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arXiv:2001.05384 · Physical review. B./Physical review. B

Modeling the spin-Peierls transition of spin- 1 2 chains with correlated states: J 1 − J 2 model, CuGeO 3 , and TTF – CuS 4 C 4 ( CF 3 ) 4

Sudip Kumar Saha, Monalisa Singh Roy, Manoranjan Kumar, Z. G. Soos

Published Feb 7, 2020• 14 citations
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Abstract

The spin-Peierls transition at ${T}_{\mathrm{SP}}$ of spin-$\frac{1}{2}$ chains with isotropic exchange interactions has previously been modeled as correlated for $T>{T}_{\mathrm{SP}}$ and mean field for $T<{T}_{\mathrm{SP}}$. We use correlated states throughout in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model with antiferromagnetic exchange ${J}_{1}$ and ${J}_{2}=\ensuremath{\alpha}{J}_{1}$ between first and second neighbors, respectively, and variable frustration $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}0.50$. The thermodynamic limit is reached at high $T$ by exact diagonalization of short chains and at low $T$ by density matrix renormalization group calculations of progressively longer chains. In contrast to mean field results, correlated states of 1D models with linear spin-phonon coupling and a harmonic adiabatic lattice provide an internally consistent description in which the parameter ${T}_{\mathrm{SP}}$ yields both the stiffness and the lattice dimerization $\ensuremath{\delta}(T)$. The relation between ${T}_{\mathrm{SP}}$ and $\mathrm{\ensuremath{\Delta}}(\ensuremath{\delta},\ensuremath{\alpha})$, the $T=0$ gap induced by dimerization, depends strongly on $\ensuremath{\alpha}$ and deviates from the BCS gap relation that holds in uncorrelated spin chains. Correlated states account quantitatively for the magnetic susceptibility of TTF-${\mathrm{CuS}}_{4}{\mathrm{C}}_{4}{({\mathrm{CF}}_{3})}_{4}$ crystals (${J}_{1}=79\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, $\ensuremath{\alpha}=0$, ${T}_{\mathrm{SP}}=12\phantom{\rule{0.16em}{0ex}}\mathrm{K}$) and ${\mathrm{CuGeO}}_{3}$ crystals (${J}_{1}=160\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, $\ensuremath{\alpha}=0.35$, ${T}_{\mathrm{SP}}=14\phantom{\rule{0.16em}{0ex}}\mathrm{K}$). The same parameters describe the specific heat anomaly of ${\mathrm{CuGeO}}_{3}$ and inelastic neutron scattering. Modeling the spin-Peierls transition with correlated states exploits the fact that $\ensuremath{\delta}(0)$ limits the range of spin correlations at $T=0$ while $T>0$ limits the range at $\ensuremath{\delta}=0$.

DOI
10.1103/physrevb.101.054411

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The spin-Peierls transition at TSP{T}_{\mathrm{SP}}TSP​ of spin-12\frac{1}{2}21​ chains with isotropic exchange interactions has previously been modeled as correlated for T>TSPT>{T}_{\mathrm{SP}}T>TSP​ and mean field for T<TSPT<{T}_{\mathrm{SP}}T<TSP​. We use correlated states throughout in the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ model with antiferromagnetic exchange J1{J}_{1}J1​ and J2=\ensuremathαJ1{J}_{2}=\ensuremath{\alpha}{J}_{1}J2​=\ensuremathαJ1​ between first and second neighbors, respectively, and variable frustration 0\ensuremath≤\ensuremathα\ensuremath≤0.500\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}0.500\ensuremath≤\ensuremathα\ensuremath≤0.50. The thermodynamic limit is reached at high TTT by exact diagonalization of short chains and at low TTT by density matrix renormalization group calculations of progressively longer chains. In contrast to mean field results, correlated states of 1D models with linear spin-phonon coupling and a harmonic adiabatic lattice provide an internally consistent description in which the parameter TSP{T}_{\mathrm{SP}}TSP​ yields both the stiffness and the lattice dimerization \ensuremathδ(T)\ensuremath{\delta}(T)\ensuremathδ(T). The relation between TSP{T}_{\mathrm{SP}}TSP​ and \ensuremathΔ(\ensuremathδ,\ensuremathα)\mathrm{\ensuremath{\Delta}}(\ensuremath{\delta},\ensuremath{\alpha})\ensuremathΔ(\ensuremathδ,\ensuremathα), the T=0T=0T=0 gap induced by dimerization, depends strongly on \ensuremathα\ensuremath{\alpha}\ensuremathα and deviates from the BCS gap relation that holds in uncorrelated spin chains. Correlated states account quantitatively for the magnetic susceptibility of TTF-CuS4C4(CF3)4{\mathrm{CuS}}_{4}{\mathrm{C}}_{4}{({\mathrm{CF}}_{3})}_{4}CuS4​C4​(CF3​)4​ crystals (J1=79K{J}_{1}=79\phantom{\rule{0.16em}{0ex}}\mathrm{K}J1​=79K, \ensuremathα=0\ensuremath{\alpha}=0\ensuremathα=0, TSP=12K{T}_{\mathrm{SP}}=12\phantom{\rule{0.16em}{0ex}}\mathrm{K}TSP​=12K) and CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​ crystals (J1=160K{J}_{1}=160\phantom{\rule{0.16em}{0ex}}\mathrm{K}J1​=160K, \ensuremathα=0.35\ensuremath{\alpha}=0.35\ensuremathα=0.35, TSP=14K{T}_{\mathrm{SP}}=14\phantom{\rule{0.16em}{0ex}}\mathrm{K}TSP​=14K). The same parameters describe the specific heat anomaly of CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​ and inelastic neutron scattering. Modeling the spin-Peierls transition with correlated states exploits the fact that \ensuremathδ(0)\ensuremath{\delta}(0)\ensuremathδ(0) limits the range of spin correlations at T=0T=0T=0 while T>0T>0T>0 limits the range at \ensuremathδ=0\ensuremath{\delta}=0\ensuremathδ=0.
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) $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$.
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.
Exact diagonalization of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperatures TTT. Density matrix renormalization group calculations of progressively larger systems are used to obtain excitations up to a cutoff WC{W}_{C}WC​ and the low-TTT thermodynamics. The hybrid approach is applied to the magnetic susceptibility $\ensuremath{\chi}(T)$ and specific heat C(T)C(T)C(T) of spin-1/21/21/2 chains with isotropic exchange such as the linear Heisenberg antiferromagnet and the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model with ferromagnetic J1<0{J}_{1}<0J1​<0 and antiferromagnetic J2>0{J}_{2}>0J2​>0. The hybrid approach is fully validated by comparison with Heisenberg antiferromagnet results. It extends ${J}_{1}\ensuremath{-}{J}_{2}$ thermodynamics down to $T\ensuremath{\sim}0.01|{J}_{1}|$ for ${J}_{2}/|{J}_{1}|\ensuremath{\ge}{\ensuremath{\alpha}}_{c}=1/4$ and is consistent with other methods. The criterion for the cutoff WC(N){W}_{C}(N)WC​(N) in systems of NNN spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific T(N)T(N)T(N) at system size NNN.
Various forms of antiferromagnetic exchange interaction among quantized spins in one-dimensional and quasi-one-dimensional lattices are examined. Primary results are reported in two real compounds, the sodium vanadate NaV2O5 and the copper germanate CuGeO3, and in a class of compounds of real materials modeled as dimerized antiferromagnetically coupled spin-1/2 chains susceptible to spin-1 impurity doping and that include such examples as the strontium cuprate Sr14Cu24O41 and vanadium pyrophosphate (VO)2P2O7. Sodium vanadate, NaV2O5, a quarter-filled two-leg ladder compound that was originally thought to be composed of magnetic legs and nonmagnetic legs was subsequently shown to be a charge ordered system below room temperature. Initial models of the material as isolated antiferromagnetically coupled spin-1/2 chains can be mapped to the subsequently resolved magnetic system of the charge ordered state with the spins on molecular orbitals of V-O-V rungs antiferromagentically coupled between neighboring rungs either on the same ladder or on neighboring ladders. Comparing the model's dynamic structure factor to inelastic neutron scattering data could distinguish between the proposed spin coupling schemes and consequently reveal the details of the interaction between the lattice and the "zig-zag" charge density as well as the magnetic stabilization of the low temperature phase. Copper germanate, the first and only known inorganic spin-Peierls compound, exhibits a strong temperature dependent exchange coupling pattern among the spin-1/2 Cu2+ chains. Above the spin-Peierls transition temperature an unmodulated antiferromagnetic nearest neighbor exchange interaction J1 competes with an antiferromagnetic next nearest neighbor interaction J2. The next nearest neighbor exchange represents an effective coupling equivalent to the net effect of all longer range exchange interactions. This frustration to the nearest neighbor spin exchange produces a distinctive magnetic susceptibility χ (T) much different from the Bonner-Fisher susceptibility of the spin-1/2 Heisenberg chain with nearest neighbor antiferromagnetic exchange only. The ratio of the second to first nearest neighbor couplings α = J2/J1 ≈ 0.36 is sufficient to open a spontaneous gap in the spin-wave excitation spectrum at low temperatures. Below TSP = 14K, the dynamical structure factor is used to fit the dimerization δ and the exchange interactions J1 and J2 to the inelastic neutron scattering data of CuGeO3 at T = 10K. It is found that both δ and α increase significantly at lower temperatures, relative to the values obtained in the high temperature phase and at the onset of dimeration at 14 K. Static structure factor calculations how Scattering inconsistent with the δ−J1−J2 model and can be attributed primarily to the phonon degrees of freedom but possibly also to the couplings between chains in the b and a directions of the crystal lattice. Structurally dimerized compounds of antiferromagnetic spin-1/2 chains possess dynamic structure factor and magnetic susceptibility features that are very sensitive to doping with magnetic and nonmagnetic impurities. It is shown that the effects of spin-1 impurities are very similar to those of nonmagnetic (S = 0) impurities if the coupling between impurity and native spin is about the same magnitude as or larger than between native S = 1/2 spins. The microscopic origins of the similarity can be appreciated with the consideration that neighboring spins to the impurity find it more energetically favorable to couple to the impurity than the other spins of the lattice. Thus the spin-1 impurity and its neighboring S = 1/2 spins decouple from the lattice, creating a nonmagnetic break in the chain. The spins that now observe a nonmagnetic cluster on one side can couple to the S = 1/2 spins on the other side with a higher AF correlation than in the pure compound since they do not have to have a fluctuating dimer resonating between two neighbors. Such enhancements to the local correlations can stabilize the global AF order in one dimensional compounds as shown in theoretical and experimental studies of CuGeO3. ED calculations of the dynamic structure factor show that S(q, ω) developes states with ω = 0 in the gap at q = π. Magnetic susceptibilites in the presense of spin-1 impurities were obtained for 80 site chains by QMC simulations and for 16 site chains by ED calculations. An impurity concentration-dependent second peak in the low temperature region reveals an increasing second maximum with T as observed in doped copper germanates and suggests that the results of the study are applicable even to phonon mediated dimerization. This is not immediately apparent because spin-phonon coupling could distort the dimerization patterns. However, experimental and theoretical results show that elastic coupling between chains can stabilize the dimerization pattern due to the impurities and the results will still apply.
Exact diagonalization (ED) of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperature TTT. Density matrix renormalization group (DMRG) calculations of progressively larger systems are used to obtain excitations up to a cutoff WCW_CWC​ and the low-TTT thermodynamics. The hybrid approach is applied to the magnetic susceptibility χ(T)χ(T)χ(T) and specific heat C(T)C(T)C(T) of spin-1/21/21/2 chains with isotropic exchange such as the linear Heisenberg antiferromagnet (HAF) and the frustrated J1−J2J_1-J_2J1​−J2​ model with ferromagnetic (F) J1<0J_1 < 0J1​<0 and antiferromagnetic (AF) J2>0J_2 > 0J2​>0. The hybrid approach is fully validated by comparison with HAF results. It extends J1−J2J_1-J_2J1​−J2​ thermodynamics down to T∼0.01∣J1∣T \sim 0.01|J_1|T∼0.01∣J1​∣ for J2/∣J1∣≥αc=1/4J_2/|J_1| \geq α_c = 1/4J2​/∣J1​∣≥αc​=1/4 and is consistent with other methods. The criterion for the cutoff WC(N)W_C(N)WC​(N) in systems of NNN spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific T(N)T(N)T(N) at system size NNN.
A modified density matrix renormalization group (DMRG) algorithm is applied to the zigzag spin-12\frac{1}{2}21​ chain with frustrated antiferromagnetic exchange J1{J}_{1}J1​ and J2{J}_{2}J2​ between first and second neighbors. The modified algorithm yields accurate results up to ${J}_{2}/{J}_{1}\ensuremath{\approx}4$ for the magnetic gap $\ensuremath{\Delta}$ to the lowest triplet state, the amplitude BBB of the bond order wave phase, the wavelength $\ensuremath{\lambda}$ of the spiral phase, and the spin correlation length $\ensuremath{\xi}$. The J2/J1{J}_{2}/{J}_{1}J2​/J1​ dependences of $\ensuremath{\Delta}$, BBB, $\ensuremath{\lambda}$, and $\ensuremath{\xi}$ provide multiple comparisons to field theories of the zigzag chain. The twist angle of the spiral phase and the spin structure factor yield additional comparisons between DMRG and field theory. Attention is given to the numerical accuracy required to obtain exponentially small gaps or exponentially long correlations near a quantum phase transition.