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Physical review. B, Condensed matter

Exact solutions in a S=12 quantum spin ladder model with pair and quartic interactions

Jeremiah H. Barry, Mark W. Meisel

Published Aug 1, 1998• 3 citations
Publisher

Abstract

The effects of multispin interactions upon quantum spin ladders are investigated theoretically. A two-leg $S=\frac{1}{2}$ quantum spin ladder is taken to have intrarung (inter-rung) quadratic (biquadratic) Heisenberg exchange-type interactions and is placed in a uniform magnetic field. Due to the commutability properties of the Hamiltonian, the magnetic canonical partition function of the model is exactly calculable by a method of rung transfer matrix. Exact solutions are then derived for thermodynamic quantities like free and internal energies, magnetization, and magnetic susceptibilities (initial and nonlinear). The zero-temperature results comprise exact solutions in the complete interaction parameter space for the ground-state energy and degeneracy, zero-field spin gap, loci of quantum phase transitions, and the critical magnetic field.

DOI
10.1103/physrevb.58.3129

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Physical review. B, Condensed matter•Oct 20, 2003•36 citations
The effects of multispin interactions upon quantum spin ladders are investigated theoretically. A two-leg S=12S=\frac{1}{2}S=21​ quantum spin ladder is taken to have intrarung (inter-rung) quadratic (biquadratic) Heisenberg exchange-type interactions and is placed in a uniform magnetic field. Due to the commutability properties of the Hamiltonian, the magnetic canonical partition function of the model is exactly calculable by a method of rung transfer matrix. Exact solutions are then derived for thermodynamic quantities like free and internal energies, magnetization, and magnetic susceptibilities (initial and nonlinear). The zero-temperature results comprise exact solutions in the complete interaction parameter space for the ground-state energy and degeneracy, zero-field spin gap, loci of quantum phase transitions, and the critical magnetic field.
We consider a two-leg S=1/2 quantum spin ladder model with two-spin (intra-rung) and four-spin (inter-rung) Heisenberg exchange interactions and a uniform magnetic field. Exact mappings are derived connecting the partition function and correlations in the three-parameter quantum model to those known in a two-parameter Ising chain. The quantum phase diagram of the ladder magnet is determined. Static correlations provide pertinent correlation lengths and underlie spatial fluctuation behaviors at arbitrary temperatures, including quantum fluctuations at absolute zero. Dynamic correlations in zero field are used to obtain an exact solution for the inelastic neutron scattering function Sxx(q, ω) at all temperatures, explicitly yielding the elementary excitation spectrum.
We study the field-induced quantum phase transitions (QPT) and the relevant magnetic properties of a spin-1/2 two-leg integrable spin ladder (ISL), of which the system parameters in bridging to the real compounds are determined by setting the extra interactions in the Hamiltonian of the ISL relative to the Heisenberg spin ladder to vanish in the expectation in the ground state (GS). Such an ISL analytically has the correct leading terms of both the critical fields of the two QPT's as in the real strongly-coupled compounds: gμ_BH_{c1}=J_{\perp}-J_{\parallel} and gμ_BH_{c2}=J_{\perp}+2J_{\parallel} in terms of the experimental leg (J_{\parallel}) and rung (J_{\perp}) interactions. The symmetric magnetization inflection point is located at gμ_BH_{IP}=J_{\perp}+J_{\parallel}/2. The magnetizations for the GS and at finite temperatures, as well as the susceptibility, show good agreements in various comparisons with the finite-site exact diagonalization, the transfer-matrix renormalization group numerical result, the perturbation theory, and the compounds (5IAP)_2CuBr_4\cdot 2H_2O, Cu_2(C_5H_{12}N_2)_2Cl_4 and (C_5H_{12}N)_2CuBr_4.
We study a frustrated two-leg spin ladder with alternate isotropic Heisenberg and Ising rung exchange interactions, whereas, interactions along legs and diagonals are Ising-type. All the interactions in the ladder are anti-ferromagnetic in nature and induce frustration in the system. This model shows four interesting quantum phases: (i) stripe rung ferromagnetic (SRFM), (ii) stripe rung ferromagnetic with edge singlet (SRFM-E), (iii) anisotropic antiferromagnetic (AAFM), and (iv) stripe leg ferromagnetic (SLFM) phase. We construct a quantum phase diagram for this model and show that in stripe rung ferromagnet (SRFM), the same type of sublattice spins (either SSS or σσσ-type spins) are aligned in the same direction. Whereas, in anisotropic antiferromagnetic phase, both SSS and σσσ-type of spins are anti-ferromagnetically aligned with each other, two nearest SSS spins along the rung form an anisotropic singlet bond whereas two nearest σσσ spins form an Ising bond. In large Heisenberg rung exchange interaction limit, spins on each leg are ferromagnetically aligned, but spins on different legs are anti-ferromagnetically aligned. The thermodynamic quantities like Cv(T)Cv(T)Cv(T), χ(T)χ(T)χ(T) and S(T)S(T)S(T) are also calculated using the transfer matrix method for different phase. The magnetic gap in the SRFM and the SLFM can be notice from χ(T)χ(T)χ(T) and Cv(T)Cv(T)Cv(T) curves.
Abstract We study a frustrated two-leg spin ladder with alternate isotropic Heisenberg and Ising rung exchange interactions, whereas, interactions along legs and diagonals are Ising-type. All the interactions in the ladder are anti-ferromagnetic in nature and induce frustration in the system. This model shows four interesting quantum phases: (i) stripe rung ferromagnetic (SRFM), (ii) stripe rung ferromagnetic with edge singlet (SRFM-E), (iii) anisotropic antiferromagnetic (AAFM), and (iv) stripe leg ferromagnetic (SLFM) phase. We construct a quantum phase diagram for this model and show that in stripe rung ferromagnet (SRFM), the same type of sublattice spins (either isotropic S -type or discrete anisotropic σ -type spins) are aligned in the same direction. Whereas, in anisotropic antiferromagnetic phase, both S and σ -type of spins are anti-ferromagnetically aligned with each other, two nearest S spins along the rung form an anisotropic singlet bond whereas two nearest σ spins form an Ising bond. In large Heisenberg rung exchange interaction limit, spins on each leg are ferromagnetically aligned, but spins on different legs are anti-ferromagnetically aligned. The thermodynamic quantities like specific heat C v ( T ), magnetic susceptibility χ ( T ) and thermal entropy S ( T ) are also calculated using the transfer matrix method for various phases. The magnetic gap in the SRFM and the SLFM can be noticed from χ ( T ) and C v ( T ) curves.
The ground-state behaviour of the frustrated quantum spin-1/2 two-leg ladder with the Heisenberg intra-rung and Ising inter-rung interactions is examined in detail. The investigated model is transformed into the quantum Ising chain with composite spins in an effective transverse and longitudinal field by employing either the bond-state representation or the unitary transformation. It is shown that the ground state of the Heisenberg–Ising ladder can be descended from three exactly solvable models: the quantum Ising chain in a transverse field, the ‘classical’ Ising chain in a longitudinal field or the spin-chain model in a staggered longitudinal–transverse field. The last model serves in evidence of the staggered bond phase with alternating singlet and triplet bonds on the rungs of a two-leg ladder, which appears at moderate values of the external magnetic field and consequently leads to a fractional plateau at a half of the saturation magnetization. The ground-state phase diagram totally consists of five ordered and one quantum paramagnetic phase, which are separated from each other either by the lines of discontinuous or continuous quantum phase transitions. The order parameters are exactly calculated for all five ordered phases and the quantum paramagnetic phase is characterized through different short-range spin–spin correlations.
We study quantum spin models with nearest- and next-nearest-neighbor exchange interactions by means of exact solutions. Depending on the representation, the models can also be viewed as spin ladders with interchain couplings and with ring biquadratic spin-spin interactions. The ground-state phase diagram is constructed as a function of the next-nearest-neighbor couplings and an external magnetic field. Several quantum phase transitions in the ground state, including the possibility of a first-order quantum phase transition, governed by the external magnetic field, are predicted. The low-energy conformal behavior of the low-lying excitations is studied in the commensurate and incommensurate phases. Finally, the properties of the models at finite temperature are studied exactly. A finite set of nonlinear integral equations determining the thermodynamic potential at arbitrary temperature and external magnetic field is derived and solved in several important cases. The results for the magnetic susceptibility and the Sommerfeld coefficient of the specific heat are presented.