Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2012 Entanglement Entropy in Ordered and Quantum Critical Systems Wenxin Ding Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES ENTANGLEMENT ENTROPY IN ORDERED AND QUANTUM CRITICAL SYSTEMS By WENXIN DING A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2012 Wenxin Ding defended this dissertation on May 11, 2012. The members of the supervisory committee were: Kun Yang Professor Directing Dissertation Phillip Bowers University Representative Nicholas Bonesteel Committee Member Jorge Piekarewicz Committee Member Peng Xiong Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with the university require- ments. ii To my beloved father, my wife, and my little angel. iii ACKNOWLEDGMENTS I would like to express my gratitude to my advisor Professor Kun Yang for his support and guidance during my graduate studies. I have learned and benefited so much from his remarkable insights in physics, without which this work can never be finished, as well as his rigorous scholarship. I am truly grateful for his patience and advice through. I also wish to thank Professors Nicholas E. Bonesteel and Alexander Seidel, with whom I am fortunately to collaborate. The inspiring discussions I had with both of them also play a key role in this work. My many thanks also go to Professors Nicholas E. Bonesteel again, Peng Xiong, Phillip Bowers, and Jorge Piekarewicz, who served in my dissertation committee, for their time and effort. iv TABLE OF CONTENTS List of Figures ...................................... vii Abstract ......................................... x 1 Introduction 1 1.1 Entanglement Entropy ............................ 3 1.1.1 Renyi,´ Tsallis Entropy, entanglement spectrum and mutual infor- mation ................................ 4 1.2 The Area Law ................................. 6 1.2.1 Corrections to the Area Law ..................... 8 1.3 Logarithmic Divergence in Entanglement Entropy .............. 9 1.4 Examples of Entanglement Entropy ..................... 10 1.4.1 Reduced Density Matrices of Free Systems . 10 1.4.2 Entanglement Entropy of 1D Free Fermions from Reduced Density Matrices ................................ 12 1.4.3 The Replica Trick ........................... 13 2 Entanglement Entropy in States with Traditional Long-Range Magnetic Or- der 15 2.1 Antiferromagnetic Spin Model and the Ground State . 16 2.2 Reduced Density Matrix and Entanglement Entropy . 17 2.3 Ferromagnetic model and its entanglement entropy . 20 2.4 Discussion ................................... 22 3 Logarithmic Divergence of Entanglement Entropy and Mutual Information in Systems with Bose-Einstein Condensation 24 3.1 Zero Temperature: Entanglement Entropy of Free Bosons . 25 3.2 Mutual Information: Analytic Study of an Infinite-Range Hopping Model . 27 3.2.1 Model, spectrum and thermodynamic properties . 28 3.2.2 Formalism and issues ......................... 30 3.2.3 Mutual information .......................... 31 3.3 Numerical Study of Mutual Information in One-dimension with Long-range Hopping .................................... 34 3.4 Summary and Discussion ........................... 41 v 4 Entanglement Entropy of Fermi Liquids via High-dimensional Bosonization 46 4.1 The Intuitive picture - a toy model ...................... 47 4.2 Multi-dimensional bosonization ....................... 50 4.2.1 Entanglement Entropy of Free Fermions . 55 4.2.2 Solution for the Fermi liquid case and non-locality of the Bogoli- ubov fields .............................. 57 4.3 Entanglement entropy from the Green’s function . 59 4.3.1 The Replica Trick and Application to 1D Free Bosonic Theory . 60 4.3.2 Geometry and Replica Boundary Conditions . 64 4.3.3 Entanglement entropy of free fermions revisited . 67 4.3.4 Differential Equations of the Green’s Functions and an Iterative Solution ................................ 71 4.3.5 Entanglement Entropy from the Iterative Solution . 73 4.4 Summary and Concluding Remarks ..................... 80 5 Conclusion and Open Questions 82 5.1 Logarithmic Entanglement Entropy and Spontaneous (Continuous) Sym- metry Breaking ................................ 82 5.2 Universal Subleading Correction to the Coefficient of the Logarithm? . 83 5.3 Experimental Prospect of Entanglement Entropy Measurement in Extended Quantum Systems ............................... 84 A Calculation of δGk/n 86 (1) A.1 δ Gk/n .................................... 86 (2) A.2 δ Gk/n .................................... 87 Bibliography ...................................... 90 Biographical Sketch ................................... 96 vi LIST OF FIGURES 2.1 Two-sublattice model: two sublattices labeled A and B interpenetrating each other. ...................................... 17 3.1 Numerical calculation of mutual information for the infinite-range hopping model for equally partitioned systems with average density n = 1. The scatters are numerical data, while the dash lines are obtainedh fromi our an- alytic results corresponding to the particular temperature. We see exactly 1 what our analytic results tell us: above TC = ln 2 the mutual information sat- 1 1 urates; below TC, S M 2 LA; at TC, M 4 LA. Note that the analytic results (dash lines) deviate from≃ the numerical≃ calculation because we only keep terms to the subleading order; terms that goes to zero [i.e., of order (1/LA)] in the thermodynamic limit are neglected. ...................O 35 3.2 Numerical calculation of TC (line + symbols) for the long-range hopping model in the thermodynamic limit. TC is measured in unit of the hopping energy t which is set to 1. As one can see, TC grows monotonically from 0 to as γ goes from 2 to 1. The divergent behavior of TC as γ 1 is a consequence∞ of the divergent bandwidth. ...................→ 38 3.3 Mutual information of the nearest neighbor hopping model plotted against subsystem size on a logarithmic scale. Average density is set to n = 1, h i and the system is equally partitioned, L = 2LA. The black dash line is S M 1 ∼ 2 ln LA. This line will be in other graphs for comparison as well. Clearly, the mutual information saturates when the system size grows large enough. 40 3.4 Mutual information of the long-range hopping model with the parameter γ = 1.7 as a function of subsystem size on a logarithmic scale, at various (inverse) temperatures. The average boson density n is set to 1, and the system is h i equally partitioned, LA = L/2. The scaling behavior for inverse temperature β = 0.5 goes as S M = 0.2405 ln LA + 0.214 (cyan dash line corresponding to the diamond data points). At the transition point, β = βC = 0.297, the scaling law is fit to be S M = 0.1226 ln LA + 0.1688 (red dash line corresponding to the right triangle data points). ......................... 42 vii 3.5 Mutual information of the long-range hopping model with the parameter γ = 1.5 as a function of subsystem size on a logarithmic scale, at various (inverse) temperatures. The average boson density n is set to 1, and the system is h i equally partitioned, LA = L/2. The mutual information for β = 1 is fit to scale as S 0.324 ln L + 0.445 (orange dash line). At β = 0.16843, we M ≃ C observer a weaker scaling behavior which is fit to be S M = 0.064 ln LA + 0.084 (red dash line). .............................. 43 3.6 Mutual information of the long-range hopping model with the parameter γ = 1.3 as a function of subsystem size on a logarithmic scale, at various (inverse) temperatures. The average boson density n is set to 1, and the system is h i equally partitioned, LA = L/2. The best fitting for β = 0.5 > βC (cyan dash line) gives S M = 0.378 ln LA + 0.2. At βC = 0.0954, the scaling behavior is fit (red dash line) as S M = 0.023 ln LA + 0.03535. 44 4.1 The toy model both in real space and in momentum space. (a) A set of parallel decoupled 1d chains of spinless free fermions (dash lines); the sub- system division is represented by the solid lines, both convex and concave geometries. (b) Fermi surfaces of the toy model. 48 4.2 Patching of the Fermi surface. The low energy theory is restricted to within a thin shell about the Fermi surface with a thickness λ kF, in the sense of renormalization. The thin shell is further divided into N≪different patches; d 1 each has a transverse dimension Λ − where d = 2, 3 is the space dimen- sions. The dimensions of the patch satisfy three conditions: (1) λ Λ min- 2 ≪ imizes inter-patch scattering; (2) Λ kF and Λ /kF λ together makes the curvature of the Fermi surface negligible.≪ (a): Division≪ of a 2D Fermi surface into N patches. Patch S is characterized by the Fermi momentum kS. (b): A patch for d = 3. The patch has a thickness λ along the normal direction and a width Λ along the transverse direction(s). 51 4.3 Origin of the bosonic commutator of patch density operators illustrated for d = 2. As shown in Eq. (4.7), the commutator is reduced to computing the difference of occupied states , i.e. the area difference below the Fermi surface, between the two θ functions (θ(S; k q) θ(S; k+ p)). The solid box indicates the original patch, or θ(S; k). The red− line− shows the Fermi surface. Both θ(S; k q) and θ(S; k + p) are denoted by dashed boxes. The occupied part in θ(S; −k + q) is denoted by blue, that of θ(S; k q) is denoted by red, and the overlapping region is denoted by yellow. Subtracting− the remaining blue area from the red, we obtain that θ(S; k q) occupies (Λ qS )qnˆS more states, which gives us the commutator.