Color Superconducting Phases of Cold Dense Quark Matter by Jeffrey Allan Bowers Abstract

180 Pages·2003·1.1 MB·English

Color Superconducting Phases of Cold Dense Quark Matter
by
Je(cid:11)rey Allan Bowers
Submitted to the Department of Physics
at the Massachusetts Institute of Technology
on May 6, 2003, in partial ful(cid:12)llment of the
requirements for the degree of
Doctor of Philosophy in Physics
Abstract
We investigate color superconducting phases of cold quark matter at densities rele-
vant for the interiors of compact stars. At these densities, electrically neutral and
weak-equilibrated quark matter can have unequal numbers of up, down, and strange
quarks. The QCD interaction favors Cooper pairs that are antisymmetric in color
and in flavor, and a crystalline color superconducting phase can occur which accom-
modates pairing between flavors with unequal number densities. In the crystalline
color superconductor, quarks of di(cid:11)erent flavor form Cooper pairs with nonzero to-
tal momentum, yielding a condensate that varies in space like a sum of plane waves.
Rotationalandtranslationalsymmetryarespontaneouslybroken. WeuseaGinzburg-
Landau method to evaluate candidate crystal structures and predict that the favored
structure is face-centered-cubic. We predict a robust crystalline phase with gaps
comparable in magnitude to those of the color-flavor-locked phase that occurs when
the flavor number densities are equal. Crystalline color superconductivity will be a
generic feature of the QCD phase diagram, occurring wherever quark matter that is
not color-flavor locked is to be found. If a very large flavor asymmetry forbids even
the crystalline state, single-flavor pairing will occur; we investigate this and other
spin-one color superconductors in a survey of generic color, flavor, and spin pairing
channels. Our predictions for the crystalline phase may be tested in an ultracold gas
of fermionic atoms, where a similar crystalline superfluid state can occur. If a layer
of crystalline quark matter occurs inside of a compact star, it could pin rotational
vortices, leading to observable pulsar glitches.
Thesis Supervisor: Krishna Rajagopal
Title: Associate Professor of Physics
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Acknowledgments
I cannot adequately express my gratitute to my mentor, teacher, collaborator, and
friend, Krishna Rajagopal, for years of patient advice and inspiration. Very special
thanks also to Mark Alford, for close collaboration and generous advice. Thanks to
Frank Wilczek for helpful discussion and guidance. Thanks to him and to Wit Busza
for serving on my thesis committee and for taking the time to review this manuscript.
Thanks to Jack M. Cheyne and Greig A. Cowan for their collaboration on the spin-
one calculations of Chapter 4 and Appendix B. Thanks to Paulo Bedaque, Michael
Forbes, Elena Gubankova, Robert Ja(cid:11)e, Chris Kouvaris, Joydip Kundu, Vincent Liu,
Dirk Rischke, Thomas Sch¨afer, Eugene Shuster, Dam Son, and Christof Wetterich for
manyenlightening conversations. This research wassupported inpart bytheU.S.De-
partment of Defense (D.O.D.) National Defense Science and Engineering Graduate
Fellowship Program, by the Kavli Institute for Theoretical Physics (KITP) Graduate
Fellowship Program, by the U.S. Department of Energy (D.O.E.) under cooperative
research agreement #DF-FC02-94ER40818, and by the National Science Foundation
under Grant No. PHY99-07949. I am grateful to the Kavli Institute for Theoreti-
cal Physics (KITP) and the Institute for Nuclear Theory (INT) at the Univeristy of
Washington for their hospitality and support during the completion of much of this
work.
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Contents
1 Introduction 7
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 The phase diagram of QCD . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 High density and color-flavor-locking . . . . . . . . . . . . . . . . . . 11
1.4 Intermediate density and unlocking . . . . . . . . . . . . . . . . . . . 14
1.5 Crystalline color superconductivity . . . . . . . . . . . . . . . . . . . 23
1.6 Single-flavor color superconductivity . . . . . . . . . . . . . . . . . . 37
1.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.7.1 Compact stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.7.2 Atomic physics . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2 Crystalline Superconductivity: Single Plane Wave 47
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 The LOFF plane wave ansatz . . . . . . . . . . . . . . . . . . . . . . 48
2.3 The gap equation and free energy . . . . . . . . . . . . . . . . . . . . 54
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.5 More general Hamiltonian and ansatz . . . . . . . . . . . . . . . . . . 68
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 Crystalline Superconductivity: Multiple Plane Waves 75
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2.1 The gap equation . . . . . . . . . . . . . . . . . . . . . . . . . 76
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3.2.2 The Ginzburg-Landau approximation . . . . . . . . . . . . . . 81
3.2.3 The free energy . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.2 One wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.3.3 Two waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3.4 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.3.5 Crystal structures with intersecting rings . . . . . . . . . . . . 99
3.3.6 \Regular" crystal structures . . . . . . . . . . . . . . . . . . . 100
3.3.7 Varying continuous degrees of freedom . . . . . . . . . . . . . 102
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4 Single Color and Single Flavor Color Superconductivity 110
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 Mean-(cid:12)eld survey of quark pairing channels . . . . . . . . . . . . . . 112
4.2.1 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2.2 Properties of the pairing channels . . . . . . . . . . . . . . . . 113
4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3 Gap calculations for the attractive diquark channels . . . . . . . . . . 118
4.4 Quasiquark dispersion relations . . . . . . . . . . . . . . . . . . . . . 122
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 Applications and Outlook 132
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Pulsar glitches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 Ultracold Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A Evaluating J and K Integrals 152
B Spin-One Calculations 156
B.1 Calculational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
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B.2 Gap equation summary . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B.2.1 Cγ and C gap equations . . . . . . . . . . . . . . . . . . . . 158
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B.2.2 C(cid:27) and C(cid:27) γ gap equations . . . . . . . . . . . . . . . . . 159
03 03 5
B.2.3 C((cid:27) (cid:6)i(cid:27) ) gap equation . . . . . . . . . . . . . . . . . . . . 159
01 02
B.2.4 Cγ gap equation . . . . . . . . . . . . . . . . . . . . . . . . . 160
3
B.2.5 Cγ γ gap equation . . . . . . . . . . . . . . . . . . . . . . . . 160
3 5
B.2.6 C(γ (cid:6)iγ ) gap equation . . . . . . . . . . . . . . . . . . . . . 161
1 2
B.2.7 Cγ γ gap equation . . . . . . . . . . . . . . . . . . . . . . . . 161
0 5
B.3 Orbital/spin content of the condensates . . . . . . . . . . . . . . . . . 162
References 166
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Chapter 1
Introduction
1.1 Overview
In this thesis we shall discuss the behavior of cold quark matter at densities that
are relevant for the interiors of compact stars. It is well known that cold dense
quark matter is unstable to the formation of a condensate of quark Cooper pairs,
making it a color superconductor. Various phases of color superconductivity have
been proposed, and in section 1.2 we review the phase diagram of QCD to provide a
larger context for a discussion of the color superconducting phases. In section 1.3 we
discuss the color-flavor-locked (CFL) color superconductor, the ground state of cold
quark matter at very high densities. In section 1.4 we describe how the CFL phase
can be disrupted at intermediate densities that are relevant for compact stars. At
these intermediate densities, neutral quark matter has unequal numbers of up, down,
and strange quarks, and a crystalline color superconducting phase is favored. The
crystallinecolorsuperconductor hastheremarkablevirtueofallowingpairingbetween
quarks with unequal Fermi surfaces. Cooper pairs with nonzero total momentum are
favored; the condensate spontaneously breaks translational and rotational invariance,
leading to gaps which vary periodically in a crystalline pattern as a superposition
of plane waves. In section 1.5 we discuss the crystalline phase and look ahead to
the detailed calculations of chapters 2 and 3 where we investigate single-plane-wave
and multiple-plane-wave crystalline phases, respectively. In section 1.6 we discuss
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single-flavor color superconductivity, which might occur when there is a very large
flavor asymmetry that forbids the crystalline state. We also preview chapter 4, which
presents alargersurvey ofvarioussingle-colorandsingle-flavor colorsuperconductors.
Many of these are spin-one phases with unusual spectra of elementary excitations.
Finally, in section 1.7 we discuss physical contexts in which the crystalline phase may
occur with observable consequences. In 1.7.1 we review the astrophysical implications
of color superconductivity for compact stars. If a layer of crystalline quark matter
occurs inside of a compact star, it could pin rotational vortices, leading to observable
pulsar glitches. In 1.7.2, we describe how a crystalline superfluid might be created
and detected in a trapped gas of ultracold fermions. These are previews of more
detailed discussions of glitches and cold atoms that appear in chapter 5.
1.2 The phase diagram of QCD
In recent years much theoretical and experimental e(cid:11)ort has been devoted to under-
standing the behavior of quantum chromodynamics (QCD) in extreme conditions of
very high temperature or density. Because QCD is asymptotically free [1], its high
temperature and high density phases are readily described in terms of quark and
gluon degrees of freedom [2]. At high temperature, the familiar hadronic phase of
QCD gives way to a decon(cid:12)ned plasma of quarks and gluons, in which all the sym-
metries of the QCD Lagrangian are unbroken [3]. This phase preceded hadronization
in the early universe, and e(cid:11)orts are underway to produce and probe this phase in
thermalized collisions of relativistic heavy ions at Brookhaven and CERN labora-
tories [4]. At low temperature and high density, on the other hand, the hadronic
phase gives way to a degenerate Fermi system of quarks. Under the influence of
the QCD interaction, quarks near the Fermi surface can bind together as Cooper
pairs, which condense by the BCS mechanism [5] to form a color superconduc-
tor [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. While not accessible in the lab-
oratory, this cold dense quark matter might occur inside compact stars, with a host
of potential astrophysical implications [19].
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