arXiv subjects

H. B. Thacker

Publications and source records attributed to H. B. Thacker.

At least 19 recordsLinked to original sources

arXiv2014

Long Range Topological Order, the Chiral Condensate, and the Berry Connection in QCD

Topological insulators are substances which are bulk insulators but which carry current via special "topologically protected" edge states. The understanding of long range topological order in these systems is built around the idea of a Berry connection, which is a gauge connection obtained from the phase of the electron wave function transported over momentum space rather than coordinate space. The phase of a closed Wilson loop of the Berry connection around the Brillouin zone defines a topological order parameter which labels discrete flux vacua. The conducting states are surface modes on the domain walls between discrete vacua. Evidence from large-$N_c$ chiral dynamics, holographic QCD, and Monte Carlo observations has pointed to a picture of the QCD vacuum that is very similar to that of a topological insulator, with discrete quasivacua labelled by $θ$ angles that differ by mod $2π$. In this picture, the domain walls are membranes of Chern-Simons charge, and the quark condensate consists of surface modes on these membranes, which are delocalized and thus support the long range propagation of Goldstone pions. The Berry phase in QED2 describes charge polarization of fermion-antifermion pairs, while in 4D QCD it describes the polarization of Chern-Simons membranes.

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arXiv2014

Bosonization and the Berry connection in two-dimensional QED

The dynamical effects of topological charge in two-dimensional QED can be expressed in terms of a topological order parameter via a Berry phase construction. The Berry phase describes the electric charge polarization of the vacuum in a manner similar to the theory of polarization in topological insulators. The topological order parameter labels discrete vacua which differ by units of electric flux. Here the associated Berry connection is explicitly constructed from the Dirac Hamiltonian eigenstates by introducing a small attractive Thirring coupling, so that there is still a stable boson in the limit of zero EM coupling. The Berry connection arises from the analytic structure of the Bethe ansatz states in complex rapidity near the free fermion point.

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arXiv2013

Topological insulators and the QCD vacuum: the theta parameter as a Berry phase

There is considerable evidence, based on large $N_c$ chiral dynamics, holographic QCD, and Monte Carlo studies, that the QCD vacuum is permeated by discrete quasivacua separated by domain walls across which the local value of the topological $θ$ parameter jumps by $\pm2π$. In the 2-dimensional $CP^{N-1}$ sigma model, a pointlike charge is a domain wall, and $θ$ describes the background electric flux and the polarization of charged pairs in the vacuum. We show that the screening process, and the role of $θ$ as an order parameter describing electric polarization, are naturally formulated in terms of Bloch wave eigenstates of the Dirac Hamiltonian in the background gauge field. This formulation is similar to the Berry phase description of electric polarization and quantized charge transport in topological insulators. The Bloch waves are quasiperiodic superpositions of localized Dirac zero modes. They define a Berry connection around the Brillouin zone of the zero mode band which describes the local polarization of vacuum pairs, analogous to its role in topological insulator theory. In 4D Yang-Mills theory, the $θ$ domain walls are 2+1-dimensional Chern-Simons membranes. The interpretation of the $θ$ parameter as a Berry phase describing the local polarization of brane-antibrane pairs follows from the descent equations of 4D Yang-Mills theory.

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arXiv2013

Topological charge membranes and Goldstone boson propagation in QCD

Both theoretical arguments and Monte Carlo observations indicate that the topological structure of the QCD vacuum consists of a laminated array of extended, coherent codimension-one membranes of alternating sign. Large-$N_c$ arguments, supported by gauge/string holography, indicate that these membranes are domain walls which separate discrete "flux vacua" with values of the topological $θ$ parameter which differ by $\pm 2π$. This exposes a close analogy with 2D U(1) gauge theory, where $θ$ can be interpreted as electric polarization, and the domain walls are pointlike charged particles. In 4D QCD, the $θ$ parameter represents background Ramond-Ramond flux, which can be interpreted as a polarization of the charged membranes in the vacuum. In this framework, the chiral condensate is formed from the quark surface modes on the membranes. Massless Goldstone boson propagation takes place due to a coordination between bulk oscillations of the polarization field $θ$ and the surface currents represented by the Chern-Simons 3-form on the brane surface. This coordination is enforced by overall gauge invariance which imposes an anomaly inflow constraint between bulk and surface currents.

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arXiv2012

Anomaly Inflow and Membranes in QCD Vacuum

We study the membrane-like structure of topological charge density and its fluctuations in the QCD vacuum. Quark zero modes are localized on the membranes and the resultant gauge anomaly is cancelled by the gauge variation of a Chern-Simons type effective action in the bulk via the anomaly inflow mechanism. The coupling between brane fluctuations, described by the rotations of its normal vector, and the Chern-Simons current provides the needed anomaly inflow to the membrane. This coupling is also related to the axial U(1) anomaly which can induce brane punctures, and consequently quark-antiquark annihilation across the brane. As the Chern-Simons current has a long-range character, together with membranes it might lead to a solution to the confinement problem.

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arXiv2012

Anomaly Inflow and Membrane Dynamics in the QCD Vacuum

Large $N_c$ and holographic arguments, as well as Monte Carlo results, suggest that the topological structure of the QCD vacuum is dominated by codimension-one membranes which appear as thin dipole layers of topological charge. Such membranes arise naturally as $D6$ branes in the holographic formulation of QCD based on IIA string theory. The polarizability of these membranes leads to a vacuum energy $\propto θ^2$, providing the origin of nonzero topological susceptibility. Here we show that the axial U(1) anomaly can be formulated as anomaly inflow on the brane surfaces. A 4D gauge transformation at the brane surface separates into a 3D gauge transformation of components within the brane and the transformation of the transverse component. The in-brane gauge transformation induces currents of an effective Chern-Simons theory on the brane surface, while the transformation of the transverse component describes the transverse motion of the brane and is related to the Ramond-Ramond closed string field in the holographic formulation of QCD. The relation between the surface currents and the transverse motion of the brane is dictated by the descent equations of Yang-Mills theory.

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arXiv2011

Chiral Quark Dynamics and the Ramond-Ramond U(1) Gauge Field

Topological excitations in the QCD vacuum take the form of coherent codimension one sheets with positive and negative sheets juxtaposed in a dipole layer. These sheets may be interpreted as Luscher's "Wilson bags," which are domain walls between discrete quasivacua labelled by a local value of the $θ$ parameter equal to $2πk$, with $k$ given by the number of units of background Ramond-Ramond flux. This picture of the vacuum is closely analogous to Coleman's description of the 2D massive Schwinger model, where $θ$ is interpreted as a background electric field, and a pointlike charged particle is a domain wall between vacua which differ by one unit of background electric flux. The main effect of the Ramond-Ramond U(1) field in low energy QCD dynamics is the generation of phenomenologically important contact terms: (1) The dominant contact term in the topological charge correlator which leads to positive topological susceptibility, (2) The $η'$ mass insertion, and (3) An $SU(N_f)\times SU(N_f)$ invariant Nambu-Jona Lasinio 4-quark interaction.

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arXiv2011

Chiral quark dynamics and topological charge: The role of the Ramond-Ramond U(1) Gauge Field in Holographic QCD

The Witten-Sakai-Sugimoto construction of holographic QCD in terms of D4 color branes and D8 flavor branes in type IIA string theory is used to investigate the role of topological charge in the chiral dynamics of quarks in QCD. The QCD theta term arises from a compactified 5-dimensional Chern-Simons term on the D4 branes. This term couples the QCD topological charge to the Ramond-Ramond $U(1)$ gauge field of IIA string theory. The nonzero topological susceptibility of pure-glue QCD can be attributed to the presence of D6 branes, which constitute magnetic sources of the RR gauge field. The topological charge of QCD is required, by an anomaly inflow argument, to coincide in space-time with the intersection of the D6 branes and the D4 color branes. This clarifies the relation between D6 branes and the coherent, codimension-one topological charge membranes observed in QCD Monte Carlo calculations. Using open-string/closed-string duality, we interpret a quark loop (represented by a D4-D8 open string loop) in terms of closed-string exchange between color and flavor branes. The role of the RR gauge field in quark-antiquark annihilation processes is discussed. RR exchange in the s-channel generates a 4-quark contact term which produces an $η'$ mass insertion and provides an explanation for the observed spin-parity structure of the OZI rule. The $(\log {\rm Det\;U})^2$ form of the $U(1)$ anomaly emerges naturally. RR exchange in the t-channel of the $q\overline{q}$ scattering amplitude produces a Nambu-Jona Lasinio interaction which may provide a mechanism for spontaneous breaking of $SU(N_f)\times SU(N_f)$.

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arXiv2010

Tachyonic crystals and the laminar instability of the perturbative vacuum in asymptotically free gauge theories

Lattice Monte Carlo studies in SU(3) gauge theory have shown that the topological charge distribution in the vacuum is dominated by thin coherent membranes of codimension one arranged in a layered, alternating-sign sandwich. A similar lamination of topological charge occurs in the 2D $CP^{N-1}$ model. In holographic QCD, the observed topological charge sheets are naturally interpreted as $D6$ branes wrapped around an $S_4$.. With this interpretation, the laminated array of topological charge membranes observed on the lattice can be identified as a "tachyonic crystal", a regular, alternating-sign array of $D6$ and $\bar{D6}$ branes that arises as the final state of the decay of a non-BPS $D7$ brane via the tachyonic mode of the attached string. In the gauge theory, the homogeneous, space-filling $D7$ brane represents the perturbative gauge vacuum, which is unstable toward lamination associated with a marginal tachyonic boundary perturbation $\propto \cos(X/\sqrt{2α'})$. For the $CP^{N-1}$ model, the cutoff field theory can be cast as the low energy limit of an open string theory in background gauge and tachyon fields $A_μ(x)$ and $λ(x)$. This allows a detailed comparison with large $N$ field theory results and provides strong support for the tachyonic crystal interpretation of the gauge theory vacuum.

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arXiv2008

Melting Instantons, Domain Walls, and Large N

Monte Carlo studies of $CP^{N-1}$ sigma models have shown that the structure of topological charge in these models undergoes a sharp transition at $N=N_c\approx 4$. For $N N_c$ it is dominated by extended, thin, 1-dimensionally coherent membranes of topological charge, which can be interpreted as domain walls between discrete quasi-stable vacua. These vacua differ by a unit of background electric flux. The transition can be identified as the delocalization of topological charge, or "instanton melting," a phenomenon first suggested by Witten to resolve the conflict between instantons and large $N$ behavior. Implications for $QCD$ are discussed.

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arXiv2008

Fractionally charged Wilson loops as a probe of $θ$-dependence in $CP^{N-1}$ sigma models: Instantons vs. large N

The behavior of Wilson loops with fractional charge is used to study the $θ$-dependence of the free energy density $ε(θ)$ for the $CP^1$, $CP^5$, and $CP^9$ sigma models in two spacetime dimensions. The function $ε(θ)$ is extracted from the area law for a Wilson loop of charge $q=θ/2π$. For $CP^1$, $ε(θ)$ is smooth in the region $θ\approxπ$ and well-described by a dilute instanton gas throughout the range $0<θ<2π$. For $CP^5$ and $CP^9$ the energy exhibits a clear cusp and evidence for discrete, degenerate vacua at $θ= π$, as expected from large $N$ arguments. For $CP^9$ the $θ$-dependence is in good quantitative agreement with the leading order large $N$ prediction $ε(θ)={1/2}χ_tθ^2$ throughout the range $0<θ<π$.

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arXiv2007

Relics of Supersymmetry in Ordinary 1-flavor QCD: Hairpin Diagrams and Scalar-Pseudoscalar Degeneracy

The large-$N_c$ orientifold planar equivalence between $\mathcal{N}=1$ SUSY Yang-Mills theory and ordinary 1-flavor QCD suggests that low-energy quark-gluon dynamics in QCD should be constrained by the supersymmetry of the parent theory. One SUSY relic expected from orientifold equivalence is the approximate degeneracy of the scalar and pseudoscalar mesons in 1-flavor QCD. Here we study the role of the $q\bar{q}$ annihilation (hairpin) contributions to the meson correlators. These annihilation terms induce mass shifts of opposite sign in the scalar and pseudoscalar channels, making degeneracy plausible. Calculations of valence and hairpin correlators in quenched lattice QCD are consistent with approximate degeneracy, although the errors on the scalar hairpin are large. We also study the role of $q\bar{q}$ annihilation in the 1- and 2-flavor Nambu-Jona Lasinio model, where annihilation terms arise from the chiral field determinant representing the axial U(1) anomaly. Scalar-pseudoscalar degeneracy for the 1-flavor case reduces to a constraint on the relative size of the anomalous and non-anomalous 4-fermion couplings.

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arXiv2006

Topological Charge and the Laminar Structure of the QCD Vacuum

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arXiv2006

Double Hairpin Diagrams and the Planar Equivalence of N=1 Supersymmetric Yang-Mills Theory and One-Flavor QCD

Recent work by Armoni, Shifman, and Veneziano suggests a large-N equivalence between supersymmetric Yang-Mills Theory and one-flavor QCD. One consequence of this "orientifold projection" is that scalar and pseudoscalar mesons in one-flavor QCD should have degenerate mass since they lie within the same Wess-Zumino supermultiplet. We use lattice calculations to investigate the mass shifts caused by "double-hairpin" annihilation diagrams in quenched QCD to test for this degeneracy. Similar quark-antiquark annihilation processes are studied in the 2-dimensional CP(N-1) model with quenched fermions.

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arXiv2006

D-branes, Wilson Bags, and Coherent Topological Charge Structure in QCD

Monte Carlo studies of pure glue SU(3) gauge theory using the overlap-based topological charge operator have revealed a laminar structure in the QCD vacuum consisting of extended, thin, coherent, locally 3-dimensional sheets of topological charge embedded in 4D space, with opposite sign sheets interleaved. Studies of localization properties of Dirac eigenmodes have also shown evidence for the delocalization of low-lying modes on effectively 3-dimensional surfaces. In this talk, I review some theoretical ideas which suggest the possibility of 3-dimensionally coherent topological charge structure in 4-dimensional gauge theory and provide a possible interpretation of the observed structure. I begin with Luscher's ``Wilson bag'' integral over the 3-index Chern-Simons tensor. The analogy with a Wilson loop as a charged world line in 2-dimensional $CP^{N-1}$ sigma models suggests that the Wilson bag surface represents the world volume of a physical membrane. The large-N chiral Lagrangian arguments of Witten also indicate the existence of multiple ``k-vacuum'' states with discontinuous transitions between k-vacua at $θ=$ odd multiples of $π$. The domain walls between these vacua have the properties of a Wilson bag surface. Finally, I review the AdS/CFT duality view of $θ$ dependence in QCD. The dual realtionship between topological charge in gauge theory and Ramond-Ramond charge in type IIA string theory suggests that the coherent topological charge sheets observed on the lattice are the holographic image of wrapped D6 branes.

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arXiv2006

Small Instantons in $CP^1$ and $CP^2$ Sigma Models

The anomalous scaling behavior of the topological susceptibility $χ_t$ in two-dimensional $CP^{N-1}$ sigma models for $N\leq 3$ is studied using the overlap Dirac operator construction of the lattice topological charge density. The divergence of $χ_t$ in these models is traced to the presence of small instantons with a radius of order $a$ (= lattice spacing), which are directly observed on the lattice. The observation of these small instantons provides detailed confirmation of Lüscher's argument that such short-distance excitations, with quantized topological charge, should be the dominant topological fluctuations in $CP^1$ and $CP^2$, leading to a divergent topological susceptibility in the continuum limit. For the $\CP$ models with $N>3$ the topological susceptibility is observed to scale properly with the mass gap. These larger $N$ models are not dominated by instantons, but rather by coherent, one-dimensional regions of topological charge which can be interpreted as domain wall or Wilson line excitations and are analogous to D-brane or ``Wilson bag'' excitations in QCD. In Lorentz gauge, the small instantons and Wilson line excitations can be described, respectively, in terms of poles and cuts of an analytic gauge potential.

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arXiv2005

Coherent Topological Charge Structure in $CP^{N-1}$ Models and QCD

In an effort to clarify the significance of the recent observation of long-range topological charge coherence in QCD gauge configurations, we study the local topological charge distributions in two-dimensional $CP^{N-1}$ sigma models, using the overlap Dirac operator to construct the lattice topological charge. We find long-range sign coherence of topological charge along extended one-dimensional structures in two-dimensional spacetime. We discuss the connection between the long range topological structure found in $CP^{N-1}$ and the observed sign coherence along three-dimensional sheets in four-dimensional QCD gauge configurations. In both cases, coherent regions of topological charge form along membrane-like surfaces of codimension one. We show that the Monte Carlo results, for both two-dimensional and four-dimensional gauge theory, support a view of topological charge fluctuations suggested by Luscher and Witten. In this framework, the observed membranes are associated with boundaries between ``k-vacua,'' characterized by an effective local value of $θ$ which jumps by $\pm 2π$ across the boundary.

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arXiv2005

D-branes and Topological Charge in QCD

The recently observed long-range coherent structure of topological charge fluctuations in QCD is compared with theoretical expectations based on the AdS/CFT brane construction of nonsupersymmetric gauge theory by Witten. Similar observations of coherent topologicalcharge structure in 2D $CP^{N-1}$ sigma models are interpreted in terms of Wilson lines representing world lines of screened electric charges. The analogy between 2D U(1) and 4D Yang-Mills theory leads to the interpretation of the observed coherent sheets of topological charge in QCD as screened ``Wilson bags'' first suggested by Luscher. The duality between the Wilson bag surface and a wrapped 6-brane in IIA string theory is discussed. The complete screening of the force between bag surfaces for integer values of the bag charge $θ/2π$ corresponds to the observation by Polchinski that the net force between d-branes from closed string exchange vanishes for quantized values of Ramond-Ramond charge.

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