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arXiv:1904.00612 · Physical review. B./Physical review. B

Sequential quantum phase transitions in J1−J2 Heisenberg chains with integer spins (S>1) : Quantized Berry phase and valence-bond solids

Shota Fubasami, Tomonari Mizoguchi, Yasuhiro Hatsugai

Published Jul 30, 2019• 9 citations
arXivHTMLPDF

Abstract

On the basis of a Berry-phase analysis, we study the ground state of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain for $S=2,3,4$. We find that changes in the Berry phase occur $S$ times for spin-$S$ systems, indicating sequential phase transitions. The sequential phase transitions are associated with the change in valence-bond configurations of the ground states. To demonstrate this numerically, we connect the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg Hamiltonian with the generalized Affleck-Kennedy-Lieb-Tasaki model and show that the different phases are connected to the different valence-bond solid states.

DOI
10.1103/physrevb.100.014438

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On the basis of a Berry-phase analysis, we study the ground state of the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg chain for S=2,3,4S=2,3,4S=2,3,4. We find that changes in the Berry phase occur SSS times for spin-SSS systems, indicating sequential phase transitions. The sequential phase transitions are associated with the change in valence-bond configurations of the ground states. To demonstrate this numerically, we connect the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg Hamiltonian with the generalized Affleck-Kennedy-Lieb-Tasaki model and show that the different phases are connected to the different valence-bond solid states.
We study quantum mixed-spin chains consisting of two integer spins S1{S}^{1}S1 and two half integer spins S2{S}^{2}S2 arrayed as ${S}^{1}\ensuremath{-}{S}^{1}\ensuremath{-}{S}^{2}\ensuremath{-}{S}^{2}$ in a unit cell with antiferromagnetic nearest-neighbor couplings J1{J}_{1}J1​ (J2){(J}_{2})(J2​) between the spins of equal (different) magnitudes. By varying the ratio of two competing couplings, $\ensuremath{\alpha}{=J}_{2}{/J}_{1},$ the systems undergo successive quantum phase transitions between different valence-bond-solid (VBS) states accompanied by the vanishing of the energy gap. By a quantum Monte Carlo simulation with an improved loop cluster algorithm, we find one critical point ${\ensuremath{\alpha}}_{c}=0.762(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two critical points ${\ensuremath{\alpha}}_{c1}=0.479(1)$ and ${\ensuremath{\alpha}}_{c2}=1.318(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain, respectively. The calculated expectation values of the Lieb-Shultz-Mattis twist operator show the characteristic features of the single VBS transition in the $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two successive VBS transitions in the $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain at these critical points.
Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic J1J_1J1​-J2J_2J2​ Chain
Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena to emerge. We consider an elementary frustrated 1D system: the spin- \frac{1}{2} 1 2 ferromagnetic ( J_1 J 1 ) Heisenberg chain with next-nearest-neighbor antiferromagnetic ( J_2 J 2 ) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at J_2/|J_1|>1/4 J 2 / | J 1 | > 1 / 4 and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous symmetry breaking through order by disorder. Quite interestingly, this VBS state has a Affleck-Kennedy-Lieb-Tasaki-type topological order. This is the first example of a frustrated spin chain in which quantum fluctuations induce gapped topological order.
We study the frustrated and dimerized ferromagnetic-antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{'}}\text{\ensuremath{-}}{J}_{2}$ chain using the density-matrix renormalization group method. Based on numerical calculations of the second derivative of energy, spin gap, spin-spin correlations, string order parameter (SOP), and entanglement spectrum (ES), we obtain the ground-state phase diagram for a wide range of ${J}_{1}^{\ensuremath{'}}/{J}_{1}$ and J2/∣J1∣{J}_{2}/|{J}_{1}|J2​/∣J1​∣ values. This phase diagram reveals a ferromagnetic phase and two distinct valence bond solid (VBS) phases. The first VBS phase, referred to as D3{\mathcal{D}}_{3}D3​-VBS, is typified by the formation of valence bonds between third-neighbor spin-1/2's, persisting as a continuation from the ${J}_{1}^{\ensuremath{'}}/{J}_{1}=1$ limit. Alternatively, the second VBS phase, referred to as mixed-VBS, exhibits a coexistence of both second- and third-neighbor valence bonds, interpreted as a continuation from the ${J}_{1}^{\ensuremath{'}}/{J}_{1}=0$ case. Remarkably, both VBS states are identified as being of Haldane type, marked by a finite SOP and twofold ES degeneracy. Unexpectedly, our analysis uncovers a significant enhancement of the valence bond stability at the boundary of the two VBS phases. This study provides an empirical demonstration of a nontrivial quantum phase transition between different topological VBS states in spin-1/2 chains. Moreover, we find that the ground state of the relevant quasi-one-dimensional material LiCuSbO4{\mathrm{LiCuSbO}}_{4}LiCuSbO4​ is classified as the D3{\mathcal{D}}_{3}D3​-VBS state. Collectively, these results mark a substantial stride forward in our comprehension of quantum phase transitions and topological states.
Using the coupled cluster method for high orders of approximation and complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2 J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.
We characterize several phases of gapped spin systems by local order parameters defined by quantized Berry phases [Y. Hatsugai, J. Phys. Soc. Jpn. 75, 123601 (2006)]. This characterization is topologically stable against any small perturbation as long as the energy gap remains finite. The models we pick up are S=1,2S=1,2S=1,2 dimerized Heisenberg chains and S=2S=2S=2 Heisenberg chains with uniaxial single-ion-type anisotropy. Analytically, we also evaluate the topological local order parameters for the generalized Affleck-Kennedy-Lieb-Tasaki model. The relation between the present Berry phases and the fractionalization in the integer spin chains are discussed as well.
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).