Loading...

Applications of Entanglement Renormalization and Non-abelian Symmetries in the Study of Strongly Correlated Systems

Saedpanah, Amir Hossein | 2021

1128 Viewed
  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 54211 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Vaezi, Mir Abolhassan
  7. Abstract:
  8. After the introduction of the framework of quantum mechanics and its experimental success, it was assumed that any physical phenomenon could be analyzed and understood accurately using quantum mechanics. But very soon we found out that in most cases, it is too hard or sometimes impossible to solve the Schrödinger equation exactly even for few-body systems. The reason is that the dimension of the Hilbert space increases exponentially with the number of particles, in a way that for a spin chain(spin-1/2), which has only two single-particle states, even using modern classical computers we are barely able to find an exact solution of a 30-40 particles system which is far smaller than the physical systems we observe their specific behaviors such as phase transition.In response to this issue, researchers developed several approximate solution techniques for many-body system’s analysis such as the Density Matrix Renormalization Group (DMRG) algorithm where we omit a part of the system’s degrees of freedom that are inaccessible for the ground state or are unnecessary and subsequently, avoid exponential increase of the Hilbert space.In DMRG algorithm, the computation error is related to the method of decreasing density matrix and its eigenvalue distribution. Eigenvalues of density matrix, as we know, determine the entanglement. It could be shown that the required number of degrees of freedom to achieve a certain accuracy depends exponentially on entanglement entropy of the system. In one dimension, due to small entanglement entropy, DMRG is very successful but by increasing the thickness of pseudo-2D systems, the accuracy decreases exponentially. In the first part of the thesis, at the first stage we try to decrease entanglement entropy using unitary transformation in order to increase the accuracy of the numerical methods based on quantum entanglement exponentially, and then, we use DMRG algorithm for the problem.In this research, we mainly focus on the unitary transformations corresponding to linear transformations of fermionic creation and annihilation operators. Therefore, using this method, we write Hubbard model in single-particle bases which have the least entanglement entropy and using DMRG algorithm in that basis, we analyze properties of the ground state of Hubbard model which has close relation with high-temperature superconductivity. Another way of decreasing the degrees of freedom is using the symmetries related to the phenomenon. Usually, abelian symmetries, due to their practical simplicity, are mostly used in numerical methods and non-abelian symmetries, due to practical complexity, are rarely used in spite of their power in decreasing the degrees of freedom. In the second part, we implement SU(2) symmetry for the Heisenberg model which is based on the Wigner-Eckart theorem and review the previous implementations. This symmetry lets us omit at least one-third of the degrees of freedom to reach the same accuracy without SU(2) symmetry in the DMRG algorithm
  9. Keywords:
  10. Entanglement Entropy ; Density Matrix Renormalization Group ; Wigner-Eckart Theorem ; Non-Abelian Symmetry ; Area Law ; Strongly Correlated Systems

 Digital Object List

 Bookmark

...see more