Replica theory for disorder-free spin-lattice glass transition on a tree -like simplex network

Our model consists of classical spins and lattice distortions on a corner-sharing $(k-1)$-dimensional simplices network with connectivity $c$ which is the number of tetrahedra sharing one lattice site. The pyrochlore lattice and kagomé lattice correspond to the cases of $(c,k)=(2,4)$ and $(c,k)=(2,3)$, respectively, as shown in Figs. 1 (a) and (b). In order to develop an exactly solvable mean-field model, we consider a tree-like graph of simplices with connectivity $c$, as shown in Figs. 1 (d) and (e) so that the presence of loops can be neglected except for the local triangles within the simplices. More precisely we consider dense limit $N\gg c\gg 1$Yoshino (2018) which greatly simplifies the theory.

We assume that spins are $M$-component classical vector spins $\bm{S}_{i}=(S_{i}^{1},S_{i}^{2},...,S_{i}^{M})$ $(i=1,2,...,N)$ subjected to a spherical constraint $\sum_{i=1}^{N}|\bm{S}_{i}|^{2}=NM$. We assume that the lattice distortions are $c$-component vectors $\bm{\sigma}=(\sigma_{i}^{1},\sigma_{i}^{2},...,\sigma_{i}^{c})$ subjected to a spherical constraint $\sum_{i=1}^{N}|\bm{\sigma}_{i}|^{2}=Nc$. Here $c$ is the connectivity of the lattice which plays the role of spacial dimension within our model. In order to obtain a non-trivial theory in the dense limit $N\gg c\gg 1$, we also assume $M\gg 1$ with,

being fixed to a value of $O(1)$.

The Hamiltonian is given by,

where $\langle i,j\rangle$ denotes the summation over the nearest-neighboring pairs, $\hat{\bm{r}}_{ij}$ is the unit vector from the $i$-th site to the $j$-th site, $J$ and $\epsilon$ are energy scales of spin-spin and lattice-lattice interactions, respectively. The parameter $\epsilon$ reflects the elastic energy cost of lattice distortion Mitsumoto et al. (2022) so that we call it rigidity in the following. The spin-exchange interaction is antiferromagnetic without lattice distortion and is modified by the lattice distortion via the Goodenough-Kanamori mechanism Goodenough (1955); Kanamori (1959). The parameter $\delta$ controls the strength of spin-lattice coupling.

Let us note that the Hamiltonian is invariant under the global reflection of the spins $\bm{S}_{i}\rightarrow-\bm{S}_{i}$ for $\forall i$. On the other hand, it is not invariant under the global reflection of the lattice distortions $\bm{\sigma}_{i}\rightarrow-\bm{\sigma}_{i}$ for $\forall i$ in general except in the absence of the spin-lattice coupling $\delta=0$.

The present model can be regarded as a high dimensional variant of the model defined in Ref. Mitsumoto et al. (2020) for the pyrochlore system. In the pyrochlore system, if the spin-lattice coupling is switched off $\delta=0$, both spin and lattice distortion remain in liquid (paramagnetic) phase down to $T=0$ due to frustration. In the presence of the coupling $\delta>0$, if the energy scale of lattice-lattice interactions is not too small, numerical simulations suggest the system exhibits a simultaneous spin and lattice glass transition at a critical temperature Mitsumoto et al. (2020).

On the corner-sharing $(k-1)$-simplices network, we can formally rewrite the Hamiltonian Eq. (2) as the summation of the potential energy running over all simplices $\triangle=1,2,\ldots,N_{\triangle}$,

with

where

is the number of simplices in the whole system, $\vec{\bm{S}}_{(\triangle)}=(\bm{S}_{1(\triangle)},\bm{S}_{2(\triangle)},...,\bm{S}_{k(\triangle)})$ and $\vec{\bm{\sigma}}_{(\triangle)}=(\bm{\sigma}_{1(\triangle)},\bm{\sigma}_{2(\triangle)},...,\bm{\sigma}_{k(\triangle)})$ represent the set of spin variables and lattice displacements belonging to a given simplex $\triangle$. The summation $\sum_{\langle i,j\rangle\in\triangle}$ represents summation over nearest neighbor pairs within a simplex $\triangle$.

The free energy of the system is given by $-\beta F=\log Z$, where $\beta$ is the inverse temperature and $Z$ represents partition function defined as, $Z=\text{Tr}_{\bm{S}}\text{Tr}_{\bm{\sigma}}e^{-\beta H}$, where $\text{Tr}_{\bm{S}}$ and $\text{Tr}_{\bm{\sigma}}$ represent traces over the spin space of the spin $\bm{S}_{i}$ and the lattice displacement $\bm{\sigma}_{i}$ for all $i$ with spherical constraint,

In the absence of the coupling $\delta=0$, we have $J_{\sigma_{i},\sigma_{j}}\rightarrow J$ so that the system decouples into two independent subsystems (see Eq. (2), Eq. (3)). Within each of the subsystem the vectorial variables simplex network interact with each other anti-ferromagnetically. In the other representation of the hamiltonian Eq. (4) with Eq. (5) we find,

where we used $\sum_{i}|\bm{S}_{i}|^{2}=NM$ and $\sum_{i}|\bm{\sigma}_{i}|^{2}=Nc$. Thus the local Hamiltonian can be minimized by requiring

Furthermore, since we are considering tree-like network of simplices, the configurations which satisfy the local sum rules Eq. (10) in all simplices $\triangle$ are the ground states of the total system. Then a Maxwellian counting argument Reimers et al. (1991); Moessner and Chalker (1998) can be made for the ground state manifold. For the spin sector, we have $NM$ degrees of freedom (disregarding the small correction due to the spin normalization condition) while the sum rule Eq. (10) imposes $N_{\triangle}M=NM(c/k)$ conditions. Then $NM(1-c/k)$ degrees of freedom remain in the ground state manifold of spins. Similarly, $\alpha NM(1-c/k)$ degrees of freedom remain in the ground state manifold of of the lattice distortions. Thus if we have,

there is a macroscopic degeneracy of the ground states in the absence of spin-lattice coupling.

The microscopic background of the spin-lattice coupled model is explained in Mitsumoto et al. (2020). Let us summarize it here and add some further remarks. The lattice distortion is a dynamical Jahn-Teller distortion caused by the orbital degeneracy of the molybdenum ions Mitsumoto et al. (2022) and favors 2-$in$-2-$out$ patterns following the ice rule. The lattice distortion changes the angle between an oxygen ion O^2- and two magnetic ions Mo⁴⁺, modifying the superexchange interaction between spins by the Goodenough-Kanamori mechanism Goodenough (1955); Kanamori (1959); Solovyev (2003); Smerald and Jackeli (2019); Mitsumoto et al. (2020). Our numerical model on a pyrochlore lattice incorporates the above two features: lattice distortions favor the ice rule, and the interaction between classical Heisenberg spins is modified by the lattice distortion patterns as antiferromagnetic for $in$-$in$ and $in$-$out$ bonds and ferromagnetic for $out$-$out$ bond.

In Ref. Mitsumoto et al. (2020), the dynamical Jahn-Teller effect was modeled phenomenologically by a simple spin-ice Hamiltonian with only nearest-neighbor interactions between lattice distortions. This was motivated by the experiment Thygesen et al. (2017) which suggested 2-$in$-2-$out$ patterns of the lattice distortions. However in Ref. Mitsumoto et al. (2022), it was found based on a detailed microscopic analysis of the Jahn-Teller effect, that the lattice model should be extended to include also second and third nearest-neighbor interactions. It was found that the ground state of this extended lattice model obeys the 2-$in$-2-$out$ rule but it exhibits a specific periodically alternating pattern, i. e. a long-ranged order. However, quite surprisingly, numerical simulations of the extended model showed that the system avoids the ordering transition even under very slow cooling and remains in a supercooled, disordered ice state down to the lowest temperature. This result suggests that we can assume an ice-like liquid state for the lattice degrees of freedom. Thus for the sake of simplicity, we just assume the nearest neighbor interactions like the simple spin-ice Hamiltonian for the lattice distortions and do not include further neighbor interactions in the present mean-field model.

In this section, we introduce the replica method, derive the replicated free-energy as a functional of the order parameters and obtain the replica symmetric (RS) solution. The details of the derivation of the free-energy functional are given in Appendix. A.

The free energy of the system can be obtained formally as,

where $Z^{n}$ is the partition function of the replicated system $a=1,2,\ldots,n$,

Here $H_{n}$ is the Hamiltonian of the replicated system,

To investigate the glass transition, we introduce the glass order parameter matrices of spins and lattice distortions as overlaps between different replicas $a$ and $b$,

For convenience, we extend the matrices to include the diagonal elements, $Q_{aa}=q_{aa}=1$, to take into account the spherical constraints.

As explained in Appendix. A, the replicated partition function can be expressed in the dense limit $N\gg c\gg 1$ as,

where we defined,

with

Note that the partition function doesn’t depend on the simplex dimension $k$ thanks to the scaling factor $1/\sqrt{k-1}$ we introduced in the Hamiltonian. The resultant free energy functional becomes,

where $Q^{*}$ and $q^{*}$ are the saddle points which verify the saddle point equations,

The stability condition is also required, namely the eigenvalues of the $n(n-1)\times n(n-1)$ Hessian matrix,

where the components of the $\frac{n(n-1)}{2}\times\frac{n(n-1)}{2}$ block matrices $H_{QQ},H_{Qq}=H_{qQ}$ and $H_{qq}$ are defined as

with $a<b,c<d$, must be non-negative in the $n\rightarrow 0$ limit. The glass susceptibility, which is associated with the nonlinear magnetic or dielectric susceptibilities, is given by the inverse matrix of the Hessian matrix,

Hereafter, we call $\chi_{QQ},\chi_{qq}$ and $\chi_{Qq}$ as the spin glass susceptibility, the lattice glass susceptibility, the cross glass susceptibility, respectively.

In the absence of the spin-lattice coupling, i.e $\delta=0$, the free-energy (see Eq. (18)-Eq. (22)) become decoupled into the spin part and lattice part. The free energy associated with them is essentially the same as that of the spherical Sherrington-Kirkpatrick (SK) model Kosterlitz et al. (1976). It is known that the latter model has a low temperature phase where the global rotational symmetry is broken. However, the replica symmetry (RS) survives marginally in the sense that the stability of the RS solution (see sec III.3) is marginally stable.

Thus in the absence of the spin-lattice coupling, the spin and lattice degrees of freedom exhibit ’pseudo glass transitions’ accompanying breaking of the global rotational symmetry but without RSB. The marginally stable RS solutions imply that the free-energy landscapes in these low temperature phases are flat as shown schematically in Fig. 2 (a) and (b).

Now, our task is to solve the above problem with the simplest ansatz called the replica symmetric (RS) ansatz. In the RS ansatz we assume that the overlap matrices $Q_{ab}$ and $q_{ab}$ are parametrized as,

where $\delta_{ab}$ is the Kronecker’s delta. Then the free energy within RS ansatz is obtained as,

The saddle point equations with respect to the order parameters $Q$ and $q$ are obtained as,

We see that $Q=q=0$ is always a solution which represents the high temperature disordered (paramagnetic or liquid) state. Below the glass transition temperatures (See Appendix. B) this solution becomes unstable.

We show in Fig. 3 the typical phase diagram, which is derived from the RS saddle point equations Eq. (29) and Eq. (30). In the absence of the spin-lattice coupling $\delta=0$, we find the spin and lattice degrees of freedom exhibit ’pseudo glass transitions’ without RSB as discussed already in sec. IV.2. We show the decoupled phase diagrams in panels (a) and (b). In the presence of the spin-lattice coupling $\delta>0$ we find a new phase which we call as ’spin-lattice glass phase’ which accompanies RSB. As shown in panels (c),(d),(e), we always find the spin-lattice glass phase at low enough temperatures as long as $\delta>0$. In rigid regime $\epsilon/J>1$, the spin-lattice glass phase emerges within the pseudo lattice glass. On the other hand, direct transition from the high temperature phase (paramagnetic and lattice liquid phase) to the spin-lattice glass phase happens if the rigidity is low $\epsilon/J<1$.

In the absence of the spin-lattice coupling, i.e $\delta=0$, the free-energy Eq. (28) as well as the saddle point equations Eq. (30) become decoupled into the spin part and lattice part. As noted in sec IV.2, the two decoupled systems are essentially the same as the spherical SK model Kosterlitz et al. (1976).

The pseudo lattice glass transition occurs at,

and below $T^{*}$, $q>0$ solution is obtained as,

where

is a dimensionless temperature that measures the distance of the temperature to the pseudo lattice glass transition temperature $T^{*}$. From the analysis of the Hessian matrix Eq. (25), we find that the $q>0$ RS solution is marginally stable, i.e., the minimum eigenvalue of the Hessian matrix is strictly zero in the pseudo lattice glass phase. Correspondingly, the glass susceptibility of the lattice distortion $\chi_{qq}$, given in Eq. (5), diverges approaching $T^{*}$ from above with the power law $|t^{*}|^{-1}$, and remain divergent within the lattice glass phase. The detailed analysis of the Hessian matrix and the glass susceptibility are presented in Appendix B and Appendix. C, respectively.

The same analysis can be repeated for the spin degrees of freedom and one easily obtain the phase diagrams as shown in Fig. 3 (a)(b). In the following we turn to the cases of finite spin-lattice coupling $\delta>0$.

Let us first discuss the case of high rigidity, i. e. the lattice-lattice coupling is strong $\epsilon/J>1$. Lowering the temperature, we first pass over the pseudo lattice glass transition temperature $T^{*}(\epsilon)$ discussed above, which does not depend on the strength of the spin-lattice coupling $\delta$. The lattice order parameter $q$ linearly grows lowering the temperature below $T^{*}$ while the spin glass order parameter $Q$ still remains zero at high enough temperatures, as shown in Fig. 4 (a). As shown in Fig. 5 (a), $\chi_{qq}$ diverges passing $T^{*}$ while $\chi_{QQ}$ and $\chi_{Qq}$ do not exhibit any singularity as expected. We also show in the figure the behaviour of the order parameters below $T_{\rm c}$ using the RS solution in broken lines. Note that the RS solution is unstable below $T_{\rm c}$ due to RSB.

By lowering the temperature further, the spin glass transition occurs at,

below which $Q>0$ solution exist. It can be seen that the phase boundary given by Eq. (34) is shifted to higher temperatures by increasing $\delta$, and it converges to $T_{\rm c}=T^{*}=\epsilon$ in the limit $\delta\rightarrow\infty$, i.e., the pseudo lattice glass phase disappears in this limit.

Below the spin glass transition temperature $T_{\rm c}$, the solutions of the order parameters are obtained from the saddle point equations Eq. (29) and Eq. (30) which can be written as,

The equations can be solved numerically as shown in Fig. 4 (a). Just below $T_{\rm c}$, the spin glass order parameter continuously increases as

where $\beta_{c}=1/T_{\rm c}$ and

measures the distance of the temperature to the spin glass transition temperature $T_{\rm c}$. On the other hand, the lattice glass order parameter is also modified reflecting the spin glass transition,

Interestingly, the minimum eigenvalue of the Hessian matrix becomes negative below $T_{\rm c}$,

This means replica symmetry breaking must occur below $T_{\rm c}$. Most probably full RSB occurs. Note that the eigenvalue is quadratically proportional to the strength of spin-lattice coupling $\delta$. Thus, spin-lattice coupling is essential for this model to exhibit the RSB.

Approaching $T_{\rm c}$ from above, the spin glass susceptibility $\chi_{QQ}$ diverges with power law $|t|^{-1}$. Below $T_{\rm c}$, the spin glass susceptibility $\chi_{QQ}$ as well as the lattice glass susceptibility decay with $|t|^{-1}$ within the RS ansatz. The cross glass susceptibility $\chi_{Qq}$ doesn’t grow above $T_{\rm c}$, but it suddenly diverges at $T_{\rm c}$ and decays with $|t|^{-1}$ below $T_{\rm c}$.

Fig.6 (a) and (c) shows the internal energy and the heat capacity for $\epsilon/J=2,\alpha=1$ and $\delta=1.5$ obtained by the RS solution (see Appendix. D). Note that the heat capacity exhibits a kink at $T^{*}$ and a cusp at $T_{\rm c}$.

If the rigidity is low $\epsilon/J<1$, the situation changes: a simultaneous spin and lattice glass transition takes place at a common transition temperature,

which doesn’t depend on the elastic energy scale $\epsilon$ nor the amplitude $\delta$ of the spin-lattice coupling. The latter implies that the spin degrees of freedom play the dominant role in this case.

Below $T_{\rm c}^{\prime}$, the non-zero order parameters $q,Q>0$ are obtained by solving Eqs. (35) and (36) numerically as shown in Fig. 4 (b). We find continuous growth of the spin and lattice glass order parameters passing $T_{\rm c}^{\prime}$ lowering the temperature. Note that both of the glass order parameters exhibit jumps slightly below $T_{\rm c}^{\prime}$. Using the dimensionless temperature,

which measures the distance of the temperature to $T_{\rm c}^{\prime}$, the order parameters $Q$ and $q$ slightly below $T_{\rm c}^{\prime}$ can be obtained as,

Comparing the two order parameters, we notice that the growth of the lattice glass order parameter $q$ is weaker than that of the spin glass order parameter $Q$. More precisely, the critical exponents are different: $q$ is proportional to $t^{2}$ while $Q$ is proportional to $t$. If we switch off the spin-lattice coupling, $\delta\rightarrow 0$, the lattice glass order parameter $q$ becomes zero, and the behavior of the spin glass order parameter $Q$ agrees with the spherical SK model Kosterlitz et al. (1976), i.e., $Q=t^{\prime}$. The spin glass order parameter $Q$ is weakly modified by the lattice glass order parameter $q$ at order $O(t^{\prime 2})$.

From the analysis of the Hessian matrix, we find that the RS solution becomes unstable below $T_{\rm c}^{\prime}$. Slightly below $T_{\rm c}^{\prime}$, the minimum eigenvalue behaves as,

which also becomes zero if $\delta=0$. The spin glass susceptibility $\chi_{QQ}$ exhibits a diverging feature approaching $T_{\rm c}$ from above with power law $|t^{\prime}|^{-1}$ while it decays as $|t|^{-2}$ below $T_{\rm c}$, reflecting $\lambda_{\rm min}$, as shown in Fig. 5 (b). In sharp contrast, the lattice glass susceptibility does not diverge at $T_{\rm c}$. Similarly to the case of the separated glass transition $\epsilon/J<1$, the cross glass susceptibility $\chi_{Qq}$ does not grow above $T_{\rm c}$, suddenly diverges at $T_{\rm c}$ and decays as $|t^{\prime}|^{-1}$ below $T_{\rm c}^{\prime}$.

Fig.6 (b) and (d) shows the internal energy and the heat capacity for $\epsilon/J=0.5,\alpha=1$ and $\delta=1.5$ using the RS solution. Reflecting the jumps of the order parameters shown in Fig. 4 (b), the energy exhibits a non-monotonic behavior, and the heat capacity diverges negatively. We believe that the latter unphysical behavior is due to the failure of the RS solution below $T_{\rm c}^{\prime}$.