1.2.1 Transverse Ising model
In this thesis, we study the one dimensional quantum transverse Ising model for the finite system and the infnite system. The transverse Ising model is the simplest quantum spin model to exhibit a quantum phase transition [21].
The hamiltonian is given by
H =−(
i
σizσzi+1+ hσix), (1.1)
where σx and σz are the Pauli matrices, and sum over i is over all sites.
For h < 1, the ground state of this system has long-range Ising order in the z-direction. For h > 1, the ground state becomes disorder in the z-direction.
A quantum phase transition occurs when the system goes from h < 1 to h > 1. The critical point is at h = 1. The transverse Ising model provides a clear evidence that the lattice is most entangled at the critical point.
1.2.2 Heisenberg model
We also investigate the antiferro Heisenberg model with the next nearest neighbor interaction. The Hamiltonian is given by
H =
i
J1Si· Si+1+ J2Si· Si+2, (1.2)
where S = ¯h2σ and J1, J2 > 0. The model is invariant under a global SU (2) rotation, so the total spin should be conserved. For J2 = 0, the model can be solved exactly by Bethe ansatz [22]. When J2 > 0, it can be solved by taking the ground state as a superposition of the nearest neighbor valence
bond states [23]. For J1, J2 > 0, there is a competition between the nearest neighbor interaction and the next nearest neighbor interaction which is called frustration. This causes the “sign problems” [24, 25] in many QMC methods which originates from the minus sign in the wave function when interchanging two fermions. An illustration of frustration is shown in Fig. 1.1.
?
Anti-parallel
Anti-parallel
Figure 1.1: An illustration of frustration. The spins tend to lie anti-parallel to minimize the energy and form a competition between the nearest neighbor interaction and the next nearest neighbor interaction.
The thesis is organized as follows. In Chapter 2, we review the construc-tion of MPS from two perspectives and discuss the connecconstruc-tion between them.
In Chapter 3, we review the Monte Carlo method and the stochatic optimiza-tion. We generalize this QMC method to the open boundary condition and study the transverse Ising model and the Heisenberg model. In chapter 4, we review the TEBD and iTEBD method in detail for imaginary time evo-lution. The infinite transverse Ising model is studied and the results imply a connection between entangelment and quantum phase transition. In chapter 5, we summarize the results and make a conclusion.
Chapter 2
Matrix Product States
In this chapter, we present the formulation of the matrix product states (MPS) from the density matrix renormalization group (DMRG) aspect and from the quantum information theory (QIT) [26] aspect. We will also discuss the notion of entanglement and use it to justify the use of MPS as a trial wave function.
2.1 MPS from the DMRG point of view
The development of the density matrix renormalization group method [5, 6]
has enabled us to analyze and understand one-dimensional quantum many body systems with high accuracy . We shall describe the main idea of DMRG without going into too much details. First of all, we start from a smaller system which is small enough that we can diagonalize its Hamiltonian. This system is labeled as the superblock and it is divided into the system block and the enviroment block. The goal is to find a set of states of the system block which can optimally represent the superblock. We construct the reduced density matrix for the system block and diagonalize it, keeping only a number of states as basis states (e. g., D states) by dropping off the states with smaller eigenvalues in the reduced density matrix, as they are less likely to be accessed. The Hamiltonian of the system block are transformed to these basis states. Then we add a single spin to the system block and use the transformed Hamiltonian together with the added spin to construct the enviroment block; thus, the new superblock can be formed. Repeat this spin-adding procedure recursively until the system reach the desired size. In practice, the eigenvalues of the density matrix decrease rapidly so that the truncation errors are small. The name density matrix renormalization group reflects the fact that we keep those most relevant states in the density matrix.
(a) (b)
ห߰ఉۄିଵٔ ȁݏۄ ȁ߰ఈۄ
Figure 2.1: (a) Add kth new spin|sk to the system block basis states |ψβk−1
containing k − 1 sites. (b) Form the new super block and construct the reduced density matrix for the new system block keeping only D states to form a new set of basis.
The theoretical foundation of the success of DMRG is pointed out by Ostlund and Rommer [11]. Their work shows that the DMRG construction is¨ closely related to position-dependent matrix product state. At each recursion step, DMRG is a particularly effective way to generate a D× D projection operator Akα,β(sk), which projects these states|ψβk−1⊗|sk to a larger system with a set of new basis states |ψαk. That is (Fig. 2.1(a) to Fig. 2.1(b))
|ψαk =
D β=1
sk
Akαβ(sk)|ψβk−1⊗ |sk. (2.1)
For a state |Ψ in an one-diensional system containing n spins with pe-riodic boundary conditions(sn+1 = s1), we can construct a matrix product state using Eq. (2.1) recursively as shown in Fig. 2.2. |Ψ is transformed into spin basis
|Ψ =
s1,s2,...,sn
tr(A1(s1)A2(s2) . . . An(sn))|s1s2. . . sn, (2.2)
where tr(· · · ) is the matrix trace. This state contains 2nD2 parameters ranther than 2nones. Furthermore, if the system has translational symmetry,
ȁȲۄ ȁȲۄିଵٔ ȁݏۄ
ȁȲۄିଶٔ ȁݏିଵۄ ٔ ȁݏۄ ȁݏଵۄ ٔ ǥ ٔ ȁݏିଵۄ ٔ ȁݏۄ
Figure 2.2: The recursive steps to transform the state into spin basis
the matrices A become independent of position. The wave function is now written
|Ψ =
s1,s2,...,sn
tr(A(s1)A(s2) . . . A(sn))|s1s2. . . sn. (2.3) This is the matrix product state we are going to use as a trial wave function to study the ground state of quantum spin systems, but before that, we would like to discuss more about the matrix product states and give the criteria for the validity of MPS as a trial wave function for such approximation.