Exploring Universal Scaling Relationship Between Ground State And Matrix Elements And Its Potential Applications

Investigating the novel phenomena of quantum many-body systems is the core of condensed matter physics.Due to the exponentially increased dimension of Hilbert space with the system size,it brings the extremely difficult to study of quantum many-body systems,especially in cases with strong correlations.De-spite the significant progresses made by conditional theoretical procedures,such as mean field theory,quantum Monte Carlo,density matrix renormalization group,there are still numerous fields in which these methods lose its powerful.Therefore,conceptual innovations are needed to break the barrier faced by theories.Obvi-ously,for a diagonalizable matrix,both eigenvalues and eigenvectors are uniquely determined by its elements.One possible idea is to establish an immediate con-nection between eigenvectors and matrix elements.If this kind of connection can be figured out,it may be an appealing method for matrix diagonalization.Enlightened by this idea,in this thesis,we focus on exploring the potential uni-versal relationship between the eigenvector and matrix elements in real symmetric matrices.The main results are summarized as following:1?The scaling relationship between the eigenvector and matrix elements.By calculating a large number of real symmetric random matrices with non-positive off-diagonal elements,we have found a universal scaling between the eigen-vector and matrix elements.Namely,each element(g_i)of the eigenvector of ground states linearly correlates with the sum(S_i)of matrix elements in the corresponding row.This correlation is valid for both dense and sparse matrices.For matrices in which the distribution width of diagonal elements is larger than the sum of off-diagonal ones,the relationship between g_iand S_iis power-law like instead of simple linear.We find an order parameter,which can describe all the scaling relationship in matrices mentioned above.We have verified the scaling correlation for three representative models,i.e.,the BCS model,the Hubbard model and quantum Ising model.The simple relationship implies a straightforward method to directly calculate the eigenvector of ground states for a matrix,which is widely needed in many fields.Finally,a Monte Carlo program is proposed for this kind of calculations.2?The influence of the signs of off-diagonal elements on ground state.We expand the results to matrices with positive off-diagonal elements and found that the linear scaling holds for any matrix with the number of non-positive elements more than that of non-negative ones.It breaks down suddenly when the number of non-negative elements overtake that of non-positive ones.Finally,we find that,the breakdown of linear relationship is corresponding to a kind of quantum phase transitions.We calculate the critical point based on the finite scaling theory.Further,based on the analysis of the eigenvalue spectrum,we confirm the same critical point.