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arXiv:2504.10774 · Physical review. B./Physical review. B

Ground-state-based model reduction with unitary circuits

Shengtao Jiang, Steven R. White

Published Jul 14, 2025• 0 citations
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Abstract

We present a method to numerically obtain low-energy effective models based on a unitary transformation of the ground state. The algorithm finds a unitary circuit that transforms the ground state of the original model to a projected wavefunction with only the low-energy degrees of freedom. The effective model can then be derived using the unitary transformation encoded in the circuit. We test our method on the one-dimensional and two-dimensional square-lattice Hubbard model at half-filling, and obtain more accurate effective spin models than the standard perturbative approach.

DOI
10.1103/h1pt-v5kz

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We present a method to numerically obtain low-energy effective models based on a unitary transformation of the ground state. The algorithm finds a unitary circuit that transforms the ground state of the original model to a projected wavefunction with only the low-energy degrees of freedom. The effective model can then be derived using the unitary transformation encoded in the circuit. We test our method on the one-dimensional and two-dimensional square-lattice Hubbard model at half-filling, and obtain more accurate effective spin models than the standard perturbative approach.
The effective low-energy models of the Hubbard model are usually derived from perturbation theory. Here we derive the effective model of the Hubbard model in spin space and t-J space using a unitary transformation from numerical optimization. We represent the Hamiltonian as Matrix product state(MPO) and represent the unitary transformation using gates according to tensor network methods. We obtain this unitary transformation by optimizing the unitary transformation between the ground state of the Hubbard model and the projection of the Hubbard model ground state into spin space and t-J space. The unitary transformation we get from numerical optimization yields effective models that are in line with perturbation theories. This numerical optimization method starting from ground state provides another approach to analyze effective low-energy models of strongly correlated electron systems.
A Quantum Monte Carlo method of calculating operator expectation values for the ground state of the nearest-neighbor Heisenberg model on large square lattices is presented, along with some comparisons of the results obtained from this method and alternative methods for 2D lattices. In the simulations the ground state is projected out from an arbitrary state and sampled in a valence\nbond basis spanning the spin singlet subspace. The projection method is then extended to be used on a simplified Hubbard model at half-filling, which is\naptly called the T0-less Hubbard model, for arbitrary relative strengths of the electron hopping and electron-electron repulsion energy parameters. Results for calculated ground state energies in 1D are presented. These calculations are performed using a spin-charge separated basis, in which the valence bonds\nare generalized to also carry charge.
We present a robust scheme to derive effective models non-perturbatively for quantum lattice models when at least one degree of freedom is gapped. A combination of graph theory and the method of continuous unitary transformations (gCUTs) is shown to efficiently capture all zero-temperature fluctuations in a controlled spatial range. The gCUT can be used either for effective quasi-particle descriptions or for effective low-energy descriptions in case of infinitely degenerate subspaces. We illustrate the method for 1d and 2d lattice models yielding convincing results in the thermodynamic limit. We find that the recently discovered spin liquid in the Hubbard model on the honeycomb lattice lies outside the perturbative strong-coupling regime. Various extensions and perspectives of the gCUT are discussed.
It is shown that a unitary transformation which eliminates the terms which couple states with different numbers of doubly occupied sites in the Hubbard model, proposed recently by MacDonald, Girvin, and Yoshioka, is conceptually equivalent to a canonical perturbation expansion, derived over a decade ago. In the half-filled case, the latter method gives an effective Hamiltonian in which many-body interactions result from even numbers of interconfigurational transitions. It is argued that it is hard to conclude anything about the nature of the ground state from the energies of the N\'eel state and various valence-bond states, as calculated by MacDonald et al., due to the importance of quantum fluctuations in a two-dimensional lattice.
We derive a low energy effective model for a two dimensional (2D) system of spinless fermions on a square lattice with repulsive short-range interactions and away from half filling. This model is a natural 2D analogue of the Luttinger model which describes interacting fermions and can be treated exactly by bosonization. It also includes fermion degrees of freedom which cannot be bosonized but, as we argue, can be treated adequately by mean field theory. Our results generalize to 2D a standard approach to 1D lattice fermion systems making them amenable to non-perturbative computation methods by a suitable continuum limit.
Working on a square lattice with an even number of sites N2{\mathit{N}}^{2}N2 and assuming a smooth connection between strong and weak coupling, we show that the ground state of the two-dimensional half-filled Hubbard model has momentum kx{\mathit{k}}_{\mathit{x}}kx​-ky{\mathit{k}}_{\mathit{y}}ky​=0, it is even under reflections, and transforms as an s wave under rotations when N/2 is even and as a d wave when N/2 is odd.