Schmittner, Sebastian E. (2015). Models for disordered phonons. PhD thesis, Universität zu Köln.

2015-01-26-phd-thesis-schmittner--PRINT-FINAL.pdf - Published Version
Bereitstellung unter der CC-Lizenz: Creative Commons Attribution Non-commercial.

Download (1MB)


We introduce novel models for disordered solids at very low temperatures with the goal of representing the physics of disorder causing local random fluctuations in the masses and inter-constituent forces. The basic model that provides the clean limit upon which the other models are built is a variant of the harmonic crystal. Neglecting the difference between longitudinal and transverse vibrational modes, known as phonons, we use a large number, n, of phonon bands. All models are constructed within a harmonic approximation for small oscillations about a stable ground state. That means that runaway dynamics are excluded, but interactions between phonons are neglected. Disorder is added by way of 14 suitable ensembles of local (hence sparse) random matrices. Each of these ensembles comes with a non-negative variance interpreted as the disorder strength. In principle, any of these disorder types can be combined, hence we obtain a model kit of disordered phonons. To assess the building blocks in our kit, we study the average density of states in the large-n limit, known as the coherent potential approximation (CPA). It is determined self-consistently by means of illustrative diagrammatic as well as rigorous supersymmetric techniques. The validity of this mean-field type approximation in three dimensions is verified by comparison with exact numerical diagonalisation. Two of our elementary models, one with pair potential (or "spring constant") and one with mass disorder, are particularly interesting. The model with strong spring constant and weak mass fluctuations is microscopically well motivated and leads to a phase transition (at finite disorder strength). A critical point terminates the weak-disorder (or Debye) phase, above which a strong-disorder phase is discovered. This novel phase features a finite density of states at zero energy. The two-parameter space of models combining both of these disorder types is called interfering mass and spring constant disorder (IMSC) model. Unlike the widely used two-level tunnelling system (TTLS) model, we do not make any a priori assumption about a finite density of scatterers at low energies. Hence we show that disorder alone is, in principle, sufficient to explain the experimental finding of the heat capacity of vitreous systems varying linearly with temperature below about 1K, first observed in 1971. The new strong-disorder phase is stable under the addition of most of the other disorder types. Assuming that the IMSC model is renormalisable and that critical points of the renormalisation group (RG) flow are correctly identified from the CPA, we predict a tentative RG flow diagram. Under these assumptions, the novel phase is also stable under the RG flow.

Item Type: Thesis (PhD thesis)
Translated title:
Modelle für ungeordnete PhononenGerman
Translated abstract:
Wir konstruieren im Rahmen dieser Arbeit neuartige Modelle fürungeordnete Festkörper bei sehr tiefen Temperaturen. Diese beschreiben harmonische Schwingungen um einen stabilen Grundzustand, deren Normalmoden als Phononen quantisieren. Stabilität bedeutet hier, dass anfänglich kleine Auslenkungen aus der Gleichgewichtslage unter Zeitentwicklung klein bleiben, was eine notwendige Voraussetzung für die Vernachlässigung der Wechselwirkungen zwischen Phononen ist. Das zugrundeliegende Modell des harmonischen Kristalls stellt den sauberen Grenzfall verschwindender Unordnung in allen Modellen dar. Wir vernachlässigen den Unterschied zwischen longitudinalen und transversalen Schwingungsmoden und verallgemeinern den harmonischen Kristall zu einem Modell mit einer beliebigen Anzahl, n, lokaler Moden, resultierend in n Phononen-Bändern. Es werden 14 verschiedene Ensembles lokaler Zufallsmatrizen zur Modellierung der Unordnung benutzt, jedes mit einer nicht negativen Varianz (Unordnungsstärke). Durch Unordnung verursachte lokale Fluktuationen in den Massen und in den harmonischen Kräften werden durch diese Zufallsvariablen möglichst wirklichkeitsnah in die Modelle einbezogen. Alle Unordnungstypen in diesem Modellbaukasten ungeordneter Phononen können beliebig kombiniert werde. Um die einzelnen Modelle hinsichtlich ihrer Realitätsnähe und Vorhersagekraft zu bewerten berechnen wir die Zustandsdichte asymptotische für große n. Dies entspricht einer als "coherent potential approximation" (CPA) bekannten Näherung. Die Berechnung mittels anschaulicher Diagrammatik liefert die selben Ergebnisse wie die rigorose supersymmetrische Methode. Die Anwendbarkeit der CPA in drei Dimensionen wird durch den Vergleich mit exakter Diagonalisierung verifiziert. Besonders interessant ist ein Modell, dass schwache Unordnung in den Massen mit starker Unordnung in den Kräften kombiniert. Dieses Modell ist mikroskopisch gut motiviert und führen zu einem (Quanten-)Phasenübergang bei endlicher Unordnungsstärke. Die (Debye) Phase schwacher Unordnung endet an einem kritischen Punkt und bei stärkerer Unordnung finden wir eine neuartige Phase mit einer endlichen und positiven Zustandsdichte bei beliebig kleiner Energie. Anders als das weit verbreitete zwei-Level Systeme (TTLS) Modell wird eine positive Zustandsdichte an Streuzentren bei kleinen Energien in unserem Modell nicht von vornherein angenommen. Wir können somit zeigen, dass Unordnung prinzipiell ohne Weiteres ausreicht um die experimentell erstmals 1971 beobachtete lineare Temperaturabhängigkeit der Wärmekapazität amorpher Festkörper bei Temperaturen unterhalb von etwa 1K zu erklären. Die neue Phase starker Unordnung ist stabil unter Hinzunahme der meisten anderen Unordnungstypen. Unter Annahme der Renormierbarkeit und korrekt identifizierter kritischer Punkte treffen wir eine vorläufige Vorhersage für den Renormierungsgruppenfluss, welcher die neue Phase ebenfalls stabilisiert.German
Schmittner, Sebastian E.sebastian.schmittner@uni-koeln.deUNSPECIFIED
Scientific advisorZirnbauer,
URN: urn:nbn:de:hbz:38-59466
Subjects: Physics
Uncontrolled Keywords:
condensed matter, mathematical physics, supersymmetry, coherent potentail approximation, disordered systems, amorphous solids, glasses, low temperature, phonons, random matrix theory, density of states, heat capacity, universalityEnglish
Faculty: Faculty of Mathematics and Natural Sciences
Divisions: Faculty of Mathematics and Natural Sciences > Department of Physics > Institute for Theoretical Physics
Language: English
Date: 2015
Date of oral exam: 12 January 2015
NameAcademic Title
Zirnbauer, MartinProf. Dr.
Altland, AlexanderProf. Dr.
Related URLs:
Funders: Deutsche Telekom Stiftung, Bonn-Cologne Graduate School of Physics and Astronomy, DPG via SFB|TR12
References: [ABDF11] G. Akemann, J. Baik, and P. Di Francesco, editors. The Oxford Handbook of Random Matrix Theory. Oxford Handbooks in Mathematics. Oxford University Press, 2011. [AHV72] P. W. Anderson, B. I. Halperin, and C. M. Varma. Anomalous low-temperature thermal properties of glasses and spin glasses. Philosophical Magazine, 25(1):1, 1972. [AM76] N. W. Ashcroft and D. N. Mermin. Solid state physics. Thomson Learning, 1976. [CKR+ 10] A. Chaudhuri, A. Kundu, D. Roy, A. Dhar, J. L. Lebowitz, and H. Spohn. Heat transport and phonon localization in mass-disordered harmonic crystals. Physical Review B, 81(6):064301, 2010. [FA86] J. J. Freeman and A. C. Anderson. Thermal conductivity of amorphous solids. Physical Review B, 34(8):5684, 1986. [HRK+ 12] M. Hassaine, M. A. Ramos, A. I. Krivchikov, I. V. Sharapova, O. A. Korolyuk, and R. J. Jim ́nez-Riob ́o. Low-temperature thermal and elastoacoustic properties of butanol glasses: Study of position isomerism effects around the boson peak. Physical Review B, 85(10):104206, 2012. [HW70] J. Harrington and C. Walker. Phonon Scattering by Point Defects in CaF2, SrF2, and BaF2. Physical Review B, 1(2):882, 1970. [JM73] W. Jones and N. H. March. Theoretical solid state physics, volume 1 and 2. Wiley-Interscience New York, 1973. [JSS83] S. John, H. Sompolinsky, and M. Stephen. Localization in a disordered elastic medium near two dimensions. Physical Review B, 27(9):5592, 1983. [LSZ06] T. Lück, H. J. Sommers, and M. R. Zirnbauer. Energy correlations for a random matrix model of disordered bosons. Journal of Mathematical Physics, 47:103304, 2006. arXiv:cond-mat/0607243. [LSZ07] P. Littelmann, H. -J Sommers, and M. R Zirnbauer. Superbosonization of invariant random matrix ensembles. Commun. Math. Phys. 283 (2008) 343. arXiv:0707.2929. [Lü09] T. Lück. Random Matrix Models for Disordered Bosons. PhD thesis, Mathematisch-Naturwissenschaftliche Fakultät der Universit ̈t zu Köln 2009. [LV13] A. J. Leggett and D. C. Vural. ”Tunneling two-level systems” model of the low-temperature properties of glasses: are ”smoking-gun” tests possible? The journal of physical chemistry. B, 117(42):12966–71, 2013. arXiv:1310.3387. [LW03] V. Lubchenko and P. G. Wolynes. The origin of the boson peak and thermal conductivity plateau in low-temperature glasses. Proceedings of the National Academy of Sciences of the United States of America, 100(4):1515, 2003. arXiv:cond-mat/0206194. [Par94] D. Parshin. Interactions of soft atomic potentials and universality of low-temperature properties of glasses. Physical Review B, 49(14):9400, 1994. [Phi72] W. A. Phillips. Tunneling states in amorphous solids. Journal of Low Temperature Physics, 7(3-4):351, 1972. [Phi87] W. A. Phillips. Two-level states in glasses. Reports on Progress in Physics, 50(12):1657, 1987. [PLT02] R. O. Pohl, X. Liu, and E. Thompson. Low-temperature thermal conductivity and acoustic attenuation in amorphous solids. Reviews of Modern Physics, 74(4):991, 2002. [Sch14] S. E. Schmittner. An adaptive high precision solver for complex non-linear equations. github:Echsecutor/rootSolver, 2014. [Sov67] P. Soven. Coherent-Potential Model of substitutional Disordered Alloys. Physical Review, 156(3):809, 1967. [SS09] M. Schechter and P. C. E. Stamp. Low temperature universality in disordered solids. Physical Review B 88, 174202, 2013 [Ste73] R. B. Stephens. Low-temperature specific heat and thermal conductivity of noncrystalline dielectric solids. Physical Review B, 8:2896, 1973. [Ste76] R. B. Stephens. Intrinsic low-temperature thermal properties of glasses. Physical Review B, 13(2):852, 1976. [SW80] L. Sch ̈fer and F. Wegner. Disordered system with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and goldstone modes. Zeitschrift für Physik B Condensed Matter, 38(2):113, 1980. [SZ10] S. E. Schmittner and M. R. Zirnbauer. Coherent potential approximation for disordered bosons. arXiv:1012.5283, 2010. [tH74] G. ’t Hooft. A planar diagram theory for strong interactions. Nuclear Physics B, 72(3):461, 1974. [TRV02] C. Tal ́n, M. Ramos, and S. Vieira. Low-temperature specific heat of amorphous, orientational glass, and crystal phases of ethanol. Physical Review B, 66(1):012201, 2002. [Tur04] M. Turlakov. Universal Sound Absorption in Low-Temperature Amorphous Solids. Physical Review Letters, 93(3), 2004. arXiv:cond-mat/0309097. [Van53] L. Van Hove. The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal. Physical Review, 89(6):1189, 1953. [Weg79] F. Wegner. Disordered system with n orbitals per site: n = ∞ limit. Physical Review B, 19(2):783, 1979. [YM73] F. Yonezawa and K. Morigaki. Coherent Potential Approximation. Progress of Theoretical Physics Supplement, 53:1, 1973. [Zir98] M. R. Zirnbauer. Riemannian symmetric superspaces and their origin in random matrix theory. Journal of Mathematics and Physics 37:4986, 1998. arXiv:math-ph/9808012 [ZP71] R. C. Zeller and R. O. Pohl. Thermal Conductivity and Specific Heat of Noncrystalline Solids. Physical Review B, 4(6):2029, 1971. [Zuk94] J. A. Zuk. Introduction to the Supersymmetry Method for the Gaussian Random-Matrix Ensembles. arXiv:cond-mat/9412060, 1994.
Refereed: Yes


Downloads per month over past year


Actions (login required)

View Item View Item