We employ a neural-network architecture based on the vision transformer (ViT) architecture to find the ground states of quantum long-range models, specifically the transverse-field Ising model for spin-1/2 chains across different interaction regimes. Harnessing the transformer's capacity to capture long-range correlations, we compute the full phase diagram and critical properties of the model in both the ferromagnetic and antiferromagnetic cases. Our findings show that the ViT maintains high accuracy across the full phase diagram. We compare these results with previous numerical studies in the literature and, in particular, show that the ViT has a superior performance than a restricted-Boltzmann-machine-like ansatz.
We employ a neural-network architecture based on the vision transformer (ViT) architecture to find the ground states of quantum long-range models, specifically the transverse-field Ising model for spin-1/2 chains across different interaction regimes. Harnessing the transformer's capacity to capture long-range correlations, we compute the full phase diagram and critical properties of the model in both the ferromagnetic and antiferromagnetic cases. Our findings show that the ViT maintains high accuracy across the full phase diagram. We compare these results with previous numerical studies in the literature and, in particular, show that the ViT has a superior performance than a restricted-Boltzmann-machine-like ansatz.
The transformer architecture has become the state-of-art model for natural language processing tasks and, more recently, also for computer vision tasks, thus defining the vision transformer (ViT) architecture. The key feature is the ability to describe long-range correlations among the elements of the input sequences, through the so-called self-attention mechanism. Here, we propose an adaptation of the ViT architecture with complex parameters to define a new class of variational neural-network states for quantum many-body systems, the ViT wave function. We apply this idea to the one-dimensional J_{1}-J_{2} Heisenberg model, demonstrating that a relatively simple parametrization gets excellent results for both gapped and gapless phases. In this case, excellent accuracies are obtained by a relatively shallow architecture, with a single layer of self-attention, thus largely simplifying the original architecture. Still, the optimization of a deeper structure is possible and can be used for more challenging models, most notably highly frustrated systems in two dimensions. The success of the ViT wave function relies on mixing both local and global operations, thus enabling the study of large systems with high accuracy.
Transformer neural networks, known for their ability to recognize complex patterns in high-dimensional data, offer a promising framework for capturing many-body correlations in quantum systems. We employ an adapted vision transformer (ViT) architecture to model quantum impurity models, optimizing it with a subspace expansion scheme that surpasses conventional variational Monte Carlo in both accuracy and efficiency. Benchmarks against matrix product states in single- and three-orbital Anderson impurity models show that these ViT-based neural quantum states achieve comparable or superior accuracy with significantly fewer variational parameters. We further extend our approach to compute dynamical quantities by constructing a restricted excitation space that effectively captures relevant physical processes, yielding accurate core-level x-ray absorption spectra. These findings highlight the potential of ViT-based neural quantum states for accurate and efficient modeling of quantum impurity models.
The simulation of interacting many-body quantum systems is a central chal- lenge in condensed matter physics due to the exponential growth of the Hilbert space with system size. In this project, we implement a Neural Network Quantum State (NQS) ansatz based on a Complex Restricted Boltzmann Machine (RBM) to solve for the ground state of the 1D Transverse Field Ising Model (TFIM). Using Variational Monte Carlo (VMC) optimization via Stochastic Reconfiguration (SR), we successfully train the network to approximate the ground state wave function across the paramagnetic-ferromagnetic phase transition. We benchmark our results against Exact Diagonalization (ED) for small systems (N = 12), demonstrating high accuracy in ground state energy estimation. Furthermore, we investigate quantum correlations and order parameters, specifically the transverse magnetization mx and longitudinal spin-spin correlations Czz(r), highlighting the RBM’s ability to capture long-range order and critical behavior near the quantum critical point hx/J = 1.
We investigate quantum phase transitions in the transverse field Ising chain with algebraically decaying long-range (LR) antiferromagnetic interactions using the variational Monte Carlo method with the restricted Boltzmann machine employed as a trial wave function ansatz. First, we measure the critical exponents and the central charge through the finite-size scaling analysis, verifying the contrasting observations in the previous tensor network studies. The correlation function exponent and the central charge deviate from the short-range (SR) Ising values at a small decay exponent α_{LR}, while the other critical exponents examined are very close to the SR Ising exponents regardless of α_{LR} examined. However, in the further test of the critical Binder ratio, we find that the universal ratio of the SR limit does not hold for α_{LR}<2, implying a deviation in the criticality. On the other hand, we find evidence of the conformal invariance breakdown in the conformal field theory (CFT) test of the correlation function. The deviation from the CFT description becomes more pronounced as α_{LR} decreases, although a precise breakdown threshold is yet to be determined.
We study the ground-state phase diagram of an unfrustrated antiferromagnetic Ising chain with longitudinal and transverse fields in the full range of interactions: from all-to-all to nearest-neighbors. First, we solve the model analytically in the strong long-range regime, confirming in the process that a mean-field treatment is exact for this model. We compute the order parameter and the correlations and show that the model exhibits a tricritical point where the phase transition changes from first to second order. This is in contrast with the nearest-neighbor limit where the phase transition is known to be second order. To understand how the order of the phase transition changes from one limit to the other, we tackle the analytically intractable interaction ranges numerically, using a variational quantum Monte Carlo method with a neural-network-based ansatz, the visual transformer. We show how the first-order phase transition shrinks with decreasing interaction range and establish approximate boundaries in the interaction range for which the first-order phase transition is present. Finally, we establish that the key ingredient to stabilize a first-order phase transition and a tricritical point is the presence of ferromagnetic interactions between spins of the same sublattice on top of antiferromagnetic interactions between spins of different sublattices. Tunable-range unfrustrated antiferromagnetic interactions are just one way to implement such staggered interactions.
We introduce a variational wave function based on neural-network quantum states (NQS) to study lattice systems whose local Hilbert space contains both spin and fermionic degrees of freedom. Our approach is based on the use of the transformer architecture, which can naturally handle composite local Hilbert spaces through a tokenization procedure closely inspired by techniques from natural language processing. The key methodological innovation is that the neural network predicts a set of fermionic orbitals that depend on the spin configuration in a backflow-inspired manner. We apply the method to the one-dimensional ancilla layer model, consisting of a chain of mobile spin- 1 / 2 fermions coupled to a two-leg spin- 1 / 2 ladder. For open boundary conditions, we achieve excellent quantitative agreement with density matrix renormalization group results across the full range of parameters considered. We find a phase in which the chain forms an effectively decoupled Luttinger liquid (LL), and an LL * phase with a distinct Fermi wave vector in which the mobile fermions are Kondo screened by one leg of the ladder, while the other leg forms the critical Bethe spin liquid. The LL * is the analog of the FL * phase in two dimensions proposed to describe the cuprate pseudogap. We also find a Luther-Emery phase, where the LL * state becomes unstable toward the formation of a spin gap. The transformer ansatz maintains comparable accuracy for periodic boundary conditions, where tensor-network methods are computationally more demanding. Moreover, we show that the approach extends naturally to fully periodic two-dimensional clusters, a regime beyond the reach of current tensor-network methods. Together, these findings establish transformer-based NQS as an accurate and scalable variational framework for correlated lattice systems with composite local Hilbert spaces and highlight its potential for studying higher-dimensional models where boundary effects and heterogeneous local structures pose significant challenges.