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Physical review. B./Physical review. B

Hybrid exact diagonalization and density matrix renormalization group approach to the thermodynamics of one-dimensional quantum models

Sudip Kumar Saha, Dayasindhu Dey, Manoranjan Kumar, Z. G. Soos

Published May 24, 2019• 12 citations
Publisher

Abstract

Exact diagonalization of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperatures $T$. Density matrix renormalization group calculations of progressively larger systems are used to obtain excitations up to a cutoff ${W}_{C}$ and the low-$T$ thermodynamics. The hybrid approach is applied to the magnetic susceptibility $\ensuremath{\chi}(T)$ and specific heat $C(T)$ of spin-$1/2$ chains with isotropic exchange such as the linear Heisenberg antiferromagnet and the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model with ferromagnetic ${J}_{1}<0$ and antiferromagnetic ${J}_{2}>0$. 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 ${W}_{C}(N)$ in systems of $N$ spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific $T(N)$ at system size $N$.

DOI
10.1103/physrevb.99.195144

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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 \ensuremathχ(T)\ensuremath{\chi}(T)\ensuremathχ(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 J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ 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 J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ thermodynamics down to T\ensuremath∼0.01∣J1∣T\ensuremath{\sim}0.01|{J}_{1}|T\ensuremath∼0.01∣J1​∣ for J2/∣J1∣\ensuremath≥\ensuremathαc=1/4{J}_{2}/|{J}_{1}|\ensuremath{\ge}{\ensuremath{\alpha}}_{c}=1/4J2​/∣J1​∣\ensuremath≥\ensuremathαc​=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.
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.
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 thermodynamic quantities (spin-spin correlation functions $\ensuremath{\langle}{\mathbf{S}}_{0}{\mathbf{S}}_{n}\ensuremath{\rangle}$, correlation length $\ensuremath{\xi}$, spin susceptibility $\ensuremath{\chi}$, and specific heat CV{C}_{V}CV​) of the frustrated one-dimensional J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg ferromagnet with arbitrary spin quantum number SSS below the quantum critical point, i.e., for J2<∣J1∣/4{J}_{2}<|{J}_{1}|/4J2​<∣J1​∣/4, are calculated using a rotation-invariant Green-function formalism and full diagonalization as well as a finite-temperature Lanczos technique for finite chains of up to N=20N=20N=20 sites. The low-temperature behavior of the susceptibility $\ensuremath{\chi}$ and the correlation length $\ensuremath{\xi}$ are well described by $\ensuremath{\chi}=\frac{2}{3}{S}^{4}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}2}+A{S}^{5/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}3/2}$ and $\ensuremath{\xi}={S}^{2}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}1}+B{S}^{1/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}1/2}$ with $A\ensuremath{\approx}1.1\ensuremath{\cdots}1.2$ and $B\ensuremath{\approx}0.84\ensuremath{\cdots}0.89$. The vanishing of the factors in front of the temperature at J2=∣J1∣/4{J}_{2}=|{J}_{1}|/4J2​=∣J1​∣/4 indicates a change of the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}$ at $T\ensuremath{\rightarrow}0$. The specific heat may exhibit an additional frustration-induced low-temperature maximum when approaching the quantum critical point. Such a maximum appears for S=1/2S=1/2S=1/2 and S=1S=1S=1, but was not found for S>1S>1S>1.
The magnetic and thermodynamic properties of spin-1∕21∕21∕2 Heisenberg diamond chains are investigated in three different cases: (a) J1{J}_{1}J1​, J2{J}_{2}J2​, J3>0{J}_{3}>0J3​>0 (frustrated), (b) J1{J}_{1}J1​, J3<0{J}_{3}<0J3​<0, J2>0{J}_{2}>0J2​>0 (frustrated), and (c) J1{J}_{1}J1​, J2>0{J}_{2}>0J2​>0, J3<0{J}_{3}<0J3​<0 (nonfrustrated), where the bond coupling Ji{J}_{i}Ji​ (i=1,2,3)>0(i=1,2,3)>0(i=1,2,3)>0 stands for an antiferromagnetic (AF) interaction, and <0<0<0 for a ferromagnetic (F) interaction. The density-matrix renormalization-group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer-matrix renormalization-group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin-correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor S(q)S(q)S(q) shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$ (a=0,1,2,3,4,5)(a=0,1,2,3,4,5)(a=0,1,2,3,4,5) for different couplings in a zero magnetic field, which, however, in the magnetic fields where the magnetization plateau with m=1∕6m=1∕6m=1∕6 pertains, exhibits the peaks only at q=0q=0q=0, $2\ensuremath{\pi}∕3$, and $4\ensuremath{\pi}∕3$, which are found to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility, and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility, and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the XXZXXZXXZ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat, and magnetization for the compound Cu3(CO3)2(OH)2{\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}Cu3​(CO3​)2​(OH)2​ are fairly compared with our TMRG results.
We calculate the thermodynamic quantities (correlation functions ⟨S0Sn⟩⟨{\mathbf{S}}_{0}{\mathbf{S}}_{n}⟩⟨S0​Sn​⟩, correlation length $\ensuremath{\xi}$, spin susceptibility $\ensuremath{\chi}$, and specific heat CV{C}_{V}CV​) of the frustrated one-dimensional spin-half ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg ferromagnet, i.e., for J2<0.25∣J1∣{J}_{2}<0.25|{J}_{1}|J2​<0.25∣J1​∣, using a rotation-invariant Green's-function formalism and full diagonalization of finite lattices. We find that the critical indices are not changed by J2{J}_{2}J2​, i.e., $\ensuremath{\chi}={y}_{0}{T}^{\ensuremath{-}2}$ and $\ensuremath{\xi}={x}_{0}{T}^{\ensuremath{-}1}$ at $T\ensuremath{\rightarrow}0$. However, the coefficients y0{y}_{0}y0​ and x0{x}_{0}x0​ linearly decrease with increasing J2{J}_{2}J2​ according to the relations ${y}_{0}=(1\ensuremath{-}4{J}_{2}/|{J}_{1}|)/24$ and ${x}_{0}=(1\ensuremath{-}4{J}_{2}/|{J}_{1}|)/4$, i.e., both coefficients vanish at J2=0.25∣J1∣{J}_{2}=0.25|{J}_{1}|J2​=0.25∣J1​∣ indicating the zero-temperature phase transition that is accompanied by a change in the low-temperature behavior of $\ensuremath{\chi}$ $(\ensuremath{\xi})$ from $\ensuremath{\chi}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ $(\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}1})$ at J2<0.25∣J1∣{J}_{2}<0.25|{J}_{1}|J2​<0.25∣J1​∣ to $\ensuremath{\chi}\ensuremath{\propto}{T}^{\ensuremath{-}3/2}$ $(\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}1/2})$ at J2=0.25∣J1∣{J}_{2}=0.25|{J}_{1}|J2​=0.25∣J1​∣. In addition, we detect the existence of an additional low-temperature maximum in the specific heat when approaching the critical point at J2=0.25∣J1∣{J}_{2}=0.25|{J}_{1}|J2​=0.25∣J1​∣.
We study the one-dimensional S = 1/2 Heisenberg model with an uniform and a staggered magnetic fields, using the dynamical density-matrix renormalization group (DDMRG) technique. The DDMRG enables us to investigate the dynamical properties of chain with lengths up to a few hundreds, and the results are numerically exact in the same sense as 'exact diagonalization' results are. Thus, we can analyze the low-energy spectrum almost in the thermodynamic limit. In this work, we calculate the dynamical spin structure factor and demonstrate the performance of the DDMRG method applying the open-end boundary conditions as well as the periodic boundary conditions.