Performance Study: Magnetization Plateau in a Mixed-Spin Chain¶

This performance example demonstrates how QS³-ED2 can efficiently handle site-dependent spin magnitudes and magnetization-sector constraints using

list_spin.dat
list_NODmax.dat

These features allow exact-diagonalization studies of mixed-spin systems, which are difficult to treat with traditional ED implementations.

As a representative benchmark, we analyze the spin-dependence of the magnetization plateau width in a one-dimensional mixed-spin chain consisting of alternating spins

\[ (S,1/2). \]

The plateau occurs at

\[ \frac{M}{M_s}=\frac{2S-1}{2S+1}. \]

Model Hamiltonian¶

We consider an antiferromagnetic mixed-spin chain described by

\[ \hat H = J\sum_{j=1}^{L} \left[ \hat{\mathbf s}_{2j}\cdot (\hat{\mathbf S}_{2j-1}+\hat{\mathbf S}_{2j+1}) - h(\hat S^z_{2j-1}+\hat s^z_{2j}) \right], \]

with periodic boundary conditions

\[ \hat{\mathbf S}_{2L+1}=\hat{\mathbf S}_1. \]

Here

\(\hat{\mathbf S}_{2j-1}\) : spin-\(S\) operator
\(\hat{\mathbf s}_{2j}\) : spin-\(1/2\) operator

The coupling constant satisfies

\[ J>0 \]

and we set

\[ J=1 \]

as the unit of energy.

The system contains \(2L\) sites corresponding to \(L\) unit cells.

Symmetry¶

The Hamiltonian is invariant under

two-site translation
site inversion

The ground state appears in the symmetry sector

momentum \(k=0\)
even inversion parity

corresponding to

M1 = 0
M2 = 0

in the ED2 input file.

Saturation magnetization¶

The saturation magnetization is

\[ M_s = \sum_{r=1}^{2L} S_r \]

For the alternating chain

\[ (S,1/2) \]

this becomes

\[ M_s = L(S+1/2) = \frac{L(2S+1)}{2}. \]

Magnetization sector¶

We denote the number of lowering operations from the fully polarized state by

\[ N_{\downarrow}. \]

In QS³-ED2 this corresponds to

NODmax

At the plateau position

\[ \frac{M}{M_s}=\frac{2S-1}{2S+1}, \]

the lowering number is

\[ N_{\downarrow}=L. \]

Therefore the plateau lies between the sectors

\[ N_{\downarrow}=L \]

and

\[ N_{\downarrow}=L-1. \]

QS³‑ED2 Input Configuration¶

For a system with \(2L\) sites

NOS    = 2L
L1     = L
L2     = 2
NO_one = 0
NO_two = 2L

The symmetry sector is specified as

M1 = 0
M2 = 0

The local spins and lowering limits are defined using

list_spin.dat
list_NODmax.dat

while the interaction pairs are specified using

list_p1.dat
list_p2.dat

Table 1: Structure of the Input Files¶

The following table summarizes the structure of the input files used in the mixed-spin chain example.

list_spin.dat	list_NODmax.dat	list_p1.dat	list_p2.dat
S	min[2S, NODmax]	3	1
0.5	1	4	2L
S	min[2S, NODmax]	5	2L − 1
0.5	1	6	2L − 2
S	min[2S, NODmax]	7	2L − 3
0.5	1	8	2L − 4
…	…	…	…
S	min[2S, NODmax]	1	3
0.5	1	2	2

Column descriptions¶

list_spin.dat
Specifies the local spin magnitude at each lattice site.

list_NODmax.dat
Defines the maximum number of lowering operations allowed at each site.
For a spin‑\(S\) site the maximum possible number of lowering operations is \(2S\), therefore the constraint is written as

min[2S, NODmax]

to ensure consistency with the physical Hilbert space.

list_p1.dat, list_p2.dat
Define the interacting site pairs used in the Hamiltonian and generate the alternating couplings along the chain.

Determining the Plateau Width¶

In a magnetic field \(h\), the ground state minimizes

\[ E_0(N_{\downarrow}) - hM(N_{\downarrow}). \]

The critical field between neighboring magnetization sectors is

\[ h_c(N_{\downarrow}) = E_0(N_{\downarrow})-E_0(N_{\downarrow}-1). \]

Thus the plateau width is obtained from the energy difference between the sectors

\(N_{\downarrow}=L\)
\(N_{\downarrow}=L-1\)

Finite‑Size Behaviour¶

Finite-size effects are extremely small.
Accurate results are obtained already for

\[ L = 2,3,\dots,8. \]

Benchmark Result¶

The numerical results

reproduce DMRG results for small \(S\)
approach the nonlinear spin‑wave prediction in the limit \(S \to \infty\).

Summary¶

This benchmark demonstrates that QS³‑ED2 can efficiently treat mixed-spin quantum systems with

heterogeneous local spins
flexible magnetization-sector constraints
symmetry-reduced Hilbert spaces

These features make QS³‑ED2 a powerful tool for investigating quantum spin systems with site-dependent spin magnitudes.