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MSc Physics Track: Theoretical Physics MASTER THESIS Finite bond dimension scaling with the corner transfer matrix renormalization group method Geert Kapteijns 10275037 July 2017 60 ECTS Supervisor: Examiner: Dr. Philippe Corboz Prof. dr. Bernard Nienhuis Institute for Theoretical Physics Abstract This thesis investigates scaling in the number of basis states kept (the bond dimension m) in approximating the partition function of two-dimensional classical models with the corner transfer matrix renormalization group (CTMRG) method. For the Ising model, it is shown that exponents and the transition temperature may be approximated with a scaling analysis in the corre- lation length defined in terms of the row-to-row transfer matrix at the (pseudo)critical point, as was suggested by Nishino et al. However, the calculated quantities show inherent deviations from the basic scaling laws, due to the spectrum of the underlying corner transfer matrix (CTM). These deviations are mitigated to some extent when we define the cor- relation length in terms of the classical analogue of the entanglement en- tropy. Scaling directly in the bond dimension m is also possible, but less accurate since the law for the correlation length ξ ∝ m κ holds only in the limit m → ∞ and does not take into account the spectrum of the CTM that is obtained. It is found that finite-m scaling and finite-size scaling yield compara- ble accuracy for critical exponents and the transition temperature. With finite-m scaling larger effective system sizes are obtainable, but finite-size approximations do not suffer from the deviations due to the CTM spec- trum and are consequently of higher quality. Therefore it is plausible that finite-size results will improve significantly if corrections to scaling are included in the fits. We also present a numerical analysis of the clock model with q = {5, 6} states, concluding that the Kosterlitz-Thouless picture is plausi- ble. We find values of the transition temperatures that are in agreement with values found by other authors. Results for the exponent η indicate that the critical temperatures found in both this study and previous work might be too close together. It is conceivable that, after considering larger systems and taking into account finite-size corrections, both critical tem- peratures and the values of η will be adjusted outwards towards their true values, thereby completely reconciling the results. Overall, we conclude that finite-m scaling is a valuable alternative to finite-size scaling within CTMRG, since larger system sizes are accessi- ble. The CTMRG analysis is itself a valuable addition to other approxi- mation methods such as Monte Carlo, yielding comparable results, while obtaining estimates from completely different principles. Furthermore it reveals information, such as the the spectrum of the transfer matrices and the central charge of the massless phase, that is not accessible otherwise. Contents Acknowledgements vi 1 Introduction 1 1.1 Statistical mechanics and phase transitions . . . . . . . . . . . 2 1.1.1 Universality . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Baxter’s method as a precursor to tensor network methods . . 3 1.3 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . 4 2 Density matrix renormalization group method 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Density matrix renormalization group . . . . . . . . . . . . . 6 2.2.1 Real-space renormalization group . . . . . . . . . . . 6 2.2.2 Single particle in a box . . . . . . . . . . . . . . . . . 7 2.2.3 Density matrix method . . . . . . . . . . . . . . . . . 7 2.2.4 Infinite-system method . . . . . . . . . . . . . . . . . 9 3 DRMG applied to two-dimensional classical lattice models 12 3.1 Statistical mechanics on classical lattices . . . . . . . . . . . . 12 3.2 Transfer matrices of lattice models . . . . . . . . . . . . . . . 13 3.2.1 1D Ising model . . . . . . . . . . . . . . . . . . . . . 13 3.2.1.1 Fixed boundary conditions . . . . . . . . . 15