# Gordon Walter Semenoff

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String theory, quantum field theory

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## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - May 2021)

In this thesis we discuss applications of quantum information theoretic concepts toquantum gravity and the low-energy regime of quantum field theories.The first part of this thesis is concerned with how quantum information spreadsin four-dimensional scattering experiments for theories coupled to quantum electrodynamicsor perturbative quantum gravity. In these cases, every scattering processis accompanied by the emission of an infinite number of soft photons or gravitons,which cause infrared divergences in the calculation of scattering probabilities.There are two methods to deal with IR divergences: the inclusive and dressedformalisms. We demonstrate that in the late-time limit, independent of the method,the hard outgoing particles are entangled with soft particles in such a way that thereduced density matrix of the hard particles is essentially completely decohered.Furthermore, we show that the inclusive formalism is ill-suited to describe scatteringof wavepackets, requiring the use of the dressed formalism. We construct theHilbert space for QED in the dressed formalism as a representation of the canonicalcommutation relations of the photon creation/annihilation algebra, and argue that itsplits into superselection sectors which correspond to eigenspaces of the generatorsof large gauge transformations.In the second part of this thesis, we turn to applications of quantum informationtheoretic concepts in the AdS/CFT correspondence. In pure AdS, we find anexplicit formula for the Ryu-Takayanagi (RT) surface for special subregions in thedual conformal field theory, whose entangling surface lie on a light cone. Theexplicit form of the RT surface is used to give a holographic proof of Markovicityof the CFT vacuum on a light cone. Relative entropy of a state on such specialsubregions is dual to a novel measure of energy associated with a timelike vector flow between the causal and entanglement wedge. Positivity and monotonicity ofrelative entropy imply positivity and monotonicity of this energy, which yields aconsistency conditions for solutions to quantum gravity.

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In the first part of this thesis we exploit supersymmetric localization to study aspects of supersymmetric gauge theories relevant to holography. In chapter 2 we study the 1/2-BPS circular Wilson loop in the totally antisymmetric representation of the gauge group in N = 4 supersymmetric Yang-Mills. We compute the first 1/N correction at leading order in ’t Hooft coupling by means of the matrix model loop equations for comparison with the 1-loop effective action of the holographically dual D5-brane. Our result suggests the need to account for gravitational backreaction on the string theory side. In chapter 3 we solve the planar N = 2* super-Yang-Mills theory at large ’t Hooft coupling again using localization on S⁴. The solution permits detailed investigation of the resonance phenomena responsible for quantum phase transitions in infinite volume, and leads to quantitative predictions for the semiclassical string dual of the N = 2* theory. The second part of the thesis deals with the Schwinger effect in scalar quantum electrodynamics and in bosonic string theory. Chapter 4 presents a detailed study of the semiclassical expansion of the world line path integral for a charged relativistic particle in a constant external electric field. It is demonstrated that the Schwinger formula for charged particle pair production is reproduced exactly by the semiclassical expansion around classical instanton solutions when the leading order of fluctuations is taken into account. By a localization argument we prove that all corrections to this leading approximation vanish and that the WKB approximation to the world line path integral is exact. Finally, in chapter 5 we analyse the problem of charged string pair creation in a constant external electric field. We find the instantons in the worldsheet sigma model which are responsible for the tunneling events, and evaluate the sigma model partition function in the multi-instanton sector in the WKB approximation. We further identify a fermionic symmetry associated with collective coordinates, which we use to localize the worldsheet functional integral onto its WKB limit, proving that our result is exact.

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Scattering amplitudes in massless gauge field theories have long been known to give rise to infrared divergent effects from the emission of very low energy gauge bosons. The traditional way of dealing with those divergences has been to abandon the idea of measuring amplitudes by only focusing on inclusive cross-sections constructed out of physically equivalent states. An alternative option, found to be consistent with the S-matrix framework, suggested to dress asymptotic states of charged particles by shockwaves of low energy bosons. In this formalism, the clouds of soft bosons, when tuned appropriately, cancel the usual infrared divergences occurring in the standard approach. Recently, the dressing approach has received renewed attention for its connection with newly discovered asymptotic symmetries of massless gauge theories and its potential role in the black hole information paradox.We start by investigating quantum information properties of scattering theory while having only access to a subset of the outgoing state. We give an exact formula for the von Neuman entanglement entropy of an apparatus particle scattered off a set of system particles and show how to obtain late-time expectation values of apparatus observables.We then specify to the case of quantum electrodynamics (QED) and gravity where the unobserved system particles are low energy photons and gravitons. Using the standard inclusive cross-section formalism, we demonstrate that those soft bosons decohere nearly all momentum superpositions of hard particles. Repeating a similar computation using the dressing formalism, we obtain an analogous result: In either framework, outgoing hard momentum states at late times are fully decohered from not having access to the soft bosons. Finally, we make the connection between our results and the framework of asymptotic symmetries of QED and gravity. We give new evidence for the use of the dressed formalism by exhibiting an inconsistency in the scattering of wavepackets in the original inclusive cross-section framework.

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In Chapter 1, we present a brief introduction to the tight-binding model of graphene and show that in the low-energy continuum limit, it can be modeled by reduced QED₂₊₁ . We then review renormalization group technique which is used in the next chapters. In Chapter 2, we consider a quantum field theory in 3+1d with the defect of a large number of fermion flavors, N. We study the next-to-leading order contributions to the fermions current-current correlation function by performing a large N expansion. We find that the next-to-leading order contributions 1/N to the current-current correlation function is significantly suppressed. The suppression is a consequence of a surprising cancellation between the two contributing Feynman diagrams. We calculate the model's conductivity via the Kubo formula and compare our results with the observed conductivity for graphene. In Chapter 3, we study graphene's beta function in large N. We use the large N expansion to explore the renormalization of the Fermi velocity in the screening dominated regime of charge neutral graphene with a Coulomb interaction. We show that inclusion of the fluctuations of the magnetic field lead to a cancellation of the beta function to the leading order in 1/N. The first non-zero contribution to the beta function turns out to be of order 1/N². We perform a careful analysis of possible infrared divergences and show that the superficial infrared divergences do not contribute to the beta function. In Chapter 4, we study the phase structure of a Φ⁶ theory in large N. The leading order of the large N limit of the O(N) symmetric phi-six theory in three dimensions has a phase which exhibits spontaneous breaking of scale symmetry accompanied by a massless dilaton. In this chapter, we show that this “light dilaton” is actually a tachyon. This indicates an instability of the phase of the theory with spontaneously broken approximate scale invariance. We rule out the existence of Bardeen-Moshe-Bander phase. In this thesis, we show that Large N expansion is a powerful tool which in regimes that the system is interacting strongly could be used as an alternative to coupling expansion scheme.

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We use the AdS/CFT correspondence (the holographic duality of gauge/gravity theory) to study exciton driven dynamical symmetry breaking in certain (2+1)-dimensional defect quantum field theories. These models can be argued to be analogs of the electrons with Coulomb interactions which occurin Dirac semimetals and the results our study of these model systems are indicative of behaviours that might be expected in semimetal systems such as monolayer and double monolayer graphene. The field theory models have simple holographic duals, the D3-probe-D5 brane system and the D3-probe-D7 brane system. Analysis of those systems yields information about the strong coupling planar limits of the defect quantum field theories. We study the possible occurrence of exciton condensates in the strong coupling limit of single-defect theories as well as double monolayer theories where we find a rich and interesting phase diagram. The phenomena which we study include the magnetic catalysis of chiral symmetry breaking in monolayers and inter-layer exciton condensation in double monolayers. In the latter case, we find a solvable model where the current-current correlations functions in the planar strongly coupled field theory can be computed explicitly and exhibit interesting behavior. Although the models that we analyze differ in detail from real condensed matter systems, we identify some phenomena which can occur at strong coupling in a generic system and which could well be relevant to the ongoing experiments on multi-monolayer Dirac semimetals. An example is the spontaneous nesting of Fermi surfaces in double monolayers. In particular, we suggest an easy to observe experimental signature of this phenomenon.

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In the first part of this thesis we explore the entanglement structure of relativistic field theories in momentum space. We discuss a Wilsonian path integral formulation and a perturbative approach. Using perturbation theory we obtain results for specific quantum field theories. These are understood through scaling and decoupling properties of field theories. Convergence of the perturbation theory taking loop diagrams into account is also discussed. We then discuss the entanglement structure in systems where Lorentz invariance is broken by a Fermi surface. The Fermi surface helps the convergence of perturbation theory and entanglement of modes near the Fermi surface is shown to be amplified, even in the presence of a large momentum cutoff. In the second part of this thesis we explore the connection between entanglement and gravity in the context of the AdS/CFT correspondence. We show that there are certain thermodynamic-like relations common to all conformal field theories, which when mapped via the AdS/CFT correspondence to the bulk are tantamount to Einstein's equations, to lowest order in the metric.

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Aspects of the AdS/CFT correspondence are studied in the pp.-wave/BMN limit. We usethe light cone string field theory to investigate energy shifts of the one and two impuritystates. In the case of two impurity states we find that logarithmic divergences, in the sumsof intermediate states, actually cancel out between the Hamiltonian and a Q-dependentcontact term”. We show how non-perturbative terms, that have previously plagued thistheory, vanish as a consequence of this cancelation. We argue from this that every order ofinternal impurities contributes to the overall energy shift and attempt to give a systematicway of calculating such sums for the case of the simplest 3-string vertex (one proposed bydiVecchia).We extend our analysis of the mass shift to the case of the most advanced 3-stringvertex (proposed by Dobashi and Yoneya). We find agreement between our string fieldtheory calculations and the leading order CFT result in the BMN limit. We also find strongsimilarities between our result and higher orders in the field theory, including, on the stringside, the disappearance of the half-integer powers which generically do not exist in the fieldtheory calculations.We also study the orbifolding of the pp-wave background which results in the discretequantization of the light-cone momentum. We present the string field theory calculation forsuch a discreet momentum case. We also observe how a particular choice of the orbifold,results in the string theory corresponding to the quantization of the finite size giant magnonon the CFT side. We study this theory in detail with particular emphasis on its superalgebra.

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##### Master's Student Supervision (2010 - 2020)

A massive quark with U(1) charge is constructed with the Higgs mechanism in N = 4supersymmetric Yang-Mills theory and is set up in a constant, uniform external electricfield such that its classical trajectory is that of constant acceleration. The leading term inthe amplitude of the trajectory in the semi-classical approximation is quadratic in the totalproper time, which is attributed to infrared divergences. We consider various methods oftreating such divergences. The inclusion of Bremsstrahlung emission desirably replacesthe quadratic dependence with a linear one, but forces us to reconsider the meaning of aglobal color charge. The method of dressing by Wilson line is shown to be unsuccessful.We finally provide an Ansatz for a Chung-like dressing factor to lowest order and showthe elimination of infrared divergences.

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In this thesis low energy excitations of perfectly dimerized trans-polyacetylene are modelled using the one-dimensional Dirac equation. The system is solved on both the half-line and segment, and the solutions are used to explore quantum phenomena. It is discovered that the zero mode of the half-line is a Majorana fermion quasiparticle. It is also found that dominate zero mode coupling to an electron on a scanning tunnelling microscope is achieved with a sufficiently large mass gap of the quantum wire. This allows scattering state excitations to be ignored in calculations in this thesis. It is also shown that the zero mode can facilitate entanglement of two electrons, each in proximity to opposite ends of a long segment of trans-polyacetylene. An algorithm is also developed which teleports the spin state of an electron on a segment of trans-polyacetylene. The quantum measurement used in this algorithm conserves fermion parity symmetry, however charge superselection is violated for three-fourths of the measurement operators. In the thermally isolated system teleportation is successful for all of the measurement operators on the ground state. However, decoherence occurs in the non-thermally isolated system due to thermal mixing of nearly degenerate states, leading to teleportation being successful for only half of the measurement operations on the thermal state.

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The purpose of the present work is to derive a classification for topologically stable Fermi surfaces for translationally invariant systems with no electron-electron interactions. To derive such a classification we introduce the necessary concepts in condensed matter and electronic band theory as well as those in mathematics such as topological spaces, building up to topological K-theory and its connections with Fredholm operators. We further compute such classes when there is only translational invariance for dimensions d = 1, 2, 3 and discuss the inclusion of other symmetries.

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We apply the AdS/CFT correspondence to study a quantum Hall system at strong coupling. Fermions at finite density in an external magnetic field are put in via gauge fields living on a stack of D5 branes in Anti-deSitter space. Under the appropriate conditions, the D5 branes blow up to form a D7 brane which is capable of forming a charge-gapped state. We add finite temperature by including a black hole which allows us to compute the low temperature entropy of the quantum Hall system. Upon including an external electric field (again as a gauge field on the probe brane), the conductivity tensor is extracted from Ohm’s law.

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The goal of this thesis is to analyse the non-abelian dual model to the defect probe D7-brane embedding in AdS₅ × S⁵[1]. The D7-brane picture can be thought of as a large number (N₅) of D5-branes growing a transverse fuzzy two-sphere, called BIon. This non-abelian solution improves our knowledge of the system by incorporating deviations in 1/N_{5}^{2} in the number of flavors. Such corrections are important from the point of view of the AdS/CFT correspondence as the CFT dual to the probe system is a candidate model for graphene, which possesses an emergent SU(4) symmetry. The main result of this work is the conductivity for the non-abelian D5 sytem. We fi nd that quantum Hall states have a non-integer transverse conductivity that depends on the number of flavor branes in the model. This deviationscales in 1/N_{5}^{2} in the number of flavor branes and vanishes in the large N₅ limit.

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In this thesis, we will study a top-down string theory holographic model of strongly interacting relativistic 2+1 dimensional fermions. We study the defect theory as examining a charged probe D7-branes/anti-branes modeland a charged probe D5-branes/anti-branes model on the thermal AdS₅×S⁵ geometry. We use the branes pair model to depict a geometrical chiral symmetry breaking. We are especially interested in the holographic magnetic effect on the flavour symmetries.

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Recently, a large amount of effort has gone towards using the AdS/CFT conjecture in condensed matter physics. First, we present a review of the conjecture, then we use the conjecture to model 2+1-dimensional fermions. We find three kinds of solutions with different kinds of discrete symmetries. We show that Chern-Simons- like electric responses, computed using a holographic model appear with the right quantized coefficients.

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We considered the k-antisymmetric representation of U(N) gauge group on two dimensional lattice space and derived the free energy by saddle point approximation in large N limit. k is a large integer comparable with N. Besides Gross-Witten phase transition[1], which happens as the coupling constant changes, we found a new phase transition in the strong coupling system that happens as k changes. The free energy of the weak coupling system is a smooth function of k under continuous limit. We have carefully selected the right saddle point solution among other possible ones. Thenumerical results match our saddle point calculations.

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After a brief review of the necessary parts of the theory of the bosonic string, a consistent pp-wave background with constant dilaton and constant three-index antisymmetric field strength is introduced. In particular, the gravitational background is the plane wave with constant coefficients, and the antisymmetric field strength is chosen such that the worldsheet theory is both diff×Weyl invariant and stable. The one-loop closed bosonic string amplitude is evaluated and shown to be modular invariant. Then the free energy of a free closed string gas is calculated, modular invariance of it is proved, and the result is shown to be equivalent to the sum of free energies for the individual particle states.

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