
Abstract = {We show that the Quantum Spin Hall Effect, a state of matter with topological
properties distinct from conventional insulators, can be realized in {HgTe}/{CdTe}
semiconductor quantum wells. By varying the thickness of the quantum well, the
electronic state changes from a normal to an "inverted" type at a critical
thickness \$d\_c\$. We show that this transition is a topological quantum phase
transition between a conventional insulating phase and a phase exhibiting the
{QSH} effect with a single pair of helical edge states. We also discuss the
methods for experimental detection of the {QSH} effect.},
Archiveprefix = {arXiv},
Author = {Bernevig, B. Andrei and Hughes, Taylor L. and Zhang, Shou-Cheng},
Citeulike-Article-Id = {5096324},
Date-Modified = {2018-12-11 23:01:17 +0530},
Day = {15},
Doi = {10.1126/science.1133734},
Eprint = {cond-mat/0611399},
Issn = {0036-8075},
Journal = {Science},
Keywords = {band\_inversion, bernevig\_ba, condensed\_matter, edge\_states, hgte\_cdte, hughes\_taylor, manybody, meron, phase\_transition, sc\_zhang, spin\_hall\_effect, spinors, topological\_order},
Month = nov,
Number = {5806},
Pages = {1757--1761},
Pmid = {17170299},
Posted-At = {2016-11-22 03:49:31},
Priority = {2},
Publisher = {American Association for the Advancement of Science},
Title = {Quantum Spin Hall Effect and Topological Phase Transition in {HgTe} Quantum Wells},
Url = {http://dx.doi.org/10.1126/science.1133734},
Volume = {314},
Year = {2006},
Bdsk-Url-1 = {http://dx.doi.org/10.1126/science.1133734}}
[28] M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X. Qi, and S. Zhang, “Quantum spin hall insulator state in HgTe quantum wells.,” Science (new york, n.y.), vol. 318, iss. 5851, p. 766–70, 2007.
[Bibtex]
@article{Konig2007Quantum,
Abstract = {Recent theory predicted that the quantum spin Hall effect, a fundamentally new quantum state of matter that exists at zero external magnetic field, may be realized in HgTe/(Hg,Cd)Te quantum wells. We fabricated such sample structures with low density and high mobility in which we could tune, through an external gate voltage, the carrier conduction from n-type to p-type, passing through an insulating regime. For thin quantum wells with well width d {\textless} 6.3 nanometers, the insulating regime showed the conventional behavior of vanishingly small conductance at low temperature. However, for thicker quantum wells (d {\textgreater} 6.3 nanometers), the nominally insulating regime showed a plateau of residual conductance close to 2e(2)/h, where e is the electron charge and h is Planck's constant. The residual conductance was independent of the sample width, indicating that it is caused by edge states. Furthermore, the residual conductance was destroyed by a small external magnetic field. The quantum phase transition at the critical thickness, d = 6.3 nanometers, was also independently determined from the magnetic field-induced insulator-to-metal transition. These observations provide experimental evidence of the quantum spin Hall effect.},
Author = {K{\"{o}}nig, Markus and Wiedmann, Steffen and Br{\"{u}}ne, Christoph and Roth, Andreas and Buhmann, Hartmut and Molenkamp, Laurens W and Qi, Xiao-Liang and Zhang, Shou-Cheng},
Date-Modified = {2018-12-14 11:27:35 +0530},
Doi = {10.1126/science.1148047},
File = {:Users/deepak/mendeley/K{\"{o}}nig et al.{\_}Quantum spin hall insulator state in HgTe quantum wells.{\_}2007.pdf:pdf},
Issn = {1095-9203},
Journal = {Science (New York, N.Y.)},
Mendeley-Groups = {QHE-BHE},
Month = {nov},
Number = {5851},
Pages = {766--70},
Pmid = {17885096},
Publisher = {American Association for the Advancement of Science},
Title = {{Quantum spin hall insulator state in HgTe quantum wells.}},
Url = {http://www.ncbi.nlm.nih.gov/pubmed/17885096},
Volume = {318},
Year = {2007},
Bdsk-Url-1 = {http://www.ncbi.nlm.nih.gov/pubmed/17885096},
Bdsk-Url-2 = {https://doi.org/10.1126/science.1148047}}`