Performance Example: Response Analysis with Fictitious Symmetry-Breaking Fields

This performance example demonstrates how QS³‑ED2 can reproduce numerical analyses of symmetry‑breaking responses in quantum spin systems near saturation magnetization.

The purpose of this page is not to present new physics, but to demonstrate that the numerical procedure used in previous studies can be practically executed using QS³‑ED2 for relatively large system sizes.

In particular, this benchmark illustrates that QS³‑ED2 enables

• calculations close to saturation magnetization
• controlled truncation of the Hilbert space using the number of down spins
• systematic analysis of responses to small transverse fields
• finite‑size scaling analysis of susceptibilities and order parameters

for lattice sizes that are difficult to treat using conventional exact‑diagonalization approaches.

The calculations below are performed for the antiferromagnetic Heisenberg model on the square lattice near the saturation field.

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Model

We consider the spin‑1/2 Heisenberg Hamiltonian

\[ \hat H = J\sum_{\langle i,j\rangle} \mathbf S_i \cdot \mathbf S_j - H \sum_i S_i^z - \delta h_u \sum_i S_i^x - \sum_i \delta h_s^{(i)} S_i^x . \]

Here

• \(J>0\) is the exchange interaction,
• \(H\) is the longitudinal magnetic field,
• \(\delta h_u\) is a uniform transverse field, and
• \(\delta h_s^{(i)}\) is a staggered transverse field.

The transverse fields are introduced as fictitious symmetry‑breaking fields in order to probe the response of the system and detect spontaneous symmetry breaking.

All calculations are performed near the saturation magnetization, where the number of down spins is small and QS³‑ED2 is particularly efficient.

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Hilbert‑Space Truncation Using the Number of Down Spins

QS³‑ED2 constructs the Hilbert space by specifying a cutoff

\[ N_{\downarrow} \]

for the number of down spins.

Because the system is studied close to the saturation magnetization, the relevant states are contained in sectors with small \(N_{\downarrow}\).

The first numerical task is therefore to determine how large the cutoff \(N_{\downarrow}\) must be when small transverse perturbations are applied.

The figure above shows the response of the transverse magnetization when small uniform and staggered transverse fields are applied at

\[ H/J = 3.7 . \]

This value corresponds to the center of the magnetization plateau intersecting this value of \(H/J\).

Using the plateau center stabilizes the numerical calculation when transverse fields are introduced.
Although the optimal longitudinal field slightly shifts for finite systems, the extrapolated value converges to \(H/J = 3.7\) in the thermodynamic limit, so the procedure remains consistent.

The numerical results show that the Hilbert‑space cutoff only needs to include

\[ N_{\downarrow}^{\prime} + 1 \]

where \(N_{\downarrow}^{\prime}\) is the number of down spins in the ground state when the transverse fields are zero.

This is physically reasonable because the transverse perturbation only mixes the ground state with nearby sectors.

Another important observation is that the response to the staggered transverse field is much larger than the response to the uniform transverse field.
This behavior is physically natural because the staggered field couples directly to the spin pattern associated with the symmetry‑broken state.

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Extraction of the Order Parameter

The second step is to estimate the spontaneous staggered magnetization in the thermodynamic limit.

Because exact diagonalization can only be performed on finite systems, we analyze the dependence of the transverse magnetization on the small staggered field.

Two fitting functions are introduced:

\[ y_1 = a\tanh(g\,\delta h_s/J) + b(\delta h_s/J) \]

\[ y_2 = y_1 + c(\delta h_s/J)^3 . \]

These functions do not have a strict theoretical justification; they are introduced as empirical fitting forms to test the stability of the extrapolation.

A key point is that in both cases the parameter

\[ a \]

corresponds to the staggered magnetization in the thermodynamic limit

\[ m_x = a \qquad (L \rightarrow \infty). \]

Thus, by fitting the finite‑field response we obtain an estimate of the spontaneous order parameter.

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Finite‑Size Scaling

The final step is to analyze the finite‑size scaling of the numerical results.

Two quantities are examined:

1. Staggered susceptibility

\[ \chi_s = \frac{1}{N\,\delta h_s} \sum_i \langle S_i^x \rangle \]

1. Spontaneous staggered magnetization extracted from the fitting parameter \(a\).

The susceptibility exhibits the scaling

\[ \chi_s \sim L^4 , \]

within the system sizes studied here. At present, we regard this as an empirical scaling observed in the present numerical data. A detailed comparison with previous theoretical and numerical studies remains to be carried out.

The finite‑size extrapolation of the order parameter yields

\[ m_x \approx 0.19 \]

at

\[ H/J = 3.79 \pm 0.03 . \]

This value appears to be compatible with previously reported numerical estimates, although a detailed literature comparison is left for future work.

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Significance for QS³‑ED2

This benchmark demonstrates that QS³‑ED2 can reproduce the numerical analysis used to identify spontaneous symmetry breaking in quantum spin systems.

In particular, QS³‑ED2 enables

• efficient calculations near saturation magnetization
• controlled truncation using the number of down spins
• numerical evaluation of response functions
• reliable finite‑size scaling analyses

for system sizes that are difficult to treat using conventional exact‑diagonalization approaches.

Therefore QS³‑ED2 provides a practical numerical platform for investigating symmetry‑breaking phenomena in strongly correlated quantum spin systems.