Quantum simulation of general spin-1/2 Hamiltonians with parity-violating fermionic Gaussian states

TL;DR

This work presents parity-violating fermionic mean-field theory (PV-FMFT), a scalable Gaussian-variational framework for simulating general spin-1/2 Hamiltonians. By extending fermionic Gaussian states to parity-violating sectors and employing the Colpa mapping to an enlarged Hilbert space, PV-FMFT preserves the structure of the PP-FMFT equations of motion while enabling treatment of parity-violating terms common in spin mappings. The authors derive explicit imaginary- and real-time EOMs for PV-FMFT, demonstrate exact results for non-interacting spin systems, and benchmark post-quench dynamics of one- and two-dimensional TFIM against MPS and discrete-Wigner methods, highlighting strengths and symmetry-breaking limitations introduced by spin-to-fermion mappings. The approach provides a practical benchmark and tool for studying quantum dynamics in spin-1/2 systems and for validating quantum simulators, with scalable performance and clear avenues for refinement through alternative mappings and basis optimization.

Abstract

We introduce equations of motion for a parity-violating fermionic mean-field theory (PV-FMFT): a numerically efficient fermionic mean-field theory based on parity-violating fermionic Gaussian states (PV-FGS). This work provides explicit equations of motion for studying the real- and imaginary-time evolution of spin-1/2 Hamiltonians with arbitrary geometries and interactions. We extend previous formulations of parity-preserving fermionic mean-field theory (PP-FMFT) by including fermionic displacement operators in the variational Ansatz. Unlike PP-FMFT, PV-FMFT can be applied to general spin-1/2 Hamiltonians, describe quenches from arbitrary initial spin-1/2 product states, and compute local and non-local observables in a straight-forward manner at the same modest computational cost as PP-FMFT -- scaling as $O(N^3)$ in the worst case for a system of $N$ spins or fermionic modes. We demonstrate that PV-FMFT can exactly capture the imaginary- and real-time dynamics of non-interacting spin-1/2 Hamiltonians. We then study the post quench-dynamics of the one- and two-dimensional Ising model in presence of longitudinal and transversal fields with PV-FMFT and compute the single site magnetization and correlation functions, and compare them against results from other state-of-the-art numerical approaches. In two-dimensional spin systems, we show that the employed spin-to-fermion mapping can break rotational symmetry within the PV-FMFT description, and we discuss the resulting consequences for the calculated correlation functions. Our work introduces PV-FMFT as a benchmark for other numerical techniques and quantum simulators, and it outlines both its capabilities and its limitations.

Figures (8)

• Figure 1: Schematic relation between spin mean-field theory (SMFT) and fermionic mean-field theory (FMFT). Each circle represents a family of Hamiltonians and states which are exactly described by a mean-field theory based on spin-1/2 product states (purple), parity-preserving fermionic Gaussian states (PP-FGS, orange), or parity-violating (PV)-FGS (green), respectively. The vast majority of previous FMFT formulations relied on PP-FGS, restricting applicability to spin systems whose fermionic representation is parity preserving. The PV-FGS we use in this work to formulate a PV-FMFT enable real- and imaginary-time evolution of arbitrary spin-1/2 and fermionic systems at low computational cost.
• Figure 2: Workflow diagram describing the steps to represent an arbitrary spin-1/2 Hamiltonian (here a two-dimensional square lattice) as a PP fermionic Hamiltonian. We consider the TFIM model where the spin-1/2 Hamiltonian is mapped to its fermionic description via the Jordan-Wigner transformation. While interaction terms (orange box) result in PP fermionic operators, local fields can lead to PV terms (blue box), which cannot be described by PP-FMFT. By introducing a single auxiliary mode ('ghost particle') we can use the Colpa mapping: previously odd fermionic monomials are turned to even fermionic monomials (dark green box), while previously even fermionic monomials are left invariant. The PV Hamiltonian prior to the mapping is therefore turned into a PP Hamiltonian, for which we formulate a FMFT based on a linear combination of two FGS of opposite parities. Thick colored arrows in the bottom two lattices represent the fermionic modes which are involved when describing the single-spin as well as spin-interaction terms under the Jordan-Wigner and Colpa mapping, assuming snake ordering (red small arrows) of the lattice.
• Figure 3: Results for the PV-FMFT applied to the random non-interacting spin Hamiltonian of Eq. \ref{['nis1']} for 7 (blue) and 8 (green) spins, respectively. (a) Reaching the ground state starting from a random PV-FGS using the FGS ZT method (full markers) and the FGS ITE method (hollow markers). (b) Single site magnetization for the PV-FMFT method (markers) and the exact evolution (solid lines) after a quench from the initial state $\ket{\mathbf 0}$.
• Figure 4: Sketch of the longitudinal field strengths $|\zeta_k|$ on the (a) one-dimensional spin chain and (b) two-dimensional square lattice. The thin red line in the 2D square lattice indicates the snake-ordering used throughout the manuscript. The central sites (orange node) experience a zero longitudinal field $|\zeta_\mathrm{c}|=0$, while all other sites experience non-zero fields that increase with distance from the center. The green sites denote the NN sites considered for the averaged connected correlation function of the central site. For the one-dimensional chain and nearest-neighbor interactions ($\alpha\rightarrow\infty$) only the boundary sites have a non-zero longitudinal field component.
• Figure 5: (a) Single-site magnetization $m_c$ and (b) connected correlation function $C^{\mathrm{NN}}_{\mathrm{c}}$ (between c and $\mathrm{c}+1$) for the TFIM with a nonzero longitudinal field and OBC on a one-dimensional chain ($N=81$) and van der Waals interaction ($\alpha=6$) and $h_x/J=1.0$ with $h_x=\Omega/2$. PV-FMFT ($ZZ$- and $XX$-representation) is compared against MPS results with a maximum bond dimension of $\xi=900$.