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arXiv:2309.13630 · Physical review. B./Physical review. B

Absence of spin liquid phase in the J1−J2 Heisenberg model on the square lattice

Xiangjian Qian, Mingpu Qin

Published Apr 3, 2024• 25 citations
arXivHTMLPDF

Abstract

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 ${J}_{2}/{J}_{1}=0.535(3)$, 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 ${J}_{2}/{J}_{1}=0.535(3)$ in the model. From the crossing of the first derivative of the energies with ${J}_{2}$ for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at ${J}_{2}/{J}_{1}=0.610(5)$.

DOI
10.1103/physrevb.109.l161103

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We perform an in-depth investigation of the phase diagram of the J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ 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).
We use the density matrix renormalization group method to calculate several energy eigenvalues of the frustrated S=1/2 square-lattice J_{1}-J_{2} Heisenberg model on 2L×L cylinders with L≤10. We identify excited-level crossings versus the coupling ratio g=J_{2}/J_{1} and study their drifts with the system size L. The lowest singlet-triplet and singlet-quintuplet crossings converge rapidly (with corrections ∝L^{-2}) to different g values, and we argue that these correspond to ground-state transitions between the Néel antiferromagnet and a gapless spin liquid, at g_{c1}≈0.46, and between the spin liquid and a valence-bond solid at g_{c2}≈0.52. Previous studies of order parameters were not able to positively discriminate between an extended spin liquid phase and a critical point. We expect level-crossing analysis to be a generically powerful tool in density matrix renormalization group studies of quantum phase transitions.
We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor J1{J}_{1}J1​ and the next-nearest-neighbor J2{J}_{2}J2​ couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width Ly=10{L}_{y}=10Ly​=10, we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.
We use the finite projected entangled pair state (PEPS) method to investigate the global phase diagram of the spin-12\frac{1}{2}21​ square-lattice ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ antiferromagnetic (AFM) Heisenberg model. The ground-state phase diagram is established with a rich variety of phases: the N\'eel AFM, gapless quantum spin liquid, valence-bond solid (VBS), stripe AFM, and incommensurate spiral phases. The nature of the VBS region is revealed, which contains a plaquette VBS and a mixed columnar-plaquette VBS, with the emergence of short-range incommensurate spin correlation in some region. The long-range incommensurate magnetic phase is also explicitly characterized as a planar spiral with incommensurate spatial periodicities. Most interestingly, there exist several multicritical points connecting different phases. These findings elucidate the true nature of the long-standing square-lattice ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ antiferromagnet at zero temperature. Our results also pave the way to accurately simulate complex two-dimensional quantum spin systems that may host nonuniform features by means of the finite PEPS.
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.
We studied the ground-state phase diagram of spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model on the square lattice with first-neighbor J1{J}_{1}J1​ and second-neighbor J2{J}_{2}J2​ antiferromagnetic interactions using both infinite density matrix renormalization group (iDMRG) and DMRG approaches. We show that a plaquette valence bond phase is realized in an intermediate region $0.50\ensuremath{\le}{J}_{2}/{J}_{1}<0.54$ between a N\'eel magnetic ordered phase at J2/J1<0.50{J}_{2}/{J}_{1}<0.50J2​/J1​<0.50 and a stripy magnetic ordered phase at ${J}_{2}/{J}_{1}\ensuremath{\ge}0.54$. The plaquette valence bond phase is characterized by finite dimer orders in both the horizontal and vertical directions. Contrary to the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model, we do not find numerical evidence for a quantum spin liquid phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model.
The nature of the intermediate ground‐state phase in the spin‐1/2 frustrated square lattice model has long been debated. We investigate the phase diagram of this model using cluster density matrix embedding theory (CDMET). Two antiferromagnetic phases are directly identified from our analysis of the ground‐state wave function: the Néel antiferromagnetic phase for J 2 ≤ 0.45 and the stripe antiferromagnetic phase for J 2 ≥ 0.65. Although our analysis of the ground state finds no direct evidence of an internal phase transition within the intermediate phase, an examination of the first excited state reveals a continuous quantum phase transition therein, with a critical point at J 2 = 0.55 that separates the intermediate phase into a plaquette valence bond solid (PVBS) and a (P + C) valence bond solid (VBS).