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arXiv:2009.05809 · Condensed Matter Physics

Anomalous thermodynamics of a quantum spin system with large residual entropy

Richter, Schulenburg, Дмитриев, Krivnov, Jürgen Schnack

Published Dec 1, 2020• 6 citations
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Abstract

In contrast to strongly frustrated classical systems, their quantum counterparts typically have a non-degenerate ground state. A counterexample is the celebrated Heisenberg sawtooth spin chain with ferromagnetic zigzag bonds $J_1$ and competing antiferromagnetic basal bonds $J_2$. At a quantum phase transition point $|J_2/J_1|=1/2$, this model exhibits a flat one-magnon excitation band leading to a massively degenerate ground-state manifold which results in a large residual entropy. Thus, for the spin-half model, the residual entropy amounts to exactly one half of its maximum value $\lim_{T\to\infty} S(T)/N = \ln2$. In the present paper we study in detail the role of the spin quantum number $s$ and the magnetic field $H$ in the parameter region around the transition (flat-band) point. For that we use full exact diagonalization up to $N=20$ lattice sites and the finite-temperature Lanczos method up to $N=36$ sites to calculate the density of states as well as the temperature dependence of the specific heat, the entropy and the susceptibility. The study of chain lengths up to $N=36$ allows a careful finite-size analysis. At the flat-band point we find extremely small finite-size effects for spin $s=1/2$, i.e., the numerical data virtually correspond to the thermodynamic limit. In all other cases the finite-size effects are still small and become visible at very low temperatures. In a sizeable parameter region around the flat-band point the former massively degenerate ground-state manifold acts as a large manifold of low-lying excitations leading to extraordinary thermodynamic properties at the transition point as well as in its vicinity such as an additional low-temperature maximum in the specific heat. Moreover, there is a very strong influence of the magnetic field on the low-temperature thermodynamics including an enhanced magnetocaloric effect.

DOI
10.5488/cmp.23.43710

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In contrast to strongly frustrated classical systems, their quantum counterparts typically have a non-degenerate ground state. A counterexample is the celebrated Heisenberg sawtooth spin chain with ferromagnetic zigzag bonds J1J_1J1​ and competing antiferromagnetic basal bonds J2J_2J2​. At a quantum phase transition point ∣J2/J1∣=1/2|J_2/J_1|=1/2∣J2​/J1​∣=1/2, this model exhibits a flat one-magnon excitation band leading to a massively degenerate ground-state manifold which results in a large residual entropy. Thus, for the spin-half model, the residual entropy amounts to exactly one half of its maximum value lim⁡T→∞S(T)/N=ln⁡2\lim_{T\to\infty} S(T)/N = \ln2limT→∞​S(T)/N=ln2. In the present paper we study in detail the role of the spin quantum number sss and the magnetic field HHH in the parameter region around the transition (flat-band) point. For that we use full exact diagonalization up to N=20N=20N=20 lattice sites and the finite-temperature Lanczos method up to N=36N=36N=36 sites to calculate the density of states as well as the temperature dependence of the specific heat, the entropy and the susceptibility. The study of chain lengths up to N=36N=36N=36 allows a careful finite-size analysis. At the flat-band point we find extremely small finite-size effects for spin s=1/2s=1/2s=1/2, i.e., the numerical data virtually correspond to the thermodynamic limit. In all other cases the finite-size effects are still small and become visible at very low temperatures. In a sizeable parameter region around the flat-band point the former massively degenerate ground-state manifold acts as a large manifold of low-lying excitations leading to extraordinary thermodynamic properties at the transition point as well as in its vicinity such as an additional low-temperature maximum in the specific heat. Moreover, there is a very strong influence of the magnetic field on the low-temperature thermodynamics including an enhanced magnetocaloric effect.
For a class of highly frustrated antiferromagnetic quantum spin lattices the ground state exhibits a huge degeneracy in high magnetic fields due to the existence of localized magnon states. For some of these spin lattices (in particular, the 1D dimer-plaquette, sawtooth and kagom\'elike chains as well as the 2D kagom\'e lattice) we calculate rigorously the ground-state entropy at the saturation field. We find that the ground-state entropy per site remains finite at saturation. This residual ground-state entropy produces a maximum in the field dependence of the isothermal entropy at low temperatures. By numerical calculation of the field dependence of the low-temperature entropy for the sawtooth chain we find that the enhancement of isothermal entropy is robust against small deviations in exchange constants. Moreover, the effect is most pronounced in the extreme quantum case of spin 12\frac{1}{2}21​.
Abstract We consider the strongly anisotropic spin-1/2 XXZ model on the sawtooth-chain lattice with ferromagnetic longitudinal interaction Jzz = ΔJ and aniferromagnetic transversal interaction Jxx = Jyy = J > 0. At Δ = −1∕2 the lowest one-magnon excitation band is dispersionless (flat) leading to a massively degenerate set of ground states. Interestingly, this model admits a three-coloring representation of the ground-state manifold [H.J. Changlani et al., Phys. Rev. Lett. 120, 117202 (2018)]. We characterize this ground-state manifold and elaborate the low-temperature thermodynamics of the system. We illustrate the manifestation of the flat-band physics of the anisotropic model by comparison with two isotropic flat-band Heisenberg sawtooth chains. Our analytical consideration is complemented by exact diagonalization and finite-temperature Lanczos method calculations. Graphical abstract
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) $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$.
Abstract. We studied the thermodynamics of the one-dimensional J1-J2 spin-1/2 Heisenberg chain for ferromagnetic nearest-neighbor bonds J1 < 0 and frustrating antiferromagnetic nextnearest-neighbor bonds J2> 0 using full diagonalization of finite rings and a second-order Green-function formalism. Thereby we focus on J2 < |J1|/4 where the ground state is still ferromagnetic, but the frustration influences the thermodynamic properties. We found that their critical indices are not changed by J2. The analysis of the low-temperature behavior of the susceptibility χ leads to the conclusion that this behavior changes from χ ∝ T −2 at J2 < |J1|/4 to χ ∝ T −3/2 at the quantum-critical point J2 = |J1|/4. Another effect of the frustration is the appearance of an extra low-T maximum in the specific heat Cv(T) for J2 � |J1|/8, indicating its strong influence on the low-energy spectrum. Introduction: In low-dimensional frustrated quantum magnets thermal and quantum fluctuations strongly influence the low-temperature physics [1,2]. Special attention has been paid to one-dimensional (1D) J1-J2 quantum Heisenberg magnets, see Ref. [3] and references therein. Recent experimental studies have shown that edge-shared chain cuprates, such as LiVCuO4, Li(Na)Cu2O2, Li2ZrCuO4, and Li2CuO2 [4–13], represent a family of quantum magnets for
We investigate in detail the influence of the violation of the strict residual-entropy hierarchy among the neighboring ground states of different orders on the low-temperature behavior of the specific heat capacity in the magnetic frustrated systems in the framework of the exactly solvable antiferromagnetic spin-1/2 Ising model with the multisite interaction in the presence of the external magnetic field on the zigzag ladder. The exact expressions for the residual entropies of all ground states of the model are found, and it is shown that when the strength of the multisite interaction is strong enough then the model exhibits a quite interesting specific situation, namely, that there exist neighboring plateau ground states together with the single-point ground state that separates them, such that the magnetization properties of all of them are different but their entropy per site is the same and equal to zero. It is shown that this fact of the violation of the strict residual-entropy hierarchy between neighboring ground states of different orders leads to the reduction of the possible number of peaks that can appear in the low-temperature behavior of the specific heat capacity in the corresponding regions of the model parametric space. In addition, it is also shown that the absence of the strict residual-entropy hierarchy among neighboring ground states of different orders changes qualitatively the behavior of the specific heat capacity as a function of the external magnetic field, namely, that the typical field-induced sharp double-peak structure in the low-temperature behavior of the specific heat capacity, which is directly related to the very existence of the highly macroscopically degenerated single-point ground states, disappears already for relatively large values of the temperature.
We investigate the spin-1/2 Heisenberg model on the delta chain (sawtooth chain) with ferromagnetic nearest-neighbor and antiferromagnetic next-neighbor interactions. For a special ratio between these interactions there is a class of exact ground states formed by localized magnons and the ground state is macroscopically degenerate. The degree of this degeneration is found analytically. It is shown that the residual entropy per spin is s0=12ln2{s}_{0}=\frac{1}{2}ln\phantom{\rule{0.16em}{0ex}}2s0​=21​ln2. An important feature of this model is a sharp decrease of the gaps for excited states with an increase of the number of magnons. These excitations give an essential contribution to the low-temperature thermodynamics. The corresponding thermodynamic quantities are calculated using full exact diagonalization of finite chains up to N=22N=22N=22. The behavior of the considered model is compared with that of the delta chain with both antiferromagnetic interactions.