Variational method for integrability-breaking Richardson-Gaudin models

We present a variational method for approximating the ground state of spin models close to (Richardson-Gaudin) integrability. This is done by variationally optimizing eigenstates of integrable Richardson-Gaudin models, where the toolbox of integrability allows for an efficient evaluation and minimization of the energy functional. The method is shown to return exact results for integrable models and improve substantially on perturbation theory for models close to integrability. For large integrability-breaking interactions, it is shown how (avoided) level crossings necessitate the use of excited states of integrable Hamiltonians in order to accurately describe the ground states of general nonintegrable models.

Figure 1

Results for the central spin model with perturbation $\mu S_i^0$. Top: Variational energy (Var.), exact ground-state energy (Exact), and first-order perturbation theory (PT) energy for different values of the perturbation strength. Bottom: Overlap of the exact ground state with the variational ground state and the ground state of the unperturbed model.

Figure 2

Variational parameters for the Hamiltonian (23). Position of $\vec{\epsilon }$ (squares) and $\vec{\lambda }$ (dots) for the variationally optimized wave function in the complex plane at different values of the perturbation strength. The white square denotes the variable ${\epsilon }_i$ associated with the level on which the perturbation is applied.

Figure 3

Results for the central spin model with perturbation $\mu {\vec{S}}_i·{\vec{S}}_j$, with $i,j=2,L-1$. Top: Variational energy, exact ground-state energy, and first-order perturbation theory energy for different values of the perturbation strength. Bottom: Overlap of the exact ground state with the variational ground state and the ground state of the unperturbed model.

Figure 4

Results for the central spin model with perturbation $\lambda {\vec{S}}_i·{\vec{S}}_j$. Top: Variational energy, exact ground-state energy, and first-order perturbation theory energy starting from the ground and excited state of the integrable model for different values of the perturbation strength. Bottom: Overlap of the exact ground state with the variational ground state and the ground state of the unperturbed model starting from both the ground and excited state of the integrable model. The perturbation has been applied to the spin with the strongest interaction with the central spin in order to maximize the effect of the perturbation.

Figure 5

Variational parameters for the Hamiltonian (25). Position of $\vec{\epsilon }$ (squares) and $\vec{\lambda }$ (dots) for the variationally optimized wave function in the complex plane at different values of the perturbation strength. The white squares denote the variables ${\epsilon }_i,{\epsilon }_j$ associated with the levels on which the perturbation is applied.

Figure 6

Results for the central spin model with perturbation $\lambda {\vec{S}}_i·{\vec{S}}_j$. Position of $\vec{\epsilon }$ (squares) and $\vec{\lambda }$ (dots) for the variationally optimized wave function starting from an excited state in the complex plane at different values of the perturbation strength. The white squares denote the variables ${\epsilon }_i,{\epsilon }_j$ associated with the levels on which the perturbation is applied.

Figure 7

Absolute value of the expectations values $\langle \overrightarrow{S_i}·{\vec{S}}_j\rangle$ for the central spin model with perturbation $\mu {\vec{S}}_i·{\vec{S}}_j$, with $i,j=2,L-1$. The expectation values are taken with respect to the exact ground state and the variational state obtained by starting from both the ground state and excited state of the unperturbed model.

Figure 8

Correlation coefficients $\langle S_i^0{S}_j^0\rangle -\langle S_i^0\rangle \langle S_j^0\rangle$ for the central spin model with perturbation $\mu {\vec{S}}_i·{\vec{S}}_j$, with $i,j=2,L-1$. These correlation coefficients are calculated for the exact ground state and the variational state obtained by starting from both the ground state and excited state of the unperturbed model.

Figure 9

Results for the inhomogeneous BCS model. Top: Variational energy, exact ground-state energy, and first-order perturbation theory for different values of the perturbation strength. Bottom: Overlap of the exact ground state with the variational ground state and the ground state of the unperturbed model.

Figure 10

Results for the inhomogeneous BCS model. Position of $\vec{\epsilon }$ (squares) and $\vec{\lambda }$ (dots) for the variationally optimized wave function in the complex plane at different values of the perturbation strength.