Variational non-gaussian approach to interacting spin-boson models

Technical Summary: Variational Non-Gaussian Approach to Interacting Spin-Boson Models

Problem Statement
Interacting spin-boson models, where discrete quantum degrees of freedom (spins) couple to one or more bosonic modes (photons or phonons), are central to contemporary quantum science, including cavity QED, circuit QED, and spin-phonon systems. A primary challenge in simulating these systems arises from the infinite-dimensional nature of the bosonic Hilbert space combined with strong many-body correlations in the spin sector. Conventional numerical methods face a trade-off: truncating the bosonic occupation basis introduces bias, while relaxing the spin sector to mean-field descriptions fails to capture essential spin-boson entanglement. Furthermore, direct tensor network approaches (like DMRG) applied to the full composite system often suffer from rapid growth in bond dimension due to the need to explicitly represent the bosonic field alongside the spins.

Methodology
The authors propose a hybrid variational framework that decouples the optimization of the bosonic sector from the spin sector. The ground state is represented within a compact non-Gaussian variational manifold defined by the ansatz:
$$|\psi_{\text{NGS}}\rangle = U_\lambda (U_{\text{GS}} |0_b\rangle \otimes |\phi_s\rangle)$$
where:

1. Bosonic Gaussian Layer ($U_{\text{GS}}$): A Gaussian transformation comprising displacement ($\Delta_R$) and squeezing ($\xi$) operators acting on the bosonic vacuum $|0_b\rangle$. This captures the bosonic fluctuations and mean-field shifts.
2. Dressing Transformation ($U_\lambda$): A unitary operator that introduces spin-boson correlations beyond the mean-field level. It acts as a spin-dependent displacement (polaron-like transformation) controlled by a variational parameter $\lambda$. This layer maps the linear spin-boson coupling into effective spin-spin interactions.
3. Effective Spin Sector ($|\phi_s\rangle$): The spin state is not a simple product state but the exact ground state of an effective spin Hamiltonian ($H_{\text{eff}}$). This Hamiltonian is obtained by averaging the full Hamiltonian over the bosonic Gaussian state and the dressing transformation.

The method employs a self-consistent optimization loop:

• For fixed bosonic parameters ($\Delta_R, \xi, \lambda$), the effective spin Hamiltonian is solved using DMRG (Density Matrix Renormalization Group) to obtain $|\phi_s\rangle$ and spin expectation values.
• These spin observables are used to update the bosonic variational parameters by minimizing the total variational energy functional.
• The loop iterates until convergence of the energy and selected observables.

This approach avoids explicit truncation of the photonic field. Instead, the bosonic degrees of freedom are variationally integrated out, mapping the problem onto a self-consistent spin model.

Key Contributions and Results
The paper benchmarks this NGS-DMRG framework against the Dicke and Dicke-Ising models, comparing results with converged reference DMRG simulations (which use explicit photon cutoffs).

1. Accuracy and Variational Improvement:

   • The non-Gaussian approach consistently yields lower ground-state energies than the Gaussian limit (mean-field), confirming the variational improvement.
   • For the Dicke model, the method accurately reproduces the superradiant quantum phase transition (QPT) and finite-size scaling, matching reference DMRG results with significantly reduced computational resources.
   • In the Dicke-Ising model, the framework successfully maps a complex phase diagram featuring:
     • Continuous (second-order) transitions between normal and superradiant phases.
     • Discontinuous (first-order) transitions and tricritical points in the ferromagnetic regime.
     • A narrow intermediate phase of coexisting antiferromagnetic order and superradiance (AFM-SP) in the antiferromagnetic regime, a feature predicted by mean-field theory but difficult to resolve numerically.
   • The method captures finite-size fluctuations and converges to the thermodynamic limit, where spin-photon entanglement vanishes, consistent with established analytical limits.

2. Computational Efficiency:

   • Bond Dimension Reduction: By treating bosons variationally, the DMRG solver operates on an effective spin-only Hamiltonian. This results in a significantly lower required MPS bond dimension ($\chi$) compared to reference DMRG, which must explicitly handle the composite spin-boson chain.
   • Runtime: Despite the overhead of the self-consistent loop, the reduced tensor network complexity leads to a substantial decrease in total execution time. The outer optimization loop demonstrates exponential convergence.
   • The advantage is most pronounced in regimes with strong correlations and complex phase boundaries (e.g., the coexistence phase), where reference DMRG requires very large bond dimensions and photon cutoffs.

Significance
The paper claims that this hybrid framework offers a distinct algorithmic pathway compared to local basis optimization (LBO) tensor network methods. By capturing photonic modes through variational parameters (displacement, squeezing, dressing) rather than iterative basis reconfiguration, the method allows the DMRG solver to focus exclusively on spin correlations.

The significance lies in providing a robust, non-perturbative numerical method capable of accessing strongly correlated spin-boson physics without eliminating or arbitrarily truncating the bosonic field. The authors demonstrate that this approach can accurately reproduce complex phase diagrams and critical behaviors in models like the Dicke-Ising chain, achieving high fidelity with reduced computational cost. The authors provide an open-source implementation (available at https://github.com/Jpedromend/NGS) that supports arbitrary-range spin-spin couplings and disorder, with potential extensions to multi-mode systems and higher-dimensional geometries via alternative solvers (e.g., PEPS, QMC).