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Assignment 4
Course: Computational Physics (PHYS4150/8150) – Prof. Zi Yang Meng
Tutor: Mr. Tim Lok Chau, Mr. Min Long
Due date: Nov. 25th, 2024
This assignment is also a project; you are encouraged to form a group of 2 to 4 people and present your work on these questions during the classes on Nov 25th and 28th. The presentation should contain your answers to the questions, how you solved the problems, and your understanding of them.
1. Density matrix
The density matrix ρ for a quantum system is ρ = |ψ⟩ ⟨ψ| where ψ is an arbitraryquantum state. For a single spin, an arbitrary state can be written as |ψ⟩ = a|0⟩ + b|1⟩, where |0⟩ and |1⟩ represent the spin up and down state, respectively. The density matrix has the form ρ = (a|0⟩ + b|1⟩)(a
∗
If we were only interested in information on the subsystem A. In principle, wecan derive a reduced density matrix which only contains the information of A.The reduced density matrix is derived by tracing out the state from subsystemB from the density matrix ρ (the eigenbasis of B is denoted as |b⟩, please do notconfuse with the amplitude b in ρ) (how about now, we don’t need to mentionsite at all?)
To compute the reduced density matrix, we write the state as |ψ⟩ = ∑ab ψab|a⟩|b⟩, where |a⟩ and |b⟩ if the basis states from subsystems A and B separately. The density matrix has the form:
1Computational Physics (PHYS4150/8150) – Assignment 4 2
Then by definition:
where [ψ] is a matrix and the matrix element is , b
is the
Hermitian conjugate of [ψ]. In the last line, we perform the matrix multiplication for two matrices.
We calculate the reduced density matrix for a spin system as an example here. An arbitrary state can be written as . We label the left spin subsystem A and the right spin subsystem B. It is easy to write the matrix . According to the formula above, the reduced density matrix of subsystem A is
The reduced density matrix contains the entanglement information between sub systems A and B (the remaining part of the system). To extract such information from the reduced density matrix, we calculate the Rényi entropy between the subsystem and the bath.
The q-th Rényi entropy is defined as:
It can be analytically reduced to the Von Neumann Entanglement entropy when
Now, consider a one-dimensional quantum spin-1/2 chain. The Hamiltonian is
defined as
where Si
is the operator of the i-th spin. N is the number of sites in the lattice and the antiferromagnetic Heisenberg interaction is J = 1. The periodic boundary condition is met by letting
For q = 1.2, 1.5, 2, 3 and N = 10, calculate the corresponding Rényi entropy as a function of subsystem size compared with the result of Von Neumann entropyComputational Physics (PHYS4150/8150) – Assignment 4 3 using the abovementioned formula. (i.e. compute the Rényi entropy of the subsystem, which only contains the left lA sites of the spin chain. Vary q and plot the dependence of Rényi entropy on q, and you can compare your result with Figure 1 ).
Figure 1: Rényi entropy with different q value.
2. Transvers field Ising model
Consider a 1D chain with periodic boundary conditions, with one spin lives on each site. Each spin can be either up (σz = 1) or down (σz = −1). The
Hamiltonian is defined as
where σi
z
is the operator measures the z component of the i-th spin. h is the strength of the transverse field, and it is on the x-direction. Also, we have σN
z
+1 = σ1
z
from the periodic boundary condition. Here, J and h are positive, so the spin chain is ferromagnetic, and the external magnetic field points toward the positive x direction.
By properties of Pauli matrices, we have
where σi
+ flips the i-th spin from down to up, and equals 0 if the i-th site is already spin up, and vice versa for σi
−. For example,
Unlike the Heisenberg model, where you need to flip two adjacent spins simultaneously, here, you only need to flip one spin each time. It is known that a quantum phase transition for the transverse field Ising model happens on hc = 1.
(a) Consider N = 10, h = 0, 0.5, 0.1, ..., 2.0 and J = 1, find the 50-th lowest energy states, plot En − E0 (E0 is the ground state energy value) versus h, and
Figure 2: Energy spectrum of Transverse Field Ising Model.
you can compare your result with Figure 2. Can you interpret the information of a quantum phase transition at hc = 1 from these data?
(b) Compute the second-order Rényi entropy of the TFIM at h = hc = 1, J = 1 with N = 10, and lA ranging from 0 to 10, and compare your result with Figure 3.
Figure 3: Rényi entropy of the Transverse field Ising Model at h = hc = 1 with N = 10.
(c) (Optional) One can exactly diagonalize the longer chain and look for the scaling behaviour of the Rényi entanglement entropy. As shown in Fig.4, we plot the second-order Rényi entropy for N = 8, 10 and 12 versus lA/N in the left panel and then rescale them according to the formula.
(b) Rényi entropy for N = 8, 10, 12 with scaled rescaled x-axis. We ignore ln(
N
π
sin( N
π
lA)) < 0 data points
In the right panel, the data follows nicely with c = 1/2, which is the central charge of this quantum Ising phase transition. Please reproduce the two graphs above and comment on your findings.
Hint for part c):
• For N = 12, you only need to calculate Rényi entropy to la = N/2 and flip the result along the x-axis since the values of Rényi entropy are symmetric. It is also applicable to N = 8, 10.
• For the fitting line in the right panel, you can fix c = 1/2 in the linear fitting. You only need to do the linear fitting to N = 12 with lA = 2, 3, ..., 6 data.
