Spin-Peierls transition of the dimer phase of the $J_1-J_2$ model: Energy cusp and CuGeO$_3$ thermodynamics

TL;DR

This paper models the spin-Peierls transition in a dimerized spin-1/2 chain, revealing how ground state degeneracy causes an energy cusp that affects thermodynamic properties, with implications for CuGeO$_3$.

Contribution

It introduces a model capturing the energy cusp effect due to ground state degeneracy in the spin-Peierls transition, explaining experimental thermodynamics of CuGeO$_3$.

Findings

Energy cusp alters dimerization temperature dependence.

Model reproduces magnetic susceptibility and specific heat anomalies.

Parameters match experimental data for CuGeO$_3$.

Abstract

The spin-Peierls transition is modeled in the dimer phase of the spin-$1/2$ chain with exchanges $J_1$, $J_2 = \alpha J_1$ between first and second neighbors. The degenerate ground state generates an energy cusp that qualitatively changes the dimerization $\delta(T)$ compared to Peierls systems with nondegenerate ground states. The parameters $J_1 = 160$ K, $\alpha = 0.35$ plus a lattice stiffness account for the magnetic susceptibility of CuGeO$_3$, its specific heat anomaly, and the $T$ dependence of the lowest gap.

Figures

Equations

$$H(\alpha ,\delta )=\sum _r\left(1+\delta (-1{)}^r\right){\vec{S}}_r\cdot {\vec{S}}_{r+1}+\alpha \sum _r{\vec{S}}_r\cdot {\vec{S}}_{r+2}.$$

$$A(T,\alpha ,\delta )=-T\ln Q(T,\alpha ,\delta ).$$

$$\frac{\delta (T,\alpha )}{{\epsilon }_d}=-{\left(\frac{\partial A(T,\alpha ,\delta )}{\partial \delta }\right)}_{\delta (T,\alpha )}.$$

$$-{\left(\frac{\partial E_0(\alpha ,\delta )}{\partial \delta }\right)}_0=B(\alpha )=$$

$$⟨{\psi }_1(\alpha )|\sum _r(-1{)}^r{\vec{S}}_r\cdot {\vec{S}}_{r+1}|{\psi }_{-1}(\alpha )⟩/N.$$

$$-\left(\frac{\partial E_0(\delta )}{\partial \delta }\right)=\frac{2\delta \left[K(q)-E(q)\right]}{\pi q^2}.$$

$$C(T,{\epsilon }_d)=C(T,\delta (T))+$$

$$\frac{\partial \delta }{\partial T}\left[{\left(\frac{\partial E(T,\delta )}{\partial \delta }\right)}_T+\frac{\delta (T)}{{\epsilon }_d}\right].$$

Full text

As pointed out by Riera and Dobry Riera and Dobry (1995), an improved fit of the CuGeO3 magnetic susceptibility $\chi(T)$ is obtained by adding an exchange $J_{2}=\alpha J_{1}$ between second neighbors. The $J_{1}-J_{2}$ model has its own literature. The frustration parameter $\alpha=0.35$ and $J_{1}=160$ K that fits the high-$T$ susceptibility Fabricius et al. (1998) is in the dimer phase that runs from the quantum critical point Okamoto and Nomura (1992) $\alpha_{c}=0.2411$ where a singlet-triplet gap $\Delta(\alpha)$ opens to the Majumdar-Gosh (MG) point Majumdar and Ghosh (1969) $\alpha=0.50$ where the exact ground state is known. But $\alpha<\alpha_{c}$ has been advocated for other reasons Castilla et al. (1995); Bouzerar et al. (1999) and while frustration is clearly important, the magnitude of $\alpha$ has not been definitively determined Uchinokura (2002). Inelastic neutron scattering Nishi et al. (1994) indicates an exchange $J_{\perp}\sim 0.1J_{1}$ between spins in adjacent chains; a 1D model may not be appropriate for CuGeO3. Another complication is that single crystals, $T_{SP}$ and properties depend somewhat on growth conditions. Most analysis is in terms of the $300$ K structure reported by Vollenkle et al. Völlenkle et al. (1967) with space group $Pbmm(D_{2h}^{5})$. The refined structure reported by Hidaka et al. Hidaka et al. (1997) has lower symmetry and larger unit cell.

We discuss in this paper the SP transition in the dimer phase of the $J_{1}-J_{2}$ model and show that the dimer phase is the key difference from previous Peierls or SP systems. The ground state of the rigid lattice is doubly degenerate with spontaneously broken inversion symmetry, while the ground state of other systems is nondegenerate. That includes the inorganic and organic systems dating back to Krogmann salts that Pouget has recently reviewed Pouget (2016); *pouget2017.

We consider the MG point where the degeneracy occurs in finite chains and $\alpha=0.35$ where the degeneracy is in the thermodynamic limit. Degeneracy has not been discussed in SP modeling of finite chains, which indeed are nondegenerate. Although $\alpha=0.35$ is motivated by CuGeO3 and we make contact below with its thermodynamics, our principal goal is the SP transition of the dimer phase of the $J_{1}-J_{2}$ model. The basic model has only three parameters: $J_{1}$ and $\alpha$ specify the spin chain. Dimerization is opposed by a harmonic lattice with linear spin-phonon coupling. A realistic model of CuGeO3 is clearly more complicated.

Energy cusp and equilibrium.—We take $J_{1}$ as the unit of energy, frustration $\alpha=J_{2}/J_{1}$ and dimerization $\delta$. The dimerized $J_{1}-J_{2}$ model for spin $S_{r}=1/2$ at site $r$ and periodic boundary conditions is

[TABLE]

The ground state of $H(\alpha,0)$ is doubly degenerate in the dimer phase. Finite $\delta$ breaks inversion symmetry at sites and increases the singlet-triplet gap $\Delta(\alpha,\delta)$ but does not change the chain length in systems with periodic boundary conditions.

In reduced units, the free energy per spin in the thermodynamic limit is

[TABLE]

The corresponding expression for finite systems has $N^{-1}\ln{Q(T,\alpha,\delta,N)}$. The Boltzmann sum in the partition function is over the eigenstates $E_{j}(\alpha,\delta,N)$ of the $N$-spin system. The adiabatic approximation for a lattice with potential energy per site $\delta^{2}/2\varepsilon_{d}$ is standard practice. The equilibrium condition that minimizes the total free energy is

[TABLE]

This relation holds for finite $N$ as well as for other Peierls systems. Dimerization $\delta(T,\alpha)$ is largest at $T=0$ and decreases to zero at $T=T_{SP}$.

The ground state energy per site, $E_{0}(\alpha,\delta,N)$, is the free energy at $T=0$. The two VB diagrams with $N/2$ singlet pairs between neighbors are the $\delta=0$ ground states at the MG point. The exact $\delta=0$ energy and slope are both $-3/8$. The $\alpha=0.35$, $\delta=0$ ground state has avoided crossings at finite $N$. The cusp in the thermodynamic limit is the amplitude $B(\alpha)$ of the bond order wave Kumar et al. (2010),

[TABLE]

The degenerate ground states $\psi_{\pm 1}(\alpha)$ are even and odd, respectively, under inversion at sites. We obtain smaller $B(0.35)=0.0783$ compared to $B(0.5)=0.375$. $B(\alpha)$ is exponentially small near $\alpha_{c}=0.2411$ and is intrinsic to the dimer phase.

Fig. 1 shows $-(\partial A/\partial\delta)$ as functions of $\delta$ at $\alpha=0.50$ and $0.35$. The thermodynamic limit is easily reached at $\alpha=0.50$ but requires $N\sim 100$ at $\alpha=0.35$, $\delta=0$. The dashed $\delta/\varepsilon_{d}$ lines are shown for $1/\varepsilon_{d}=15$ and $24$, which leads respectively to $\delta(0)=0.025$ and $0.016$ at $\alpha=0.50$. The $\alpha=0.35$ lines intersect at smaller $\delta(0)$. By contrast, the right-hand side of Eq. 3 is proportional to $\delta$ in Peierls systems with a nondegenerate ground state. For example, the ground state of a half-filled band of noninteracting spinless fermions leads to

[TABLE]

Here $q^{2}=1-\delta^{2}$ and $K(q)$, $E(q)$ are complete elliptic integrals of the first and second kind. The fermion result in Fig. 1 is qualitatively different since it passes through the origin. Nevertheless, finite $\delta(0)$ is guaranteed for finite $\varepsilon_{d}$ after dividing both sides of Eq. 3 by $\delta$ because $K(q)$ has a logarithmic divergence at $\delta=0$. The half-filled tight binding (Hückel) band has $4\delta$ instead of $2\delta$ in Eq. 5. Previous Peierls or SP systems Jacobs et al. (1976); *bray1983; Pouget (2016); *pouget2017 have nondegenerate ground states; the $E_{0}(\delta)$ expansion has only even powers of $\delta$.

SP transition.—Dimerization at finite $T$ follows from the equilibrium condition. The $T=0.03$ or $0.05$ lines in Fig. 1 intersect $\delta/\varepsilon_{d}$ at smaller $\delta(T)$, while the $T=0.07$ line for $\alpha=0.35$ passes through the origin. We obtain the thermodynamics of $H(\alpha,\delta)$ by exact diagonalization (ED) to $N=24$ for $\delta=0$ and $N=20$ for $\delta>0$. We use the density matrix renormalization group (DMRG) for the ground state properties of larger systems whose thermodynamics is found by a combination of ED and DMRG Saha et al. (2019). The energy spectrum ${E(\delta,N)}$ is truncated at a cutoff $W_{C}(\delta,N)$ based on $S(T,\delta,N)/T$, where $S$ is the entropy per site. $S(T,\delta,N)$ is a lower bound on the thermodynamic limit, $S(T,\delta)$, due to finite size gaps at low $T$ and truncation at high $T$. The maximum of $S(T,\delta,N)/T$ at $T^{\prime}(\delta,N)$ is the best approximation of $S(T^{\prime},\delta)$ for the truncated spectrum. Accordingly, we keep the lowest few thousand states needed to converge, or almost to converge, the maximum of $S(T,\delta,N)/T$.

Fig. 2 shows $\delta(T)$ vs $T$ at $\alpha=0.35$ and $N=32$. Each point is an intersection as in Fig. 1. Comparison with results for $N=20$ and $24$, as well as fewer points at $N=48$ and $64$, indicate that $N=32$ is close to the thermodynamic limit. As expected, $\delta(0)$ and $T_{SP}$ decrease with increasing stiffness $1/\varepsilon_{d}$. Similar $\delta(T)$ vs. $T$ curves with higher $T_{SP}$ are found at $\alpha=0.50$.

The choice $1/\varepsilon_{d}=11.1$ returns $T_{SP}=0.09=14.4$ K, in the observed range for CuGeO3 crystals. The calculated dimerization is $\delta(0)=0.025$. Fig. 3 shows that $\delta(T)/\delta(0)$ vs. $T/T_{SP}$ falls on a single line for all curves in Fig. 2. There is little dependence on $\varepsilon_{d}$ in stiff lattices with small $\delta(0)$. The ratio of the gaps $\Delta(\alpha,\delta(T))$ at $T$ and [math] in Fig. 3 is evaluated at $\delta(T)$ along the $1/\varepsilon_{d}=11.1$ line. The gap ratio, not shown, for $\Delta(\alpha)$ at $T\geq T_{SP}$ is less than $3\%$ in the thermodynamic limit.

We include in Fig. 3 gap ratios from inelastic neutron scattering Nishi et al. (1994); Lussier et al. (1996); Martin et al. (1996). The dashed line is $\delta(T)/\delta(0)$ for a half-filled band of noninteracting fermions, spinless or spin-1/2, with small $\delta(0)$ and, more importantly in the present context, gap $4\delta(T)$. The SP transition of the HAF, approximations and connections to BCS are discussed in Ref. Jacobs et al., 1976; *bray1983. The neutron data disagreed with general expectations. The scaled gaps are instead consistent with an SP transition in the dimer phase.

The data in Fig. 3 are specific to CuGeO3. We turn to thermodynamics for two reasons. One is to seek additional evidence for the characteristic $\delta(T)$ of the dimer phase. The other is to propose that the dimer phase of the $J_{1}-J_{2}$ model is the proper starting point for CuGeO3 modeling.

Susceptibility and specific heat.—The $\chi(T)$ data Fabricius et al. (1998) in Fig. 4 from $T=10$ to $950$ K was kindly provided in digital form by Professor Lorenz. ED for $N=18$ reaches the thermodynamic limit at $T<T_{max}=56$ K, the maximum that was used to fix the parameters Fabricius et al. (1998) $J_{1}=160$ K, $\alpha=0.35$ and (from ESR) $g=2.256$. ED to $N=24$ reaches the limit by $T\sim 25$ K. DMRG results are shown as dashed lines that terminate at the solid point $T^{\prime}(N)$ that are the best approximation to the thermodynamic limit for the truncated energy spectrum. The limit for $N=48$ holds for $T>10$ K, and $\chi(T,0)$ closely matches the $T>T_{SP}$ data. Bouzerar et al. Bouzerar et al. (1999) discuss reasons for choosing $\alpha$ slightly less than $\alpha_{c}$ in Eq. 1, which leads Castilla et al. (1995) to $T_{m}>100$ K. They address the issue by treating the observed Nishi et al. (1994) interchain exchange $J_{\perp}\sim 0.1J_{1}$ at a mean field level; even so their $\chi(T)$ fit (Fig.1, ref. Bouzerar et al., 1999) is rather approximate below $100$ K.

The calculated $\chi(T,\varepsilon_{d})$ at $T<T_{SP}=14.4$ K is evaluated with $\delta(T)$ along the $1/\varepsilon_{d}=11.1$ line in Fig. 2. The thermodynamic limit is easily reached for dimerized systems, here with $\delta(0)=0.025$. Data Hase et al. (1993a); *haseprb1993 up to $10$ K are shown as crosses. The $10$ K points do not match. The $T<T_{SP}$ fit is for the basic 1D model, Eq. 1, with isotropic exchange.

The measured Lorenz et al. (1996); Liu et al. (1995) specific heat $C(T)$ to $20$ K is shown in Fig. 5 as the entropy derivative $S^{\prime}=C/T$. The dashed line is the lattice (Debye) contribution, $AT^{2}$, with Liu et al. (1995) $A=0.32$ mJ mole*-1K-4*. The anomaly is sharper and better resolved than in organic SP systems Wei et al. (1977). The equilibrium $C(T,\varepsilon_{d})$ has two contributions below $T_{SP}$,

[TABLE]

The first term is evaluated at $\delta(T)$ along the $1/\varepsilon_{d}=11.1$ line in Fig. 2. The second term tracks $\delta(T)$ variations. The calculated $C(T,\varepsilon_{d})/T$ and $C(T,0)/T$ are shown in Fig. 5. The low-$T$ behavior at $\delta=0$ is a finite size effect. Since the gap $\Delta(\alpha,N)>\Delta(\alpha)$ initially decreases $C(T,\alpha,N)/T$, entropy conservation requires increased $C(T,\alpha,N)/T$ before converging from above to the thermodynamic limit. The $N=48$ and $24$ gaps are smaller and larger, respectively, than $N=32$. The thermodynamic limit is almost reached by $12$ K for $N=32$.

The $C(T,\delta(T))/T$ term generates the curve labeled (a) in Fig. 5. The $\partial\delta(T)/\partial T$ term is mainly responsible for the sharp anomaly. It was computed by fitting $\delta(T)$ in Fig. 3 to a smooth curve and taking the derivative. The area $C(T,\varepsilon_{d})/T$ to $T_{SP}$ is within $5\%$ of the accurately known area for $\delta=0$. The anomaly is directly related to $\delta(T)$ of the dimer phase. The adiabatic and mean field approximations for the lattice enforce $\delta=0$ for $T>T_{SP}$; this oversimplification is a general problem for modeling any transition and has long been recognized. The measured area (or entropy) at $20$ K is within a few percent of the $\delta=0$ entropy. Lattice fluctuations above $T_{SP}$ imply reduced entropy below $T_{SP}$. Overall, the anomaly is fit almost quantitatively.

Conclusions.—We have analyzed the SP transition of the dimer phase of the $J_{1}-J_{2}$ model. Dimerization $\delta>0$ lifts the degeneracy and generates a cusp $B(\alpha)$ at $\delta=0$ in the ground-state energy $E_{0}(\delta)$. The cusp differentiates the dimer phase from previously studied Peierls and SP models with nondegenerate ground states. Such models have $E_{0}^{\prime}(0)=0$ and transitions due to divergent curvatures $E_{0}^{\prime\prime}(0)$. Their transitions illustrate shared features among otherwise quite dissimilar quasi-1D systems.

The equilibrium condition, Eq. 3, for dimerization returns a different $\delta(T)$ when $E_{0}(\delta)$ has a cusp. The temperature dependence of $\delta(T)/\delta(0)$ or of the scaled gap in Fig. 3 is characteristic of the dimer phase. The specific heat anomaly in Fig. 5 is directly related to $\partial\delta(T)/\partial T$. The parameters $J_{1}=160$ K and $\alpha=0.35$ in Eq. 1 plus the stiffness $1/\varepsilon_{d}=11.1$ in Eq. 3 account for several features of the SP transition of CuGeO3. We are applying the basic model to other data and think that the dimer phase is the starting point for detailed modeling these quasi-1D spin-$1/2$ chains.