David K. Campbell is one of the founders of “nonlinear science”—the systematic study of inherently nonlinear phenomena in the natural world—and has made contributions to many subfields of physics, ranging from quantum field theory through nuclear and condensed matter physics to computational and mathematical physics. A central theme of his work is the role of nonlinear, localized excitations—”solitons,” “breathers,” “polarons/bipolarons,” “Intrinsic Localized Modes”—in diverse physical systems, but accompanying this theme have been many variations and extensions that have influenced both theory and experiment. His principal contributions are grouped below in two broad topical areas: nonlinear science and novel states of matter. The numbers in brackets indicate the respective publications in the bibliography accompanying his Curriculum Vitae.

- 1Nonlinear Science
- 1.1Nonlinear Excitations in Field Theory and Condensed Matter Physics
- 1.2Interactions of Solitary Waves in Non-Integrable Models
- 1.3"Discrete Breathers" and Intrinsic Localized Modes
- 1.4Reviews and Educational Articles on Nonlinear Science
- 2Novel States of Matter
- 2.1Pion Condensation in Nuclear Physics and Astrophysics
- 2.2Competing Ground States in Strongly Correlated Electronic Systems in Reduced Dimensions
- 2.3Graphene and Related Quasi-2D Electronic Membranes
- 2.4Bose Einstein Condensates (BECs)

# Nonlinear Science

## Nonlinear Excitations in Field Theory and Condensed Matter Physics

Campbell’s initial contribution to nonlinear science was a study of the nature of localized, nonlinear excitations in a model relativistic field theory—the “sigma” model—in one space dimension and time. Campbell extended the (then) recently developed “inverse scattering” technique to this theory [14, 15], which he showed was a caricature of both the “SLAC bag” and of a meson model of nuclear physics [18]. Generalizing an earlier result due to Faddeev, he proved the important “trace identities” for the Dirac equation and, using these results, established the existence of two types of nonlinear excitations, corresponding, in the nuclear physics model, to normal nuclei and to Lee-Wick “abnormal” nuclei. He further showed that these localized nonlinear excitations could, in certain instances, become unstable to quantum tunneling decay from one type of excitation to another [26]. Apart from providing critical insight into the power (and limitations) of semi-classical/TDHF methods, this work was a direct precursor of the later, widely studied (3+1)-dimensional “skyrmion” models of nuclear physics.

- “Exact Classical Solutions of the Two-Dimensional Sigma Model”,
*Phys. Lett.***B64**, 187-190 (1976) - “A Semi-Classical Analysis of Bound States in the Two-Dimensional Sigma Model”,
*Phys. Rev. D***14**, 2093-2116 (1976), with Y.-T. Liao. - “Nuclear Physics in One Dimension,” pp. 673-716 in
*Nuclear Physics with Heavy Ions and Mesons*, R. Balian, M. Rho, and G. Ripka, eds., (North Holland, 1978). - “Tunneling Decay of Unstable Particles in a Quantum Field Theoretic Model,”
*Annals of Physics***129**249-272 (1980).

In the early 1980s, Campbell recognized a correspondence between these relativistic field theories in one space dimension and time and condensed matter physics models for highly anisotropic, quasi-one-dimensional solid state materials, including the chain-like conducting polymers such as polyacetylene [27, 30, 31]. Using this correspondence, Campbell (with several different collaborators) predicted the existence of a wide range of nonlinear excitations—”kink” solitons, “two-band” polarons, bipolarons, “breathers,” polar-excitons—and used their properties to explain and interpret experimental results in conducting polymers. His work on many aspects of optical absorption—for example, singlet and triplet optical absorption, photo-induced absorption, photo-conductivity, and photo-luminescence [34,41]—was particularly influential, both experimentally and theoretically. Specifically, the work on optical properties of polarons and bipolarons influenced the experimental studies that led to the development of polymeric light-emitting diodes, which have been produced commercially. To compare with real materials, Campbell recognized the need to go beyond the elegant but limiting analytic results of the original (1+1)-dimensional models assuming independent electrons and explored numerically the effects of electron-electron correlations in these materials [43, 83] and developing a fully (3+1)-dimensional model for polyacetylene [71,78].

- “Solitons in Polyacetylene and Relativistic Field Theory Models,”
*Phys. Rev. B***24**, 4859–4862 (1981), with A. R. Bishop. - “Soliton Excitations in Polyacetylene and Relativistic Field Theory Models,”
*Nuc. Phys. B***200(FS4)**, 297–328 (1982), with Alan Bishop. - “Polarons in Quasi-One-Dimensional Systems,”
*Phys. Rev. B***26**, 6862–6874 (1982), with A. R. Bishop and K. Fesser. - “Optical Absorption from Polarons in a Model of Polyacetylene,”
*Phys. Rev. B***27**, 4804–4825 (1983), with K. Fesser and A. R. Bishop. - “Breathers and Photoinduced Absorption in Polyacetylene,”
*Phys. Rev. Lett.***52**, 671–674 (1984), with A. R. Bishop, P. S. Lomdahl, B. Horovitz, and S. R. Phillpot. - D. K. Campbell, T. A. DeGrand, and S. Mazumdar, “Soliton Energetic in Peierls-Hubbard Models,”
*Phys. Rev. Lett.***52**, 1717–1720 (1984). - D. K. Campbell, J. Tinka Gammel, and E. Y. Loh, Jr., “Modeling electron-electron interactions in reduced-dimensional materials: Bond charge Coulomb repulsion and dimerization in Peierls-Hubbard models,”
*Phys. Rev. B***42**, 475–492 (1990). - “Three-Dimensional Structure and Structure and Intrinsic Defects in trans-Polyacetylene,”
*Phys. Rev. Lett.***62**, 2012–2015 (1989), with P. Vogl. - “First-Principles Calculations of the Three-Dimensional Structure and Intrinsic Defects in trans-Polyacetylene,”
*Phys. Rev. B***41**, 12797-12817 (1990), with Peter Vogl.

## Interactions of Solitary Waves in Non-Integrable Models

In addition to his extensive work on the physical applications of “solitons,” Campbell made an important contribution to the understanding of the interactions of nonlinear waves. True solitons, which exist only in completely integrable Hamiltonian systems, are famous (indeed, were named) for the remarkable property of emerging unchanged from interactions despite their highly nonlinear nature. In contrast, there is no general behavior (or theory) of the interactions of other solitary waves, which exist in a wide variety of nonlinear systems. In a series of papers beginning in the early 1980s, Campbell (with several colleagues) addressed this problem using both analytic and computational techniques. His most important result in this area was the discovery of a surprising “resonance” phenomenon in the scattering interactions of kink-like solitary waves in a non-integrable classical field theory (the “F4” theory) [36] and, later, in other non-integrable theories [37,53]. Campbell developed a heuristic theory—the “resonance energy exchange mechanism”—which both explained the initial numerical simulations and predicted the existence, later confirmed, of an intricate, fractal-like hierarchy of higher resonances. In these papers, Campbell also developed a “collective coordinate” approach, which was subsequently widely used by the nonlinear community. These results remain of interest, as shown by an article validating Campbell’s heuristic approach by a controlled asymptotic analysis [601].

- “Resonance Structure in Kink-Antikink Interactions in F4 Theory,”
*Physica D***9**, 1–32 (1983), with J. F. Schonfeld and C. A. Wingate. - “Kink-Antikink Interactions in a Modified Sine-Gordon Model,”
*Physica D***9**, 33–51 (1983), with M. Peyrard. - “Kink-Antikink Interactions in the Double Sine-Gordon Equation,”
*Physica D***19**, 165–205 (1986), with M. Peyrard and P. Sodano. - Roy H. Goodman and Richard Haberman, “Chaotic Scattering and n-bounce resonance in solitary wave interactions,”
*Phys. Rev. Lett.***98**, 104103 (2007).

## “Discrete Breathers” and Intrinsic Localized Modes

Campbell’s third major research contribution to our understanding of nonlinear, localized excitations concerned the existence of spatially extended, time periodic (“breather”) solutions to nonlinear equations. Using a combination of numerical simulations and asymptotic perturbative results, Campbell showed (in collaboration with Peyrard [77]) that while continuum non-integrable theories should not exhibit “breathers,” discrete models—such as those described by solid state lattices—could indeed support such excitations: this was the first mention in the literature of the subject of “discrete breathers.” Together with references [57] and [108], which showed how nonlinearity itself could produce localized excitations in a dimer and on a lattice (respectively), these results were direct precursors of the current flood of articles on “discrete breathers/intrinsic localized modes” (DBs/ILMs) (see the discussion in [Ex9]). The observations of ILMs in experiments ranging from optical wave guides through antiferromagnetic solids to micromechanical arrays have confirmed the theoretical predictions of Campbell and others. More recently, Campbell has studied ILMs in Bose-Einstein Condensates (BECs) trapped in optical lattices.

- “Self-trapping on a nonlinear dimer: Time-Dependent Solutions of a discrete nonlinear Schrödinger Equation,”
*Phys. Rev. B***34**, 4959–4961 (1986), with V. M. Kenrke. - “Chaos and Order in Non-Integrable Model Field Theories,” 305–334 in
*CHAOS/XAOC: Soviet-American Perspectives on Nonlinear Science*, D. K. Campbell, ed. (A.I.P., New York, 1990), with M. Peyrard. - “Peierls-Nabarro Potential Barrier for Highly Localized Nonlinear Modes,”
*Phys. Rev. E***48**, 3077–3081 (1993), with Yuri S. Kivshar. - “Localizing Energy Through Nonlinearity and Discreteness,”
*Physics Today*, 43–49 (January 2004), with Sergej Flach and Yuri Kivshar. - “Transfer of BECs through discrete breathers in an optical lattice,”
*Phys. Rev. A***82**, 053604 (2010).

## Reviews and Educational Articles on Nonlinear Science

Beyond his individual research contributions, Campbell has been an influential leader in the development and popularization field of “nonlinear science.” He led the efforts to establish the Center for Nonlinear Studies (CNLS) at Los Alamos National Laboratory, one of the first centers focused on nonlinear phenomena, and served as its initial co-Director (1981) and later its Director (1985–1992). For these efforts, he received the Laboratory’s “Distinguished Performance Award.” At the CNLS and elsewhere, he organized numerous international meetings on nonlinear phenomena, including the influential series of meetings sponsored by the American Institute of Physics that brought together experts in nonlinear science from the US and the (then) Soviet Union (1989–1993). He served as the US representative to the NATO Special Program Panel on Chaos, Order, and Patterns: Aspects of Nonlinearity (1988–1993). He is the founding Editor-in-Chief of the American Institute of Physics journal *Chaos: An Interdisciplinary Journal of Nonlinear Science* (1990–present). He served as Chair of the American Physical Society’s Group on Statistical and Nonlinear Physics. Through his numerous invited talks at international meetings, his four years of lecturing at the Santa Fe Summer School (see lecture notes [72]), his edited volumes, his influential overview articles [64, 403, 404, 414], and his published software [301], Campbell helped establish the interdisciplinary organizing principles—the paradigms of solitons, chaos, patterns, and adaptation—that have defined the research agenda in nonlinear science for more than two decades. For these and related efforts, Campbell was awarded the 2010 Julius Edgar Lilienfeld Prize of the American Physical Society.

- D. K. Campbell, “Chaos: Chto Delat?,”
*Nuc. Phys. B (Proc. Suppl.)***2**, 541–562 (1987). - D. K. Campbell, “Introduction to Nonlinear Phenomena,” pp. 3-105 in
*Lectures in the Sciences of Complexity*, SFI Studies in the Sciences of Complexity, D. Stein, ed., (Addison-Wesley Longman, 1989). - David Campbell, Doyne Farmer, Jim Crutchfield, and Erica Jen, “Experimental Mathematics: The Role of Computation in Nonlinear Science,”
*Comm. Of the Assoc. for Comp. Machinery***28**, 374–384 (1985). - D. K. Campbell, “Nonlinear Science: From Paradigms to Practicalities,”
*Los Alamos Science***15**, 218–262 (1987); reprinted in*From Cardinals to Chaos: Reflections on the Life and Legacy of Stanislaw Ulam*(Cambridge University Press, 1989). - David K. Campbell, Andrew Charneski, Gamaliel Lodge, Sebastian M. Marotta, Gary Tam, and Tom Tanury,
*Nonlinear Dynamics: A Mathematica Lab Notebook*, (published as a DVD by Cambridge University Press, November 2012) - Adilson E. Motter and David K. Campbell, “Chaos at Fifty,”
*Physics Today*27–33 (May 2013).

# Novel States of Matter

## Pion Condensation in Nuclear Physics and Astrophysics

Campbell’s training in theoretical elementary particle physics provided a strong background in both quantum field theory and symmetries. In the mid 1970s (in collaboration with Dashen, Manassah, and others), he applied the field theoretic techniques of chiral symmetry to the phenomenon of “pion condensation,” a (proposed) exotic state of high density nuclear or neutron-rich matter [11, 12]. Apart from providing a general framework for pion condensation, Campbell’s work also proposed observational tests for this phenomenon based on neutron star cooling rates [16] and established the link between it and other exotic nuclear states, such as the “abnormal” matter proposed by Lee and Wick. His review article (with Baym) [22] strongly influenced later studies of matter at high densities, including recent work on quark matter, “kaon”-condensates, and soliton stars.

- “Chiral Symmetry and Pion Condensation: I. Model-dependent Results,”
*Phys. Rev. D***12**, 979–1009 (1975), with R. F. Dashen and J. T. Manassah. - “Chiral Symmetry and Pion Condensation: II. General Formalism,”
*Phys. Rev. D***12**, 1010–1025 (1975), with R. F. Dashen and J. T. Manassah. - “Beta Decay of Pion Condensates as a Cooling Mechanism for Neutron Stars,”
*Astrophys. J***216**, 77–85 (1977), with O. E. Maxwell, G.E. Brown, R. F. Dashen, and J. T. Manassah. - “Chiral Symmetry and Pion Condensation,” pp. 1031–1094 in Vol. III of
*Mesons in Nuclei*, M. Rho and D. Wilkinson, eds. (North Holland, 1979), with Gordon Baym.

In the early 80s Campbell was among the first in the physics community to argue for the importance of electron-electron interaction effects in quasi-one- and quasi-two-dimensional novel electronic materials. Accordingly, he (working with a sequence of collaborators, including Mazumdar, DeGrand, and others) focused much of his research on the interplay between nonlinearity and many-body effects in a range of novel materials, including conducting polymers, organic superconductors, antiferromagnets, and high Tc superconductors. The effective reduced dimensionality and strong electron-electron interactions both enhance the consequences of nonlinearity and generally invalidate mean field theory, making essential the use of full, non-perturbative many-body methods. Campbell, with several collaborators, used both analytical (strong coupling, Bethe Ansatz) and numerical (Lanczos “exact” diagonalization and quantum Monte Carlo) methods to obtain a number of experimentally relevant results. In [43] (with DeGrand and Mazumdar), he demonstrated the stability of the dimerized ground state and the persistence of soliton excitations in the presence of strong electron-electron interactions. With Loh and Gammel, Campbell studied “off-diagonal” electron-electron interactions and showed that, contrary to earlier assertions, these effects did not destroy dimerization/bond order for physically consistent values of the parameters [83]. With Baeriswyl and Mazumdar, he provided a comprehensive overview [95] of the theoretical understanding of conducting polymers, treating electron-phonon and electron-electron interactions on an equal basis and reconciling the differing perspectives of chemists and physicists. In a series of papers with several colleagues, Campbell explored the relations among the many exotic broken symmetry phases-charge density waves (CDW), spin density waves (SDW), and bond order waves (BOW) that exist in charge-transfer solids and related materials [163, 169, 174]. In particular, he explained [150] the nature of a “previously mysterious” mixed density wave state that occurs at low temperature in a wide range of organic charge transfer solids, including many that become superconducting under pressure. More recently, again with several colleagues, Campbell studied “striped” phases in high-temperature superconductors [190] and developed a “Multiscale Functional Renormalization Group” (MFRG) approach to study a variety of strongly correlated systems [184,185,192]. Further MFRG studies are one of Campbell’s present research interests [209].

- “Soliton Energetics in Peierls-Hubbard Models,”
*Phys. Rev. Lett.***52**, 1717–1720 (1984), with T. A. DeGrand and S. Mazumdar. - “Modeling electron-electron interactions in reduced-dimensional materials: Bond charge Coulomb repulsion and dimerization in Peierls-Hubbard models,”
*Phys. Rev. B***42**, 475–492 (1990), with J. T. Gammel and E. Y. Loh, Jr. - “An Overview of the Theory of π-Conjugated Polymers,” pp. 7–134 in
*Conjugated Conducting Polymers*, H. Kiess, ed. (Springer Verlag, 1992), with D. Baeriswyl and S. Mazumdar. - S. Mazumdar, R. T. Clay and D. K. Campbell, “Bond and charge density waves in the isotropic interacting two-dimensional quarter-filled band and the insulating state proximate to organic superconductivity,”
*Phys. Rev. B***62**, 13400–13425 (2000). - Pinaki Sengupta, Anders Sandvik and D. K. Campbell, “Bond-order-wave phase and quantum phase transitions in the one-dimensional extended Hubbard model,”
*Phys. Rev. B***65**, 155113/1-18 (2002). - R. T. Clay, S. Mazumdar, and D. K. Campbell, “Pattern of charge ordering in quasi-one-dimensional charge transfer solids,”
*Phys. Rev. B***67**, 115121/1-9 (2003). - “Theory of Coexisting Charge- and Spin-Density Waves in (TMTTF)
_{2}Br, (TMTSF)_{2}PF_{6}, and α-(BEDT-TTF)_{2}MHg(SCN)_{4}“*Phys. Rev. Lett.***82**, 1522–1525 (1999), with S. Mazumdar, S. Ramasesha, and R. T. Clay. - S. W. Tsai, A.H.C. Neto, R. Shankar, and D. K. Campbell, “Renormalization-group approach to strong-coupled superconductors,”
*Phys. Rev. B***72**, 054531 (2005). - K. M. Tam, S. W. Tsai, and D. K. Campbell, “Functional renormalization group analysis of the half-filled one-dimensional extended Hubbard model,”
*Phys. Rev. Lett.***96**036408 (2006). - D. X. Yao, E. W. Carlson, and D. K. Campbell. Magnetic excitations of stripes near a quantum critical point,”
*Phys. Rev. Lett.***97**, 017003 (2006). - Ka-Ming Tam, S.-W. Tsai, D. K. Campbell, and A. H. Castro Neto, “Retardation effects in the Holstein-Hubbard chain at half-filling,”
*Phys. Rev. B***75**, 161103(R) (2007). - Ka-Ming Tam, Shan-Wen Tsai, and David K. Campbell, “Dominant superconducting fluctuations in the 1D extended Holstein-extended Hubbard model,”
*Phys. Rev. B*(in press, 2014).

## Graphene and Related Quasi-2D Electronic Membranes

Campbell’s long-time interest in novel electronic materials has in recent years led him to study graphene and related quasi-two-dimensional electronic membranes, such as MoS_{2}. This is one of Campbell’s present research interests. A few selected publications include:

- S. Viola-Kusminsky, Johan Nilsson, D. K. Campbell, and A. H. Castro Neto, “Electronic Compressibility of a graphene bilayer,”
*Phys. Rev. Lett.***100**, 106805 (2008). - S. Viola-Kusminsky, D. K. Campbell, and A. H. Castro Neto, “Electron-electron interactions in graphene bilayers,”
*EPL***85**, 58005 (2009). - S. Viola Kusminsky, D. K. Campbell, A. H. Castro Neto, F. Guinea, “Pinning of a 2D Membrane on top of a patterned substrate: the case of graphene,”
*Phys. Rev. B***83**, 165405 (2011). - Zenan Qi, D. A. Bahamon, Vitor M. Pereira, Harold S. Park, D. K. Campbell, and A. H. Castro Neto, “Resonant Tunneling in Graphene Pseudomagnetic Quantum Dots,”
*Nano Lett.***13**, 2692–2697 (2013).

## Bose Einstein Condensates (BECs)

Most recently, Campbell has studied Bose Einstein Condensares (BECs) trapped in optical lattices. He and his collaborators have focused on understanding the effects of ILMs predicted to exist by the Gross-Pitaevsky (mean-field) model for these systems and the extent to which these effects persist in a fully quantum treatment of the problem. This is one of Campbell’s present research interests.

- Holger Hennig, Jerome Dorignac, and David K. Campbell, “Transfer of BECs through discrete breathers in an optical lattice,”
*Phys. Rev. A***82**, 053604 (2010). - Holger Hennig, Dirk Witthaut, and David K Campbell, “Global phase space of coherence and entanglement in a double-well Bose-Einstein condensate,”
*Phys. Rev. A***86**, 051640(R) (2012). - Tadeusz Pudlik, Holger Hennig, D. Witthaut, and David K. Campbell, “Dynamics of entanglement in a dissipative Bose-Hubbard dimer,”
*Phys. Rev. A***88**, 063606 (2013).