arXiv subjects

Federico Becca

Publications and source records attributed to Federico Becca.

At least 37 records · Page 2Linked to original sources

arXiv2023

Static and dynamical signatures of Dzyaloshinskii-Moriya interactions in the Heisenberg model on the kagome lattice

Motivated by recent experiments on Cs$_2$Cu$_3$SnF$_{12}$ and YCu$_{3}$(OH)$_{6}$Cl$_{3}$, we consider the ${S=1/2}$ Heisenberg model on the kagome lattice with nearest-neighbor super-exchange $J$ and (out-of-plane) Dzyaloshinskii-Moriya interaction $J_D$, which favors (in-plane) ${\bf Q}=(0,0)$ magnetic order. By using both variational Monte Carlo (based upon Gutzwiller-projected fermionic wave functions) and tensor-network approaches (built from infinite projected-entangled pair/simplex states), we show that the ground state develops a finite magnetization for $J_D/J \gtrsim 0.03 - 0.04$, while the gapless spin liquid remains stable for smaller values of the Dzyaloshinskii-Moriya interaction. The relatively small value of $J_D/J$ for which magnetic order sets in is particularly relevant for the interpretation of low-temperature behaviors of kagome antiferromagnets, including ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$. In addition, we assess the spin dynamical structure factors and the corresponding low-energy spectrum, by using the variational Monte Carlo technique. The existence of a continuum of excitations above the magnon modes is reported within the magnetically ordered phase, similarly to what has been detected by inelastic neutron scattering on Cs$_{2}$Cu$_{3}$SnF$_{12}$.

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arXiv2022

A real-space many-body marker for correlated ${\mathbb Z}_2$ topological insulators

Taking the clue from the modern theory of polarization [R. Resta, Rev. Mod. Phys. {\bf 66}, 899 (1994)], we identify an operator to distinguish between ${\mathbb Z}_2$-even (trivial) and ${\mathbb Z}_2$-odd (topological) insulators in two spatial dimensions. Its definition extends the position operator [R. Resta and S. Sorella, Phys. Rev. Lett. {\bf 82}, 370 (1999)], which was introduced in one-dimensional systems. We first show a few examples of non-interacting models, where single-particle wave functions are defined and allow for a direct comparison with standard techniques on large system sizes. Then, we illustrate its applicability for an interacting model on a small cluster, where exact diagonalizations are available. Its formulation in the Fock space allows a direct computation of expectation values over the ground-state wave function (or any approximation of it), thus allowing us to investigate generic interacting systems, such as strongly-correlated topological insulators.

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arXiv2022

Charge-density waves in kagome-lattice extended Hubbard models at the van Hove filling

The Hubbard model on the kagome lattice is presently often considered as a minimal model to describe the rich low-temperature behavior of AV$_{3}$Sb$_{5}$ compounds (with A=K, Rb, Cs), including charge-density waves (CDWs), superconductivity, and possibly broken time-reversal symmetry. Here, we investigate, via variational Jastrow-Slater wave functions, the properties of its ground state when both onsite $U$ and nearest-neighbor $V$ Coulomb repulsions are considered at the van Hove filling. Our calculations reveal the presence of different interaction-driven CDWs and, contrary to previous renormalization-group studies, the absence of ferromagnetism and charge- or spin-bond order. No signatures of chiral phases are detected. Remarkably, the CDWs triggered by the nearest-neighbor repulsion possess charge disproportionations that are not compatible with the ones observed in AV$_{3}$Sb$_{5}$. As an alternative mechanism to stabilize charge-bond order, we consider the electron-phonon interaction, modeled by coupling the hopping amplitudes to quantum phonons, as in the Su-Schrieffer-Heeger model. Our results show the instability towards a tri-hexagonal distortion with $2\times 2$ periodicity, in a closer agreement with experimental findings.

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arXiv2022

Symmetric projected entangled-pair states analysis of a phase transition in coupled spin-1/2 ladders

Infinite projected entangled-pair states (iPEPS) have been introduced to accurately describe many-body wave functions on two-dimensional lattices. In this context, two aspects are crucial: the systematic improvement of the {\it Ansatz} by the optimization of its building blocks, i.e., tensors characterized by bond dimension $D$, and the extrapolation scheme to reach the "thermodynamic" limit $D \to \infty$. Recent advances in variational optimization and scaling based on correlation lengths demonstrated the ability of iPEPS to capture the spontaneous breaking of a continuous symmetry in phases such as the antiferromagnetic (Néel) phase with high fidelity, in addition to valence-bond solids which are already well described by finite-$D$ iPEPS. In contrast, systems in the vicinity of continuous quantum phase transitions still present a challenge for iPEPS, especially when non-abelian symmetries are involved. Here, we consider the iPEPS Ansatz to describe the continuous transition between the (gapless) antiferromagnet and the (gapped) paramagnet that exists in the $S=1/2$ Heisenberg model on coupled two-leg ladders. In particular, we show how accurate iPEPS results can be obtained down to a narrow interval around criticality and analyze the scaling of the order parameter in the Néel phase in a spatially anisotropic situation.

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arXiv2022

Stripes in the extended $t-t^\prime$ Hubbard model: A Variational Monte Carlo analysis

By using variational quantum Monte Carlo techniques, we investigate the instauration of stripes (i.e., charge and spin inhomogeneities) in the Hubbard model on the square lattice at hole doping $δ=1/8$, with both nearest- ($t$) and next-nearest-neighbor hopping ($t^\prime$). Stripes with different wavelengths $λ$ (denoting the periodicity of the charge inhomogeneity) and character (bond- or site-centered) are stabilized for sufficiently large values of the electron-electron interaction $U/t$. The general trend is that $λ$ increases going from negative to positive values of $t^\prime/t$ and decreases by increasing $U/t$. In particular, the $λ=8$ stripe obtained for $t^\prime=0$ and $U/t=8$ [L.F. Tocchio, A. Montorsi, and F. Becca, SciPost Phys. {\bf 7}, 21 (2019)] shrinks to $λ=6$ for $U/t\gtrsim 10$. For $t^\prime/t<0$, the stripe with $λ=5$ is found to be remarkably stable, while for $t^\prime/t>0$, stripes with wavelength $λ=12$ and $λ=16$ are also obtained. In all these cases, pair-pair correlations are highly suppressed with respect to the uniform state (obtained for large values of $|t^\prime/t|$), suggesting that striped states are not superconducting at $δ=1/8$.

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arXiv2022

Accuracy of Restricted Boltzmann Machines for the one-dimensional $J_1-J_2$ Heisenberg model

Neural networks have been recently proposed as variational wave functions for quantum many-body systems [G. Carleo and M. Troyer, Science 355, 602 (2017)]. In this work, we focus on a specific architecture, known as Restricted Boltzmann Machine (RBM), and analyse its accuracy for the spin-1/2 $J_1-J_2$ antiferromagnetic Heisenberg model in one spatial dimension. The ground state of this model has a non-trivial sign structure, especially for $J_2/J_1>0.5$, forcing us to work with complex-valued RBMs. Two variational Ansätze are discussed: one defined through a fully complex RBM, and one in which two different real-valued networks are used to approximate modulus and phase of the wave function. In both cases, translational invariance is imposed by considering linear combinations of RBMs, giving access also to the lowest-energy excitations at fixed momentum $k$. We perform a systematic study on small clusters to evaluate the accuracy of these wave functions in comparison to exact results, providing evidence for the supremacy of the fully complex RBM. Our calculations show that this kind of Ansätze is very flexible and describes both gapless and gapped ground states, also capturing the incommensurate spin-spin correlations and low-energy spectrum for $J_2/J_1>0.5$. The RBM results are also compared to the ones obtained with Gutzwiller-projected fermionic states, often employed to describe quantum spin models [F. Ferrari, A. Parola, S. Sorella and F. Becca, Phys. Rev. B 97, 235103 (2018)]. Contrary to the latter class of variational states, the fully-connected structure of RBMs hampers the transferability of the wave function from small to large clusters, implying an increase of the computational cost with the system size.

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arXiv2021

The Hubbard model on triangular $N$-leg cylinders: chiral and non-chiral spin liquids

The existence of a gapped chiral spin liquid has been recently suggested in the vicinity of the metal-insulator transition of the Hubbard model on the triangular lattice, by intensive density-matrix renormalization group (DMRG) simulations [A. Szasz, J. Motruk, M.P. Zaletel, and J.E. Moore, Phys. Rev. X {\bf 10}, 021042 (2020)]. Here, we report the results obtained within the variational Monte Carlo technique based upon Jastrow-Slater wave functions, implemented with backflow correlations. As in DMRG calculations, we consider $N$-leg cylinders. For $N=4$ and in the presence of a next-nearest neighbor hopping, a chiral spin liquid emerges between the metal and the insulator with magnetic quasi-long-range order. Within our approach, the chiral state is gapped and breaks the reflection symmetry. By contrast, for both $N=5$ and $N=6$, the chiral spin liquid is not the state with the lowest variational energy: in the former case a nematic spin liquid is found in the entire insulating regime, while for the less frustrated case with $N=6$ the results are very similar to the one obtained on two-dimensional clusters [L.F. Tocchio, A. Montorsi, and F. Becca, Phys. Rev. B {\bf 102}, 115150 (2020)], with an antiferromagnetic phase close to the metal-insulator transition and a nematic spin liquid in the strong-coupling regime.

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arXiv2021

Gutzwiller-projected states for the $J_1$-$J_2$ Heisenberg model on the kagome lattice: achievements and pitfalls

We assess the ground-state phase diagram of the $J_1$-$J_2$ Heisenberg model on the kagome lattice by employing Gutzwiller-projected fermionic wave functions. Within this framework, different states can be represented, defined by distinct unprojected fermionic Hamiltonians that include hopping and pairing terms, as well as a coupling to local Zeeman fields to generate magnetic order. For $J_2=0$, the so-called U(1) Dirac state, in which only hopping is present (such as to generate a $π$-flux in the hexagons), has been shown to accurately describe the exact ground state [Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B 87, 060405 (2013); Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Phys. Rev. X 7, 031020 (2017)]. Here, we show that its accuracy improves in presence of a small antiferromagnetic super-exchange $J_2$, leading to a finite region where the gapless spin liquid is stable; then, for $J_2/J_1=0.11(1)$, a first-order transition to a magnetic phase with pitch vector ${\bf q}=(0,0)$ is detected, by allowing magnetic order within the fermionic Hamiltonian. Instead, for small ferromagnetic values of $|J_2|/J_1$, the situation is more contradictory. While the U(1) Dirac state remains stable against several perturbations in the fermionic part (i.e., dimerization patterns or chiral terms), its accuracy clearly deteriorates on small systems, most notably on $36$ sites where exact diagonalization is possible. Then, upon increasing the ratio $|J_2|/J_1$, a magnetically ordered state with $\sqrt{3} \times \sqrt{3}$ periodicity eventually overcomes the U(1) Dirac spin liquid. Within the ferromagnetic $J_{2}$ regime, evidence is shown in favor of a first-order transition at $J_2/J_1=-0.065(5)$.

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arXiv2021

Effects of spin-phonon coupling in frustrated Heisenberg models

The existence and stability of spin-liquid phases represent a central topic in the field of frustrated magnetism. While a few examples of spin-liquid ground states are well established in specific models (e.g. the Kitaev model on the honeycomb lattice), recent investigations have suggested the possibility of their appearance in several Heisenberg-like models on frustrated lattices. An important related question concerns the stability of spin liquids in the presence of small perturbations in the Hamiltonian. In this respect, the magnetoelastic interaction between spins and phonons represents a relevant and physically motivated perturbation, which has been scarcely investigated so far. In this work, we study the effect of the spin-phonon coupling on prototypical models of frustrated magnetism. We adopt a variational framework based upon Gutzwiller-projected wave functions implemented with a spin-phonon Jastrow factor, providing a full quantum treatment of both spin and phonon degrees of freedom. The results on the frustrated $J_1-J_2$ Heisenberg model on one- and two-dimensional (square) lattices show that, while a valence-bond crystal is prone to lattice distortions, a gapless spin liquid is stable for small spin-phonon couplings. In view of the ubiquitous presence of lattice vibrations, our results are particularly important to demonstrate the possibility that gapless spin liquids may be realized in real materials.

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arXiv2021

Gapless spin liquids in disguise

We show that gapless spin liquids, which are potential candidates to describe the ground state of frustrated Heisenberg models in two dimensions, become trivial insulators on cylindrical geometries with an even number of legs. In particular, we report calculations for Gutzwiller-projected fermionic states on strips of square and kagome lattices. By choosing different boundary conditions for the fermionic degrees of freedom, both gapless and gapped states may be realized, the latter ones having a lower variational energy. The direct evaluation of static and dynamical correlation functions, as well as overlaps between different states, allows us to demonstrate the sharp difference between the ground-state properties obtained within cylinders or directly in the two-dimensional lattice. Our results shed light on the difficulty to detect bona fide gapless spin liquids in such cylindrical geometries.

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arXiv2020

Variational wave functions for the spin-Peierls transition in the Su-Schrieffer-Heeger model with quantum phonons

We introduce variational wave functions to evaluate the ground-state properties of spin-phonon coupled systems described by the Su-Schrieffer-Heeger model. Quantum spins and phonons are treated on equal footing within a Monte Carlo sampling, and different regimes are investigated. We show that the proposed variational Ansatz yields good agreement with previous density-matrix renormalization group results in one dimension and is able to accurately describe the spin-Peierls transition. This variational approach is neither constrained by the magnetoelastic-coupling strength nor by the dimensionality of the systems considered, thus allowing future investigations in more general cases, which are relevant to spin-liquid and topological phases in two spatial dimensions.

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arXiv2020

Magnetic and spin-liquid phases in the frustrated $t-t^\prime$ Hubbard model on the triangular lattice

The Hubbard model and its strong-coupling version, the Heisenberg one, have been widely studied on the triangular lattice to capture the essential low-temperature properties of different materials. One example is given by transition metal dichalcogenides, as 1T$-$TaS$_2$, where a large unit cell with $13$ Ta atom forms weakly-coupled layers with an isotropic triangular lattice. By using accurate variational Monte Carlo calculations, we report the phase diagram of the $t-t^\prime$ Hubbard model on the triangular lattice, highlighting the differences between positive and negative values of $t^\prime/t$; this result can be captured only by including the charge fluctuations that are always present for a finite electron-electron repulsion. Two spin-liquid regions are detected: one for $t^\prime/t<0$, which persists down to intermediate values of the electron-electron repulsion, and a narrower one for $t^\prime/t>0$. The spin-liquid phase appears to be gapless, though the variational wave function has a nematic character, in contrast to the Heisenberg limit. We do not find any evidence for non-magnetic Mott phases in the proximity of the metal-insulator transition, at variance with the predictions (mainly based upon strong-coupling expansions in $t/U$) that suggest the existence of a weak-Mott phase that intrudes between the metal and the magnetically ordered insulator.

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arXiv2020

Investigation of the Néel phase of the frustrated Heisenberg antiferromagnet by differentiable symmetric tensor networks

The recent progress in the optimization of two-dimensional tensor networks [H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Phys. Rev. X ${\bf 9}$, 031041 (2019)] based on automatic differentiation opened the way towards precise and fast optimization of such states and, in particular, infinite projected entangled-pair states (iPEPS) that constitute a generic-purpose ${\it Ansatz}$ for lattice problems governed by local Hamiltonians. In this work, we perform an extensive study of a paradigmatic model of frustrated magnetism, the $J_1-J_2$ Heisenberg antiferromagnet on the square lattice. By using advances in both optimization and subsequent data analysis, through finite correlation-length scaling, we report accurate estimations of the magnetization curve in the Néel phase for $J_2/J_1 \le 0.45$. The unrestricted iPEPS simulations reveal an $U(1)$ symmetric structure, which we identify and impose on tensors, resulting in a clean and consistent picture of antiferromagnetic order vanishing at the phase transition with a quantum paramagnet at $J_2/J_1 \approx 0.46(1)$. The present methodology can be extended beyond this model to study generic order-to-disorder transitions in magnetic systems.

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arXiv2020

Gapless spin liquid and valence-bond solid in the $J_1-J_2$ Heisenberg model on the square lattice: insights from singlet and triplet excitations

The spin-$1/2$ $J_1-J_2$ Heisenberg model on the square lattice represents one of the simplest examples in which the effets of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for $J_2/J_1 \approx 0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48 \lesssim J_2/J_1 \lesssim 0.54$) and a valence-bond solid (for $0.54 \lesssim J_2/J_1 \lesssim 0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A.W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)] reconcile the contraddicting results obtained within different approaches over the years.

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arXiv2019

Dynamical properties of Néel and valence-bond phases in the $J_1-J_2$ model on the honeycomb lattice

By using a variational Monte Carlo technique based upon Gutzwiller-projected fermionic states, we investigate the dynamical structure factor of the antiferromagnetic $S=1/2$ Heisenberg model on the honeycomb lattice, in presence of first-neighbor ($J_1$) and second-neighbor ($J_2$) couplings, for ${J_2 < 0.5 J_1}$. The ground state of the system shows long-range antiferromagnetic order for ${J_2/J_1 \lesssim 0.23}$, plaquette valence-bond order for ${0.23 \lesssim J_2/J_1 \lesssim 0.36}$, and columnar dimer order for ${J_2/J_1 \gtrsim 0.36}$. Within the antiferromagnetic state, a well-defined magnon mode is observed, whose dispersion is in relatively good agreement with linear spin-wave approximation for $J_2=0$. When a nonzero second-neighbor super-exchange is included, a roton-like mode develops around the $K$ point (i.e., the corner of the Brillouin zone). This mode softens when $J_2/J_1$ is increased and becomes gapless at the transition point, $J_2/J_1 \approx 0.23$. Here, a broad continuum of states is clearly visible in the dynamical spectrum, suggesting that nearly-deconfined spinon excitations could exist, at least at relatively high energies. For larger values of $J_2/J_1$, valence-bond order is detected and the spectrum of the system becomes clearly gapped, with a triplon mode at low energies. This is particularly evident for the spectrum of the dimer valence-bond phase, in which the triplon mode is rather well separated from the continuum of excitations that appears at higher energies.

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arXiv2019

Neural Gutzwiller-projected variational wave functions

Variational wave functions have enabled exceptional scientific breakthroughs related to the understanding of novel phases of matter. Examples include the Bardeen-Cooper-Schrieffer theory of superconductivity, the description of the fractional quantum Hall effect through the Laughlin state, and Feynman's variational understanding of large-scale quantum effects in liquid Helium. More recently, Gutzwiller-projected wave functions, typically constructed from fermionic degrees of freedom, have been employed to examine quantum spin models in the presence of competing interactions, where exotic phases with no spontaneous symmetry breaking and fractional excitations may exist. In this work, we investigate the aforementioned fermionic wave functions supplemented with neural networks, specifically with the so-called restricted Boltzmann machine (RBM), to boost their accuracy and obtain reliable approximations to the ground state of generic spin models. In particular, we apply our neural augmented fermionic construction to the description of both magnetically ordered and disordered phases of increasing complexity, including cases where the ground state displays a non-trivial sign structure. Even though the RBM state is by far more effective for Néel states endowed with a particularly simple sign structure, it provides a significant improvement over the original fermionic state in highly frustrated regimes where a complex sign structure is anticipated, thus marking the path to an understanding of strongly-correlated spin models on the lattice via neural Gutzwiller-projected variational wave functions.

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arXiv2019

Optimization of infinite projected entangled pair states: The role of multiplets and their breaking

The infinite projected entangled pair states (iPEPS) technique [J. Jordan {\it et al.}, Phys. Rev. Lett. {\bf 101}, 250602 (2008)] has been widely used in the recent years to assess the properties of two-dimensional quantum systems, working directly in the thermodynamic limit. This formalism, which is based upon a tensor-network representation of the ground-state wave function, has several appealing features, e.g., encoding the so-called area law of entanglement entropy by construction; still, the method presents critical issues when dealing with the optimization of tensors, in order to find the best possible approximation to the exact ground state of a given Hamiltonian. Here, we discuss the obstacles that arise in the optimization by imaginary-time evolution within the so-called simple and full updates and connect them to the emergence of a sharp multiplet structure in the "virtual" indices of tensors. In this case, a generic choice of the bond dimension $D$ is not compatible with the multiplets and leads to a symmetry breaking (e.g., generating a finite magnetic order). In addition, varying the initial guess, different final states may be reached, with very large deviations in the magnetization value. In order to exemplify this behavior, we show the results of the $S=1/2$ Heisenberg model on an array of coupled ladders, for which a vanishing magnetization below the critical interladder coupling is recovered only for selected values of $D$, while a blind optimization with a generic $D$ gives rise to a finite magnetization down to the limit of decoupled ladders.

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arXiv2019

Dynamical structure factor of the $J_1-J_2$ Heisenberg model on the triangular lattice: magnons, spinons, and gauge fields

Understanding the nature of the excitation spectrum in quantum spin liquids is of fundamental importance, in particular for the experimental detection of candidate materials. However, current theoretical and numerical techniques have limited capabilities, especially in obtaining the dynamical structure factor, which gives a crucial characterization of the ultimate nature of the quantum state and may be directly assessed by inelastic neutron scattering. In this work, we investigate the low-energy properties of the $S=1/2$ Heisenberg model on the triangular lattice, including both nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ super-exchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models. For $J_2=0$, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations at $K$ points (i.e., at the corners of the Brillouin zone). The strong renormalization of the magnon branch (also including roton-like minima around the $M$ points, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a non-trivial magnetic $π$-flux threading half of the triangular plaquettes. When increasing the frustrating ratio $J_2/J_1$, we detect a progessive softening of the magnon branch at $M$, which eventually becomes gapless within the spin-liquid phase. This feature is captured by the band structure of the unprojected wave function (with $2$ Dirac points for each spin component). In addition, we observe an intense signal at low energies around the $K$ points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids.

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