skB(sk)⊗ B(sk); also, when the operator ˜O acts on the kth site, we have T˜[A](k) =
skA(sk)⊗ ˜A(sk) or ˜T[B](k) =
skB(sk)⊗ ˜B(sk). Because the matrices and vectors are all kept in the normalization condition, Ψ|Ψ = 1, the expectation value is written
ˆO = tr(T[A](1)T[B](2)· · · ˜T[m](k)· · · T[A](n−1)T[B](n)), (4.22) where m = A or B depending on the odd site or the even site. Furthermore, the cyclic property of the trace can be used to simplify the calculation.
However, in this case, using the above wave function is only an approxi-mation because there is no inforapproxi-mation about the periodic boundary during the process of update. Take the simplest case for example, consider a two spins system, the schmidt rank is two. On the other hand, the above update scheme gives a schmidt rank D regardless of the lattice size. To fully account for the boundary effect, one should use matrices and vectors which depend on position. However, in this scheme, we need to update the matrices and vectors site by site and a lot of computing time and memory is requried. The QMC algorithm developed in Chapter 3 avoids this porblem and there is no explicit truncation during the update. The boundary effect incorporates into the wave fuction automatically.
4.5 Imaginary time evolution algorithm
1. Decompose the Hamiltonian into two non-commute groups F and G.
The operators in each group commutes.
2. Perform the Suzuki-Trotter decomposition to the imaginary time evo-lution operator e−τH.
e−τH = [e−δ(F +G)]τδ ≈ [e−δF/2e−δGe−δF/2]τδ where δ > 0 and δ/τ 1.
3. Apply the evolution operator evolving imaginary time δ 1 to the state.
|Ψtimag+δ ≈ e−δF/2e−δGe−δF/2|Ψtimag.
4. Perform the nomalization procedure.
5. Go back to 3 until it reaches time τ or until it reaches a optimal state.
1We use progressively decreasing values of δ ∈ {0.1, 0.01, 0.001, ...} to reach a better convergence.
4.6 Infinite transeverse Ising model
In this section, we investigate the infinite transverse Ising model by iTEBD.
The infinite transverse Ising model can be solved exactly via femioniza-tion [31]. Comparing the simulafemioniza-tion results with the exact solufemioniza-tions, it shows that the imaginary time evolution with MPS provides accurate approxima-tion for the ground state.
1.8 2.0
Figure 4.4: The distribution of the singular values λi for different h.
Fig. 4.4(a) shows the distribution of the singular values λi as a function of h. Recall that λi can be viewed as a measure of entanglement. We can compare it with the single site entanglement which is obtained from the
von Neumann entropy S calculated from the single site density matrix ρ1, S =−tr(ρ1logρ1). The single site entanglement can be viewed as a measure of how entangled the lattice is [32]. In Fig. 4.5 we plot the exact single site entanglement, and we also plot the exact nearest site correlation [31], C12z = σz1σ2z − σ1zσz2. The two show similar behavoir. The decreasing rate of the distribution of λi shows a qualitative agreement with the single site entanglement. For a fixed h, we observe that the singular values decrease exponentially and then after some value i, the decreasing rate becomes slower.
Furthermore, as h approaches to 1, the decreasing rate of the singular values also becomes slower, indicating that the system has a maximum entanglement at the critical point [12, 32].
0.0 0.5 1.0 1.5 2.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
entanglement
h
entaglement
nearest site correlation
Figure 4.5: The single site entanglement and the nearest site correlation.
Fig. 4.6 shows a typical convergence process of the energy. We compare the ground state energy with the exact solutions. It is shown in Fig. 4.7. With D = 1, the wave function is approximated as a product state which fails to account for the entanglement. This result corresponds to the molecular field theory [33] which is a mean field theory. With D > 1, as D grows larger, the ground state energy obtained by the imaginary time evolution converges to the exact energy. There are two main errors. One is the error from Suzuki-Trotter decomposition, and one is the error from the truncation steps in the updating process. Comparing Fig. 4.4(a) with Fig. 4.7, the cuvrve of the relative error of energy has a similar shape with the decreasing rate of λi as a function of h. This indcates that the main error comes from the truncation
error.
In Fig. 4.8, we plot the magnetization and compare them with the exact solution. The D = 1 case corresponds to the mean field results which ap-proximate the state as a product state, and the dotted line is obtained from the molecular field theory. In the case of D = 10, the magnetization σx shows a good agreement with the exact solution. Although the magnetiza-tion σz cannot be obtained via fermionization because of the spin inversion invariance of the hamiltoian, it can be obtained from the large−k limit of the correlation function σ1zσzk [31]. Fig. 4.9 shows that modest values of D can produce good approximations of the ground state, and as D becomes larger, the drop of the magnetization approaches the critical value h = 1.
Comparing Fig. 4.5 with Fig. 4.8, the magnetization of the product state approximation begins to deviate as soon as the entanglement begins to grow.
Also, the critical point in the product state approximation is not at h = 1 but h = 2. These results show that entanglement is essential to the quantum phase transition. The product state aprroximation fails to account for the correct characteristics of the system when it is near the critical point.
0 200 400 600 800 1000 1200 -1.282
-1.280 -1.278 -1.276 -1.274 -1.272 -1.270 -1.268
energy
update times
h=1.01; D=10 exact
Figure 4.6: Energy convergence process.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 10-10
10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
ΔE / E
h
D=1 D=2 D=4 D=10
Figure 4.7: Energy convergence as a function of h for different D.
0.0 0.5 1.0 1.5 2.0 2.5
Figure 4.8: Magnetization as a function of h. The dotted lines are the molecular field results.
0.90 0.95 1.00 1.05 1.10
0.0
Figure 4.9: Magnetization as a function of h for different D.
Chapter 5 Conclusions
In this thesis, we have reviewed the construction of MPS from two per-spectives: desity matrix renormalization group (DMRG) and quantum in-formation theory (QIT). The connection between these two perspectives are shown leading to a clear picture of the physical meaning, which relates to en-tanglement, about the effectiveness of turncation steps in DMRG. Next, We develope two algorithms in detail within MPS ansatz which can deal with one dimensional quantum spin systems. First, we develope quantum Monte Carlo simulation with stochastic optimization. The notion behind this method is easy to understand. By rewriting the equation of expectation values, the sampling of states becomes classical-like. We show the convergence behav-ior of this method with the periodic and open boundary conditions. The results show excellent agreement with exact solutions even at critical point.
We apply this method to a detailed study of the transverse Ising model.
In addition, this method is free from sign-problem. By combining different symmetries, we calculate the gound state and the first few excited states of the Heisenberg model with the next nearest enighbor interaction, and the re-sults give a clear evidence that the peak of the energy levels comes from the crossing of different symmetry states. Second, we develop the time-evolving block decimation algorithm (TEBD) and gives a detailed procedure for the imaginary tme evolution. We discuss the feasibility to apply TEBD to the periodic boundary condition. We also apply the infinite TEBD to the infi-nite transverse Ising model. The distribution of the Schmidt coefficients as a function of h is shown. Comparing with the convergence behavior of the en-ergy for different D, we identify the main error source in the imaginary time evolution to be the trucation error. Finally, comparing the single site en-tanglement, the nearest site correlation and the phase diagram, these results imply that the entanglement can be used as an indicator to the quantum phase transition.
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