[Could not find the bibliography file(s)
Table of Contents
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
- 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.
-
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.
-
$SO(5)$ 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.
Phase diagram of two cuprate superconductors where the external variable is the dopant concentration $n$ on the left and the pressure $p$ on the right. When the dopant concentration (or pressure) is low the system exhibits an anti-ferromagnetic (AF) phase at low temperatures. As the dopant concentration (or pressure) is increased, the AF phase disappears and after going through a region referred to as the “pseudogap” phase, the system enters a superconducting (SC) phase.
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:
\begin{equation}\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}
\end{equation}
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 :
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.$\Lambda < 0 $ $\Lambda = 0 $ $\Lambda > 0 $ Lorentzian Anti de Sitter (AdS)
$ SO(3,2) $
Minkowski
$ ISO(3,1)$de Sitter (dS)
$SO(4,1)$Riemannian Hyperbolic
$ SO(4,1) $Euclidean
$ISO(4)$Spherical
$ SO(5) $
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:
\begin{equation}\label{eqn:bf-action-interactions}
S_{B^2 F} = \int d^4 x~ \Tr\left[B \wedge F – \theta~B \wedge B \right]
\end{equation}
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:
\begin{equation}\label{eqn:bf-effective-lambda}
\theta = \frac{G \Lambda}{6}
\end{equation}
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:
\begin{equation}\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)
\end{equation}
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:
\begin{equation}\label{eqn:cosmo-length}
l = \sqrt{\frac{3}{\Lambda}}
\end{equation}
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:
\begin{equation}\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)
\end{equation}
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:
\begin{equation}\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)
\end{equation}
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}$ Rotations $S_x$
$S_y$
$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]. -
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.
-
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.
February 15, 1963 – December 1, 2018
Rest in Peace
@book{Jain2007Composite,
abstract = {This book was first published in 2007. 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.},
annotation = {Published: Hardcover},
author = {Jain, Jainendra},
doi = {10.1017/CBO9780511607561},
edition = {1},
isbn = {978-0-511-60756-1},
journal = {Composite Fermions},
keywords = {composite_fermions,condensed_matter,hall_effect,manybody,nonequilibrium},
month = apr,
publisher = {Cambridge University Press},
title = {Composite Fermions},
volume = {9780521862},
year = {2007},
bdsk-url-1 = {https://doi.org/10.1017/CBO9780511607561}}![[doi]](https://i0.wp.com/www.quantumofgravity.com/blog/wp-content/plugins/papercite/img/external.png?resize=10%2C10&ssl=1){kind=small}
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},
doi = {10.1103/PhysRevLett.68.2386},
file = {/Volumes/Data/owncloud/root/research/zotero_pdfs/Lee;Kivelson;Zhang_Quasiparticle charge and the activated conductance of a quantum Hall liquid_1992.pdf},
issn = {0031-9007},
journal = {Physical Review Letters},
keywords = {classic,condensed_matter,duality,insulator,kivelson,law_of_corresponding_states,manybody,nosource,phase_diagram,quantum_hall_effect,sc_zhang,selection_rule,transport-coefficients},
month = apr,
pages = {2386--2389},
publisher = {American Physical Society},
title = {Quasiparticle Charge and the Activated Conductance of a Quantum {{Hall}} Liquid},
volume = {68},
year = {1992},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevLett.68.2386}}[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-Hai and Zhang, Shou-Cheng},
doi = {10.1103/PhysRevB.46.2223},
issn = {0163-1829},
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,
number = {4},
pages = {2223--2238},
publisher = {American Physical Society},
title = {Global Phase Diagram in the Quantum {{Hall}} Effect},
volume = {46},
year = {1992},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevB.46.2223}}[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 renormalization group flow, we derive many properties of both the integer and fractional quantum Hall effects including: universality; the selection rule {\textbar}p1q2-p2q1{\textbar} = 1 for transitions between quantum Hall states characterized by filling factors 1 = p1/q1 and 2 = p2/q2; critical values of the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalization group flow lead to the semicircle rule for transitions between Hall plateaux.},
archiveprefix = {arXiv},
author = {Dolan, Brian P},
doi = {10.1088/0305-4470/32/21/101},
eprint = {cond-mat/9805171},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Dolan_Duality and the modular group in the quantum Hall effect_1999.pdf},
issn = {0305-4470},
journal = {Journal of Physics A: Mathematical and General},
keywords = {bp_dolan,conductivity,duality,modular_group,phase_transition,quantum_hall_effect,renormalization-group,semicircle_law,sl2z},
month = jan,
number = {21},
pages = {L243},
title = {Duality and the Modular Group in the Quantum {{Hall}} Effect},
volume = {32},
year = {1999},
bdsk-url-1 = {https://doi.org/10.1088/0305-4470/32/21/101}}[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{\textbackslash} 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},
doi = {10.1103/PhysRevLett.75.4126},
eprint = {cond-mat/9502060},
issn = {00319007},
journal = {Physical Review Letters},
keywords = {cuprate_superconductors,demler_eugene,empirical_data,experiment,gap,high_temperature,hubbard_model,neutron,nosource,resonance,scattering,so5,superconductors,zhang_sc},
month = feb,
number = {22},
pages = {4126--4129},
pmid = {10059821},
title = {Theory of the Resonant Neutron Scattering of High-{{Tc}} Superconductors},
volume = {75},
year = {1995},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevLett.75.4126}}[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 \${\textbackslash}pi\$ 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},
author = {Meixner, Stefan and Hanke, Werner and Demler, Eugene and Zhang, Shou-Cheng},
date-modified = {2024-08-25 13:14:11 +0530},
doi = {10.1103/PhysRevLett.79.4902},
eprint = {cond-mat/9701217v1},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Meixner et al_Finite-Size Studies on the SO(5) Symmetry of the Hubbard Model_1997.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},
number = {24},
pages = {4902--4905},
title = {Finite-{{Size Studies}} on the {{SO}}(5) {{Symmetry}} of the {{Hubbard Model}}},
volume = {79},
year = {1997},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevLett.79.4902}}[Bibtex]
@article{Rabello1998Microscopic,
abstract = {[PDF]},
archiveprefix = {arXiv},
author = {Rabello, S and Kohno, H and Demler, E and Zhang, S},
date-modified = {2024-08-25 13:14:13 +0530},
doi = {10.1103/PhysRevLett.80.3586},
eprint = {cond-mat/9707027v1},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Rabello et al_Microscopic Electron Models with Exact SO (5) Symmetry_12.pdf;/Users/deepak/ownCloud/root/research/zotero_pdfs/Rabello et al_Microscopic Electron Models with Exact SO (5) Symmetry_1998.pdf},
issn = {0031-9007},
journal = {Physical Review Letters},
keywords = {antiferromagnetism,condensed matter,d-wave,demler_eugene,duality,exact_solution,hamiltonian,many body,manybody,nonlinear-sigma-model,nosource,phase_diagram,prl,quantum gravity,quantum_gravity,sc_zhang,so5,superconductivity,symmetry,symmetry_breaking,zhang_s_c},
month = jul,
number = {5},
pages = {0--3},
title = {Microscopic {{Electron Models}} with {{Exact SO}} (5) {{Symmetry}}},
volume = {80},
year = {1998},
bdsk-url-1 = {https://doi.org/10.1103/PhysRevLett.80.3586}}[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.},
archiveprefix = {arXiv},
author = {Romano, Joseph D.},
date-modified = {2024-08-25 13:14:14 +0530},
doi = {10.1007/BF00758384},
eprint = {gr-qc/9303032},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Romano_Geometrodynamics vs_1993.pdf},
issn = {00017701},
journal = {General Relativity and Gravitation},
keywords = {adm_formulation,ashtekar_variables,chern_simons,connection_dynamics,geometrodynamics,loop_quantum_gravity,lqg,palatini},
number = {8},
pages = {759--854},
title = {Geometrodynamics vs. Connection Dynamics},
type = {Journal Article},
volume = {25},
year = {1993},
bdsk-url-1 = {https://doi.org/10.1007/BF00758384}}[Bibtex]
@article{Ashtekar2004Background,
abstract = {The goal of this article 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 article 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 article 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 article 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.},
archiveprefix = {arXiv},
author = {Ashtekar, Abhay and Lewandowski, Jerzy},
date-modified = {2024-08-25 13:14:04 +0530},
doi = {10.1088/0264-9381/21/15/R01},
eprint = {gr-qc/0404018},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Ashtekar;Lewandowski_Background Independent Quantum Gravity_2004.pdf},
issn = {0264-9381},
journal = {Classical and Quantum Gravity},
keywords = {geometry,lecture-notes,lqg,quantum-gravity,review},
number = {15},
pages = {125},
publisher = {IOP Publishing},
title = {Background {{Independent Quantum Gravity}}: {{A Status Report}}},
type = {Journal Article},
urldate = {2011-01-20},
volume = {21},
year = {2004},
bdsk-url-1 = {https://doi.org/10.1088/0264-9381/21/15/R01}}[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},
eprint = {1010.5822},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Randono_Gauge Gravity_2010.pdf},
keywords = {einstein_cartan_theory,gauge_gravity_duality,hartle_hawking_state,macdowell-mansouri,quantum_tunnelling,randono,symmetry_breaking},
month = oct,
title = {Gauge {{Gravity}}: A Forward-Looking Introduction},
year = {2010}}[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},
doi = {10.1088/0264-9381/27/21/215019},
eprint = {1005.1294},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Randono_Gravity from a fermionic condensate of a gauge theory_2010.pdf},
issn = {0264-9381},
keywords = {cartan-geometry,desitter,fermionic_condensate,lorentz_group,manybody,spontaneous_symmetry_breaking},
month = may,
pages = {16},
title = {Gravity from a Fermionic Condensate of a Gauge Theory},
volume = {215019},
year = {2010},
bdsk-url-1 = {https://doi.org/10.1088/0264-9381/27/21/215019}}[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{\textbackslash}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},
eprint = {hep-th/0311163},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Smolin;Starodubtsev_General relativity with a topological phase_2003.pdf},
keywords = {High Energy Physics - Theory},
month = nov,
number = {5},
pages = {8},
title = {General Relativity with a Topological Phase: An Action Principle},
year = {2003}}[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 `spin 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.},
date-modified = {2024-08-25 13:14:04 +0530},
doi = {10.1007/3-540-46552-9_2},
eprint = {gr-qc/9905087},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Baez_An Introduction to Spin Foam Models of Quantum Gravity and BF Theory_1999.pdf},
issn = {978-3-540-67112-1},
keywords = {baez,bf-theory,intertwiner,introduction,pedagogical,quantum_gravity,simplicial_geometry,spin-foams},
month = may,
pages = {1--55},
title = {An {{Introduction}} to {{Spin Foam Models}} of {{Quantum Gravity}} and {{BF Theory}}},
year = {1999},
bdsk-url-1 = {https://doi.org/10.1007/3-540-46552-9_2}}[Bibtex]
@article{Bernevig2006Quantum,
abstract = {We show that the quantum spin Hall (QSH) effect, a state of matter with topological properties distinct from those of conventional insulators, can be realized in mercury telluride-cadmium telluride semiconductor quantum wells. When the thickness of the quantum well is varied, 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 methods for experimental detection of the QSH effect.},
archiveprefix = {arXiv},
author = {Bernevig, B. A. and Hughes, T. L. and Zhang, S.-C.},
doi = {10.1126/science.1133734},
eprint = {cond-mat/0611399},
file = {/Users/deepak/ownCloud/root/research/zotero_pdfs/Bernevig;Hughes;Zhang_Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells_2006.pdf},
isbn = {1095-9203 (Electronic) 0036-8075 (Linking)},
issn = {0036-8075},
journal = {Science},
keywords = {band_inversion,bernevig_ba,condensed_matter,edge_states,hgte_cdte,hughes_taylor,manybody,meron,nosource,phase_transition,sc_zhang,spin_hall_effect,spinors,topological_order},
month = nov,
number = {5806},
pages = {1757--1761},
pmid = {17170299},
publisher = {American Association for the Advancement of Science},
title = {Quantum {{Spin Hall Effect}} and {{Topological Phase Transition}} in {{HgTe Quantum Wells}}},
volume = {314},
year = {2006},
bdsk-url-1 = {https://doi.org/10.1126/science.1133734}}[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 {$<$} 6.3 nanometers, the insulating regime showed the conventional behavior of vanishingly small conductance at low temperature. However, for thicker quantum wells (d {$>$} 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 = {2024-08-25 13:14:10 +0530},
doi = {10.1126/science.1148047},
eprint = {17885096},
eprinttype = {pubmed},
file = {/Users/deepak/ownCloud/root/research/mendeley/K{\"o}nig et al_Quantum spin hall insulator state in HgTe quantum wells_2007.pdf},
issn = {1095-9203},
journal = {Science (New York, N.Y.)},
keywords = {_tablet},
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.},
urldate = {2018-12-14},
volume = {318},
year = {2007},
bdsk-url-1 = {https://doi.org/10.1126/science.1148047}}