Phase Diagram and Real-Time Dynamics of the Semiclassical Kondo Lattice on a Zigzag Ladder
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Phase Diagram and Real-Time Dynamics of the Semiclassical Kondo Lattice on a Zigzag Ladder Dissertation with the aim of achieving a doctoral degree at the Faculty of Mathematics, Informatics and Natural Sciences Department of Physics at the University of Hamburg submitted by Lena-Marie Woelk Hamburg, June 2019 Gutachter der Dissertation: Prof. Dr. Michael Potthoff Prof. Dr. Alexander Lichtenstein Zusammensetzung der Prüfungskommission: Prof. Dr. Michael Potthoff Prof. Dr. Alexander Lichtenstein Prof. Dr. Daniela Pfannkuche Prof. Dr. Michael A. Rübhausen Dr. Elena Y. Vedmedenko Vorsitzende/r der Prüfungskommission: Prof. Dr. Michael A. Rübhausen Datum der Disputation: 18.09.2019 Vorsitzender Fach-Promotionsausschusses PHYSIK: Prof. Dr. Michael Potthoff Leiter des Fachbereichs PHYSIK: Prof. Dr. Wolfgang Hansen Dekan der Fakultät MIN: Prof. Dr. Heinrich Graener ii Acknowledgements I am forever thankful to Michael Potthoff for always being supportive, kind, and an all around ex- cellent supervisor as well as a continuous source of inspiration. I would also like to sincerely thank Prof. Alexander Lichtenstein and all other members of my evaluation committee, for kindly agreeing to evaluate my work. I am grateful to all former and current members of Gruppe Potthoff for instruc- tive and enjoyable lunch breaks, stimulating discussions and always being ready for cake breaks. In particular, I would like to thank Mohammad Sayad, for helpful discussions and the many coffee breaks as well as Matthias Peschke, for a fruitful and enjoyable cooperation and also for letting me win at Doppelkopf. I would also like to thank Roman Rausch and Christian Gramsch for making sharing an office not only very instructive through many discussions but also great fun. Furthermore, Iwouldlike to express my gratitude to the Physnet, especially Martin Stieben, for being so very helpful and patient while troubleshooting my program. I am so very grateful to my family and friends for invaluable moral support. Most of all, I would like to thank Pablo Woelk, who was my rock at all times and without whom I would not be where I am now. If this work could be dedicated, it would be to him. iii iv Abstract The equilibrium phase diagram of the Kondo lattice model with classical spins on the so-called zigzag ladder, which is a minimal example of frustration in one dimension, is derived as a function of exchange coupling constant J and measure of frustration j. It is found to contain the well-known antiferromag- netic phase, an incommensurate spiral phase with varying pitch angle, and a novel spin-dimerized phase. The results are compared to perturbation theory in both the strong and weak coupling limit and found to agree well, with the exception of the dimerized phase which is absent in the perturba- tive approaches. A comparison to results obtained by density-matrix renormalization group (DMRG) reveals many similarities to the quantum-mechanical model. Both models predict the existence of the dimerized phase, whereas the DMRG-phase diagram contains further features absent in the classical approximation. Next, the real-time dynamics of the system is analyzed following a quench, i.e. a sudden change in parameter. Two qualitatively different energy regions are identified. For low quench energies, the system is non-ergodic and remains in the initial spin configuration for all times. The corresponding energy threshold is reminiscent of the Fermi-Pasta-Ulam (FPU) paradox known from classical dynam- ics. After exploring the (non-)integrability of the model, the ergodicity threshold is explained usinga linear approximation in the equations of motion describing spin-wave-like excitations. For higher quench energies, in particular when crossing the equilibrium phase boundary, the dynamics is ergodic. The time scale of thermalization is found to be highly energy-dependent. Starting from an initial spiral configuration, the gradual emergence of long range dimer order can be seen. Abovea certain critical energy, however, this long range dimer order breaks down. This thermal transition is found stable in the limit of larger lattice sizes. v vi Kurzzusammenfassung Das Gleichgewichtsphasendiagramm des Kondo Gitter Modells mit klassischen Spins auf der soge- nannten Zickzack-Leiter, welche ein Minimalbeispiel für Frustration in einer Dimension ist, wird hergeleitet als Funktion der Austausch Kopplung J und Frustration j. Es beinhaltet die bekannte antiferromagnetische Phase, eine inkommensurable Spiralphase mit variierendem Winkel und eine neuartige spin-dimerisierte Phase. Die Ergebnisse werden mit Störungstheorie sowohl im Limes starker als auch schwacher Kopplung verglichen und stimmen gut überein, mit Ausnahme der dimerisierten Phase, die von der Störungstheorie nicht beschrieben wird. Anschließend wird das Phasendiagramm verglichen mit Ergebnissen, die mit density-matrix renormalization group (DMRG) berechnet wurden. Es sind viele Gemeinsamkeiten zwischen semiklassischem und quantenmecha- nischem Modell festzustellen. Beide sagen die dimerisierte Phase vorher, allerdings beinhaltet das DMRG-Phasendiagramm noch andere Phänomene, die in der klassischen Approximation fehlen. Dann wird die Echtzeitdynamik des Systems nach einem Quench, also der plötzlichen Änderung eines Parameters, analysiert. Es werden zwei qualitativ unterschiedliche Energiebereiche identifiziert. Für Quenches mit niedriger Energie ist das System nicht ergodisch und verbleibt für immer in der An- fangsspinkonfiguration. Die entsprechende Energieschwelle erinnert an das Fermi-Pasta-Ulam (FPU) Paradoxon, bekannt aus der klassischen Dynamik. Nach einer Untersuchung der (Nicht-) Integrabilität des Modells, wird die Ergodizitätsschwelle mithilfe einer linearen Näherung der Bewegungsgleichun- gen erklärt, die spinwellenartige Anregungen beschreibt. Für Quenches mit höherer Energie, insbesondere wenn Phasengrenzen überquert werden, ist die Dy- namik ergodisch. Die Zeitskala der Thermalisierung ist stark energieabhängig. Ausgehend von einer anfänglichen spiralen Spinordnung kann der graduelle Aufbau einer langreichweitigen Dimerisierung- sordnung beobachtet werden. Oberhalb einer kritischen Energie bricht diese Ordnung jedoch zusam- men. Dieser thermische Übergang ist stabil im Limes größerer Systeme. vii viii Contents 1. Introduction 1 1.1. Emergence, Frustration and the Kondo Lattice ....................... 1 1.2. Moving on: Dynamics and Dynamical Phase Transitions ................. 3 2. The Kondo Lattice 5 2.1. Introduction to the Kondo Lattice Model .......................... 5 2.2. Equilibrium Spin Configurations on the Zigzag Ladder .................. 7 2.2.1. Spin Configurations With Constant Pitch Angle .................. 8 2.2.2. Dimerized Spin Configurations ........................... 11 2.3. Perturbative Approaches ................................... 15 2.3.1. Weak Coupling (RKKY) Limit ............................ 15 2.3.2. Strong Coupling Limit ................................ 17 3. Equilibrium Phase Diagram 23 3.1. The Ground State as a Function of J and j ......................... 24 3.2. Perturbative Approaches ................................... 28 3.2.1. Strong Coupling Regime ............................... 28 3.2.2. Weak Coupling Regime - RKKY ........................... 32 3.2.3. Perturbation Theory Around t1 = 0 ........................ 37 3.3. Finite Size Analysis ...................................... 38 3.4. Comparison to DMRG Results ................................ 40 4. Real-Time Dynamics 45 4.1. Method and Formalism .................................... 46 4.1.1. Equations of Motion ................................. 46 4.1.2. Initial Conditions and Numerical Remarks ..................... 48 4.2. Ergodicity Threshold ..................................... 51 4.2.1. Numerical Observations ............................... 53 4.2.2. The FPU Problem and Proximity to Integrability ................. 62 4.2.3. Integrability of the Classical J1-J2- Heisenberg Model .............. 63 4.2.4. Linear Approximation and Spin-Wave-Like Excitations .............. 70 ix Contents 4.3. Dimerization Transition ................................... 81 4.3.1. Phase Transitions ................................... 81 4.3.2. Spontaneous Symmetry Breaking and Long Range Order ............ 84 4.3.3. Thermalization .................................... 90 4.3.4. Emergence of Long Range Dimer Order ...................... 93 5. Conclusions, Summary and Perspectives 107 − A. Heisenberg J1 J2 Model on a Zigzag Ladder 111 B. Perturbation Theory in t1 117 Bibliography 127 List of Publications 133 Eidesstattliche Versicherung / Declaration on oath 135 x 1. Introduction 1.1. Emergence, Frustration and the Kondo Lattice Studying a single water molecule does not convey the enormous complexity of the properties of a macroscopic body of water. It cannot be explained by considering solely one oxygen and two hydrogen atoms on their own, that the same lake can be used in summer for swimming and in winter for ice skating. Since John Dalton published his model of the atom as an indivisible entity in 1808 in his work A New System of Chemical Philosophy, much has been corrected and by looking closer and closer ever new fundamental particles were discovered. Still, even with all this knowledge, the interplay between many atoms together and the qualities, which result, often remain a mystery. As P.W. Anderson put it so eloquently in his famous 1972 paper More is different: “… we can see how the whole becomes not only more than but very different