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arXiv:1309.7542 · Physical Review B

Decoupled phase of frustrated spin- 1 2 antiferromagnetic chains with and without long-range order in the ground state

Manoranjan Kumar, Z. G. Soos

Published Oct 14, 2013• 17 citations
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Abstract

The quantum phases of one-dimensional spin $s=1/2$ chains are discussed for models with two parameters, frustrating exchange $g={J}_{2}>0$ between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at $g=0$ has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin $S={S}_{A}+{S}_{B}$, have singlet GS for any $g$, $\ensuremath{\alpha}\ensuremath{\ge}0$ and decouple at $1/g=0$ to linear Heisenberg antiferromagnets on sublattices $A$ and $B$ of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap ${E}_{m}$ to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at ${g}_{c}=1/4\mathrm{ln}2$ from a GS with perfect LRO to a decoupled phase with ${S}_{A}={S}_{B}=0$ for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.

DOI
10.1103/physrevb.88.134412

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The quantum phases of one-dimensional spin s=1/2s=1/2s=1/2 chains are discussed for models with two parameters, frustrating exchange g=J2>0g={J}_{2}>0g=J2​>0 between second neighbors and normalized nonfrustrating power-law exchange with exponent \ensuremathα\ensuremath{\alpha}\ensuremathα and distance dependence r\ensuremath−\ensuremathα{r}^{\ensuremath{-}\ensuremath{\alpha}}r\ensuremath−\ensuremathα. The ground state (GS) at g=0g=0g=0 has a long-range order (LRO) for \ensuremathα<2\ensuremath{\alpha}<2\ensuremathα<2 and long-range spin fluctuations for \ensuremathα>2\ensuremath{\alpha}>2\ensuremathα>2. The models conserve total spin S=SA+SBS={S}_{A}+{S}_{B}S=SA​+SB​, have singlet GS for any ggg, \ensuremathα\ensuremath≥0\ensuremath{\alpha}\ensuremath{\ge}0\ensuremathα\ensuremath≥0 and decouple at 1/g=01/g=01/g=0 to linear Heisenberg antiferromagnets on sublattices AAA and BBB of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin \ensuremath⟨SA2\ensuremath⟩\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}\ensuremath⟨SA2​\ensuremath⟩, the magnetic gap Em{E}_{m}Em​ to the lowest triplet state, and the excitation E\ensuremathσ{E}_{\ensuremath{\sigma}}E\ensuremathσ​ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at gc=1/4ln2{g}_{c}=1/4\mathrm{ln}2gc​=1/4ln2 from a GS with perfect LRO to a decoupled phase with SA=SB=0{S}_{A}={S}_{B}=0SA​=SB​=0 for g\ensuremath≥4/\ensuremathπ2g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}g\ensuremath≥4/\ensuremathπ2 and no correlation between spins in different sublattices. The model with \ensuremathα=1\ensuremath{\alpha}=1\ensuremathα=1 has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with \ensuremathα\ensuremath≥2\ensuremath{\alpha}\ensuremath{\ge}2\ensuremathα\ensuremath≥2 are discussed in terms of GS degeneracy where E\ensuremathσ(g)=0{E}_{\ensuremath{\sigma}}(g)=0E\ensuremathσ​(g)=0, excited state degeneracy where E\ensuremathσ(g)=Em(g){E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)E\ensuremathσ​(g)=Em​(g), and \ensuremath⟨SA2\ensuremath⟩\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}\ensuremath⟨SA2​\ensuremath⟩. The decoupled phase at large frustration has nondegenerate GS for any exponent \ensuremathα\ensuremath{\alpha}\ensuremathα and excited states related to sublattice excitations.
Exact diagonalization of finite spin-1/2 chains with periodic boundary conditions is applied to the ground state (GS) of chains with ferromagnetic (F) exchange J1<0{J}_{1}<0J1​<0 between first neighbors, antiferromagnetic (AF) exchange ${J}_{2}=\ensuremath{\alpha}{J}_{1}>0$ between second neighbors, and axial anisotropy $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$. In zero field, the GS is in the Sz=0{S}_{z}=0Sz​=0 sector for the relevant parameters and is doubly degenerate at multiple points ${\ensuremath{\gamma}}_{m}=({\ensuremath{\alpha}}_{m},{\ensuremath{\Delta}}_{m})$ in the $\ensuremath{\alpha}$, $\ensuremath{\Delta}$ plane. Degeneracy under inversion at sites or spin-parity or both leads, respectively, to a bond order wave (BOW), to staggered magnetization, or to vector chiral (VC) order. Exact results up to N=28N=28N=28 spins directly yield order parameters and spin correlation functions whose weak NNN dependencies allow inferences about infinite chains. The high-spin GS at J2=0{J}_{2}=0J2​=0 changes discontinuously at ${\ensuremath{\gamma}}_{1}=(\ensuremath{-}1/4,1)$ to a singlet in the isotropic ($\ensuremath{\Delta}=1$) chain. The transition from high to low spin $S(\ensuremath{\alpha},\ensuremath{\Delta})$ is continuous for $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{B}=0.95\ifmmode\pm\else\textpm\fi{}0.01$ on the degeneracy line ${\ensuremath{\alpha}}_{1}(\ensuremath{\Delta})$. The GS has staggered magnetization between ${\ensuremath{\Delta}}_{A}=0.72$ and ${\ensuremath{\Delta}}_{B}$ and a BOW for $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{A}$. When both inversion and spin-parity are reversed at ${\ensuremath{\gamma}}_{m}$, the correlation functions C(p)C(p)C(p) for spins separated by ppp sites are identical. C(p)C(p)C(p) minima are shifted by $\ensuremath{\pi}/2$ from the minima of VC order parameters at separation ppp, consistent with right- and left-handed helices along the zzz axis and spins in the xyxyxy plane. Degenerate GSs of finite chains are related to quantum phase diagrams of extended $\ensuremath{\alpha}$, $\ensuremath{\Delta}$ chains, with good agreement for order parameters along the line ${\ensuremath{\alpha}}_{1}(\ensuremath{\Delta})$. Degenerate GSs limit a VC phase to intermediate $\ensuremath{\alpha}$ and $\ensuremath{\Delta}$ where $S(\ensuremath{\alpha},\ensuremath{\Delta})$ varies rapidly but continuously, in contrast to many-body treatments in which VC phases extend over a larger range of parameters.
The spin-1/2 chain with isotropic exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1 and has a singlet ground state (GS) for J1/J2 ⩾ -4. Its rich quantum phase diagram supports gapless, gapped, commensurate (C), incommensurate (IC) and other phases. Critical points J1/J2 are evaluated using exact diagonalization and density matrix renormalization group calculations. The wave vector qG of spin correlations is related to GS degeneracy and obtained as the peak of the spin structure factor S(q). Variable qG indicates IC phases in two J1/J2 intervals, [-4, - 1.24] and [0.44, 2], and a C-IC point at J1/J2 = 2. The decoupled C phase in [-1.24, 0.44] has constant qG = π/2, nondegenerate GS, and a lowest triplet state with broken spin density on sublattices of odd and even numbered sites. The lowest triplet and singlet excitations, E m and Eσ, are degenerate in finite systems at specific frustration J1/J2. Level crossing extrapolates in the thermodynamic limit to the same critical points as qG. The S(q) peak diverges at qG = π in the gapless phase with J1/J2 > 4.148 and quasi-long-range order (QLRO(π)). S(q) diverges at ±π/2 in the decoupled phase with QLRO(π/2), but is finite in gapped phases with finite-range correlations. Numerical results and field theory agree at small J2/J1 but disagree for the decoupled phase with weak exchange J1 between sublattices. Two related models are summarized: one has an exact gapless decoupled phase with QLRO(π/2) and no IC phases; the other has a single IC phase without a decoupled phase in between.
The static structure factor S(q) of frustrated spin-1/2 chains with isotropic exchange and a singlet ground state (GS) diverges at wave vector q_m when the GS has quasi-long-range order (QLRO) with periodicity 2π/q_m but S(q_m) is finite in bond-order-wave (BOW) phases with finite-range spin correlations. Exact diagonalization and density matrix renormalization group (DMRG) calculations of S(q) indicate a decoupled phase with QLRO and q_m = π/2 in chains with large antiferromagnetic exchange between second neighbors. S(q_m) identifies quantum phase transitions based on GS spin correlations.
The spin-1/2 chain with isotropic Heisenberg exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1. Its quantum phase diagram has critical points at fixed J1/J2 between gapless phases with nondegenerate ground state (GS) and quasi-long-range order (QLRO) and gapped phases with doubly degenerate GS and spin correlation functions of finite range. In finite chains, exact diagonalization (ED) estimates critical points as level crossing of excited states. GS spin correlations enter in the spin structure factor S(q) that diverges at wave vector qm in QLRO(q(m)) phases with periodicity 2π/q(m) but remains finite in gapped phases. S(q(m)) is evaluated using ED and density matrix renormalization group (DMRG) calculations. Level crossing and the magnitude of S(q(m)) are independent and complementary probes of quantum phases, based respectively on excited and ground states. Both indicate a gapless QLRO(π/2) phase between -1.2 < J1/|J2| < 0.45. Numerical results and field theory agree well for quantum critical points at small frustration J2 but disagree in the sector of weak exchange J1 between Heisenberg antiferromagnetic chains on sublattices of odd and even-numbered sites.
The antiferromagnetic ${J}_{1}\ensuremath{-}{J}_{2}$ model is a spin-1/2 chain with isotropic exchange J1>0{J}_{1}>0J1​>0 between first neighbors and ${J}_{2}=\ensuremath{\alpha}{J}_{1}$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})>0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\ensuremath{\alpha}=0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\ensuremath{\alpha}$ followed by density matrix renormalization group calculations returns the entropy density $S(T,\ensuremath{\alpha},N)$ and magnetic susceptibility $\ensuremath{\chi}(T,\ensuremath{\alpha},N)$ of progressively larger systems up to N=96N=96N=96 or 152 spins. Convergence to the thermodynamic limit, $S(T,\ensuremath{\alpha})$ or $\ensuremath{\chi}(T,\ensuremath{\alpha})$, is demonstrated down to $T/J\ensuremath{\sim}0.01$ in the sectors $\ensuremath{\alpha}<1$ and $\ensuremath{\alpha}>1$. $S(T,\ensuremath{\alpha})$ yields the critical points between gapless phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})>0$ and gapped phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})=0$. The ${S}^{\ensuremath{'}}(T,\ensuremath{\alpha})$ maximum at ${T}^{*}(\ensuremath{\alpha})$ is obtained directly in chains with large $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\ensuremath{\alpha})$ down to T=0T=0T=0 indicates power-law deviations ${T}^{\ensuremath{-}\ensuremath{\gamma}(\ensuremath{\alpha})}$ from $exp[\ensuremath{-}\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})/T]$ with exponent $\ensuremath{\gamma}(\ensuremath{\alpha})$ that increases with $\ensuremath{\alpha}$. The $\ensuremath{\chi}(T,\ensuremath{\alpha})$ analysis also yields power-law deviations, but with exponent $\ensuremath{\eta}(\ensuremath{\alpha})$ that decreases with $\ensuremath{\alpha}$. Spin correlation functions account for $S(T,\ensuremath{\alpha})$ differences between frustration $\ensuremath{\alpha}<1$ within a chain and $\ensuremath{\alpha}>1$ between HAFs on sublattices. $S(T,\ensuremath{\alpha})$ and the spin density $\ensuremath{\rho}(T,\ensuremath{\alpha})=4T\ensuremath{\chi}(T,\ensuremath{\alpha})$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\ensuremath{\alpha})/\ensuremath{\rho}(T,\ensuremath{\alpha})$ for T<0.10T<0.10T<0.10. Remarkably, the ratio decreases (increases) with TTT in chains with large (small) $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$.
The frustrated ladder with alternate ferromagnetic(F) exchange −JF-J_F−JF​ and AF exchange JAJ_AJA​ to first neighbors and F exchange −JL-J_L−JL​ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of 2N2N2N spins-1/2 with periodic boundary conditions. The ground state is a singlet (S=0)(S = 0)(S=0) and the singlet-triplet gap εT\varepsilon_TεT​ is finite for the exchanges considered. Spin-1/2 string correlation functions g1(N)g_1(N)g1​(N) and g2(N)g_2(N)g2​(N) are defined for an even number NNN of consecutive spins in systems with two spins per unit cell; the ladder has string order g2(∞)>0g_2(\infty)> 0g2​(∞)>0 and g1(∞)=0g_1(\infty) = 0g1​(∞)=0. The minimum N∗N^*N∗ of g2(N)g_2(N)g2​(N) is related to the range of ground-state spin correlations. Convergence to g2(∞)g_2(\infty)g2​(∞) is from below, and g1(N)g_1(N)g1​(N) decreases exponentially for N≥N∗N \geq N^*N≥N∗. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of JFJ_FJF​, JLJ_LJL​ and JAJ_AJA​ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the J1−J2J_1-J_2J1​−J2​ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at JL=JF/2≤JAJ_L = J_F/2 \leq J_AJL​=JF​/2≤JA​, that is reversed on interchanging −JF-J_F−JF​ and JAJ_AJA​. Inversion symmetry is spontaneously broken in the dimer phase of the J1−J2J_1-J_2J1​−J2​ model where the Kekulé diagrams are the doubly degenerate ground states at J2/J1=1/2J_2/J_1 = 1/2J2​/J1​=1/2.