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arXiv:2610.26135

Evidence of valence bond solid and quantum paramagnetism in the $J_1$--$J_2$--$J_3$ Heisenberg model from pseudofermion functional renormalization group

GÜN GÜNAL, Erhai Zhao, Ahmet Keleş

Published Oct 19, 2026• 0 citations
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Abstract

Motivated by recent tensor network studies reporting a quantum spin liquid (QSL) within the quantum paramagnetic (QP) regime, we investigate numerically the zero-temperature phase diagram of the $J_1$--$J_2$--$J_3$ Heisenberg model on the square lattice using the pseudofermion functional renormalization group (pf-fRG). Thanks to improved frequency resolution and larger lattice truncations, we are able to explore a significantly larger parameter space than previous pf-fRG work to map out the phase boundary of an extended QP region. We further resolve possible bond order within the QP regime by measuring the dimer response to small deformations in $J_1$, i.e. bond anisotropy that breaks the point-group symmetry of the Hamiltonian. The functional renormalization group flow of the dimer susceptibility suggests that the QP window splits into two: a region where it grows continuously down to the lowest energy scale to signal valence bond solid (VBS) order, and another region where it reaches a peak at a finite scale and is subsequently suppressed to indicate a QSL.

DOI
10.48550/arxiv.2610.26135

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Evidence of valence bond solid and quantum paramagnetism in the J1J_1J1​--J2J_2J2​--J3J_3J3​ Heisenberg model from pseudofermion functional renormalization group
Motivated by recent tensor network studies reporting a quantum spin liquid (QSL) within the quantum paramagnetic (QP) regime, we investigate numerically the zero-temperature phase diagram of the J1J_1J1​--J2J_2J2​--J3J_3J3​ Heisenberg model on the square lattice using the pseudofermion functional renormalization group (pf-fRG). Thanks to improved frequency resolution and larger lattice truncations, we are able to explore a significantly larger parameter space than previous pf-fRG work to map out the phase boundary of an extended QP region. We further resolve possible bond order within the QP regime by measuring the dimer response to small deformations in J1J_1J1​, i.e. bond anisotropy that breaks the point-group symmetry of the Hamiltonian. The functional renormalization group flow of the dimer susceptibility suggests that the QP window splits into two: a region where it grows continuously down to the lowest energy scale to signal valence bond solid (VBS) order, and another region where it reaches a peak at a finite scale and is subsequently suppressed to indicate a QSL.
We present a detailed functional renormalization group analysis of spin-12\frac{1}{2}21​ dipolar Heisenberg model on square lattice. This model is similar to the well-known ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model and describes the pseudospin degrees of freedom of polar molecules confined in deep optical lattice with long-range anisotropic dipole-dipole interactions. Previous study of this model based on tensor network ansatz indicates a paramagnetic ground state for certain dipole tilting angles which can be tuned in experiments to control the exchange couplings. The tensor ansatz formulated on a small cluster unit cell is inadequate to describe the spiral order, and therefore the phase diagram at high azimuthal tilting angles remains undetermined. Here, we obtain the full phase diagram of the model from numerical pseudofermion functional renormalization group calculations. We show that an extended quantum paramagnetic phase is realized between the N\'eel and stripe/spiral phases. In this region, the spin susceptibility flows smoothly down to the lowest numerical renormalization group scales with no sign of divergence or breakdown of the flow, in sharp contrast to the flow towards the long-range-ordered phases. Our results provide further evidence that the dipolar Heisenberg model is a fertile ground for quantum spin liquids.
We use our recently developed functional renormalization group (FRG) approach for quantum spin systems to investigate the phase diagram of the frustrated J1J2J3{J}_{1}{J}_{2}{J}_{3}J1​J2​J3​ quantum Heisenberg model on a cubic lattice. From a simple truncation of the hierarchy of FRG flow equations for the irreducible spin vertices, which retains only static spin fluctuations and neglects the flow of the four-spin interaction, we can estimate the critical temperature with a similar accuracy as the numerically more expensive pseudofermion FRG. In the regime where the ground state exhibits either ferromagnetic or antiferromagnetic order, a more sophisticated truncation including the renormalization of the four-spin interaction as well as dynamic spin fluctuations reveals the underlying renormalization group fixed point and yields critical temperatures which deviate from the accepted values by at most 44%4.
The pseudofermion functional renormalization group (pf-FRG) has been put forward as a semianalytical scheme that, for a given microscopic spin model, allows to discriminate whether the low-temperature states exhibit magnetic ordering or a tendency toward the formation of quantum spin liquids. However, the precise nature of the putative spin-liquid ground state has remained hard to infer from the original (single-site) pf-FRG scheme. Here, we introduce a cluster pf-FRG approach, which allows for a more stringent connection between a microscopic spin model and its low-temperature spin-liquid ground states. In particular, it allows to calculate spatially structured fermion bilinear expectation values on spatial clusters, which are formed by splitting the original lattice into several sublattices, thereby allowing for the positive identification of a family of bilinear spin-liquid states. As an application of this cluster pf-FRG approach, we consider the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)$ Heisenberg model on a square lattice, which is a paradigmatic example for a frustrated quantum magnet exhibiting quantum spin-liquid behavior for intermediate coupling strengths. In the well-established large-NNN limit of this model, we show that our approach correctly captures the emergence of the $\ensuremath{\pi}$-flux spin-liquid state at low temperatures. For small NNN, where the precise nature of the ground state remains controversial, we focus on the widely studied case of N=2N=2N=2, for which we determine the low-temperature phase diagram near the strongly frustrated regime after implementing the fermion-number constraint by the flowing Popov-Fedotov method. Our results suggest that the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model does not support the formation of a fermion bilinear spin-liquid state.
We study the phase diagram and the dynamical spin structure factor of the spin-1/2 J1−−J3{J}_{1}--{J}_{3}J1​−−J3​ Heisenberg model on the square lattice using density matrix renormalization group, exact diagonalization (ED), and cluster perturbation theory (CPT). By extrapolating the order parameters and studying the level crossings of the low-lying energy and entanglement spectra, we obtain the phase diagram of this model and identify a narrow region of quantum spin liquid (QSL) phase followed by a plaquette valence-bond solid (PVBS) state in the intermediate region, whose nature has been controversial for many years. More importantly, we use CPT and ED to study the dynamical spin structure factor in the QSL and the PVBS phase. In the QSL phase, the high-energy magnon mode completely turns into some dispersive weak excitations around the XXX and MMM points. For the PVBS phase, the low-energy spectrum is characterized by a gapped triplet excitation, and at the high energy, we find another branch of dispersive excitation with broad continua, which is unlike the plaquette phase in the $2\ifmmode\times\else\texttimes\fi{}2$ checkerboard model. In the latter case, the second branch of excitation is nearly flat due to the weak effective interactions between the local excitations of the plaquettes. And in the J1−−J3{J}_{1}--{J}_{3}J1​−−J3​ Heisenberg model, the uniform interactions and the spontaneously translational symmetry breaking of the PVBS phase make the difference in the excitation spectra.
We perform an in-depth investigation of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice. We take advantage of density matrix renormalization group and fully augmented matrix product states methods and reach unprecedented accuracy with large bond dimensions. We utilize excited-level crossing analysis to pinpoint the phase transition points. It was believed before that there exists a narrow spin liquid phase sandwiched by the N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through careful finite-size scaling of the level crossing points, we find a direct phase transition between the N\'eel AFM and VBS phases at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3), suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3) in the model. From the crossing of the first derivative of the energies with J2{J}_{2}J2​ for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at J2/J1=0.610(5){J}_{2}/{J}_{1}=0.610(5)J2​/J1​=0.610(5).
The spin-1/21/21/2 J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on the square lattice represents one of the simplest examples in which the effects 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 ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), 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 contradicting results obtained within different approaches over the years.