Randomness-induced quantum spin liquid behavior in the $s=$1/2 bond-random Heisenberg antiferromagnet on the pyrochlore lattice

TL;DR

This study demonstrates that bond randomness in a $s=1/2$ Heisenberg antiferromagnet on the pyrochlore lattice induces a gapless quantum spin liquid state, providing insights into experimental observations of such states.

Contribution

It reveals that randomness can induce a gapless quantum spin liquid in the pyrochlore lattice, a novel mechanism for QSL formation in this system.

Findings

Randomness induces a gapless quantum spin liquid state.

The state is identified as a random-singlet state.

Implications for experimental materials like Lu$_2$Mo$_2$O$_5$N$_2$.

Abstract

We investigate the zero- and finite-temperature properties of the bond-random $s=1/2$ Heisenberg antiferromagnet on the pyrochlore lattice by the exact diagonalization and the Hams--de Raedt methods. We find that the randomness induces the gapless quantum spin liquid (QSL) state, the random-singlet state. Implications to recent experiments on the mixed-anion pyrochlore-lattice antiferromagnet Lu$_2$Mo$_2$O$_5$N$_2$ exhibiting gapless QSL behaviors are discussed.

Equations

$$\mathcal{H}=J\sum _{⟨i,j⟩}{j}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j,$$

$${\bar{q}}^2=\frac{1}{N^2}\sum _{i,j}{\left[{⟨{\mathbf{S}}_i\cdot {\mathbf{S}}_j⟩}^2\right]}_J,$$

$$S_{\mathbf{q}}(\omega )$$

Full text

One promising scenario might be that the randomness or inhomogeneity, either of extrinsic or intrinsic origin, induces the gapless QSL-like state Watanabe ; Kawamura ; Shimokawa ; Uematsu ; Uematsu2 . The present authors and collaborators have demonstrated that such a randomness-induced gapless QSL-like state, called the “random-singlet state,” is stabilized for various 2D frustrated $s=1/2$ Heisenberg models, including the triangular Watanabe ; Shimokawa , kagome Kawamura ; Shimokawa , $J_{1}-J_{2}$ square Uematsu2 and $J_{1}-J_{2}$ honeycomb Uematsu magnets, as long as the randomness is moderately strong. The origin of such (effective) randomness could be of variety, e.g., the intrinsic ones like the dynamical freezing of the charge (dielectric) degrees of freedom in case of $\kappa$-ET and dmit salts and the slowing down of the proton motion in case of Cat salt, or the extrinsic ones like the possible Jahn-Teller distortion accompanied by the the random substitution of Zn2+ by Cu2+ in case of herbertsmithite and the random occupation of Te/W, in case of Sr2Cu(Te1-xWx)O6. The role of randomness on the possible QSL-like behavior was also studied recently by other groups for various 2D Heisenberg models Kimchi ; Kimchi2 ; Sheng ; Guo .

The random-singlet state may roughly be pictured as the random covering of singlet-dimers to minimize the total energy for a given random distribution of the exchange couplings $\{J_{ij}\}$, as schematically illustrated in the inset of Fig. 1(c). Such dimer covering on the random lattice, however, is highly nontrivial, leading to not only nearest-neighbor (NN) singlets but also further-neighbor singlets (represented by red dimers with their strength denoted by their thickness), to nearly-free spins failed to form singlets, the so-called “orphan spins” (represented by arrows), and even to resonances between distinct singlet-dimers (represented by blue dimers). Low-energy excitations of the state would be (i) local singlet-to-triplet excitations of weakly-bound singlets, (ii) recombinations of nearly-degenerate singlet-dimer coverings, and (iii) fluctuations of orphan spins, etc. Reflecting the spatially random character of the state, there would be no characteristic energy scale in these excitations, as postulated in a phenomenological theory of Ref. AndHalVar . Such an analogy yields the $T$-linear low-$T$ specific heat for the random-singlet state Uematsu2 .

The next question to be addressed might be whether the QSL state is ever possible in 3D, whatever its physical origin. Generally, 3D magnets tend to be subject to less fluctuations and to stronger ordering tendency than in 2D even on frustrated lattices. In this connection, the pyrochlore lattice, known to be a highly frustrated 3D lattice, might provide a promising stage.

An intensively studied example might be Tb2Ti2O7 possessing a modest amount of Ising-like $\langle 111\rangle$ anisotropy Gardner ; Fennell ; Lhotel ; Fritsch ; Takatsu . This material has widely been regarded as a quantum analog of the classical spin ice, i.e., quantum spin ice Gardner ; Fennell ; Takatsu . Meanwhile, it turns out that even a minute amount of off-stoichiometry induces a sharp specific-heat peak signaling a thermodynamic phase transition Takatsu . Furthermore, the onset of the glass-like spin freezing was observed in samples both with and without a sharp specific-heat peak Lhotel ; Fritsch . In any case, the underlying physics of Tb2Ti2O7 would be that of the anisotropic spin ice.

One may then ask what is the situation in more isotropic Heisenberg-like pyrochlore magnets. Recently, an interesting Heisenberg-like pyrochlore magnet exhibiting the QSL behavior down to the low temperature of 0.5K was reported in the mixed-anion antiferromagnet Lu2Mo2O5N2 Pattfield , where the magnetism is borne by $s=1/2$ spins of Mo5+ 4$d^{1}$, in contrast to its precursor oxide Lu2Mo2O7 where the magnetism is borne by $s=1$ spins of Mo4+ 4$d^{2}$. The oxynitride Lu2Mo2O5N2 was observed to exhibit gapless QSL behaviors characterized by the $T$-linear specific heat and the broad dynamical structure factor, in contrast to the oxide Lu2Mo2O7 which exhibits a spin-glass (SG) freezing and the low-$T$ specific heat proportional to $T^{2}$. An interesting observation is that the oxynitride inevitably contains a significant amount of quenched disorder derived from the random occupation of O2-/N3- anions. Hence, somewhat counter-intuitively, the introduction of randomness into the nominally disorder-free SG state apparently induces the QSL state Pattfield . Lu2Mo2O5N2 was recently investigated theoretically by means of the density functional theory and the pseudofermion functional renormalization group method, suggesting the importance of the third-neighbor interaction $J_{3}$ in stabilizing the QSL state, while the analysis was basically that of the homogeneous system and the effect of O/N randomness was not considered explicitly Iqbal .

Under such circumstances, we wish to investigate in the present Letter the role of randomness in the quantum Heisenberg antiferromagnet on the 3D pyrochlore lattice. We consider the bond-random $s=1/2$ isotropic Heisenberg model on the pyrochlore lattice with the AF NN interaction. The Hamiltonian is given by

[TABLE]

where ${\bm{S}}_{i}=(S_{i}^{x},S_{i}^{y},S_{i}^{z})$ is an $s=1/2$ spin operator at the $i$-th site, and the sum $\Braket{i,j}$ is taken over all NN pairs on the lattice, while $j_{ij}\geq 0$ is the random variable obeying the bond-independent uniform distribution between $[1-\Delta,1+\Delta]$ with $0\leq\Delta\leq 1$. We consider the NN interaction only just for simplicity, and put $J=1$ as the energy unit. The parameter $\Delta$ represents the extent of the randomness: $\Delta=0$ corresponds to the regular case and $\Delta=1$ to the maximally random case when the interaction is to be kept AF.

The corresponding regular model with $\Delta=0$ has been studied extensively by various numerical methods. There is a consensus that the system remains disordered even at $T=0$ without any spin LRO, while the nature of the nonmagnetic ground state still remains unclear Chandra . Whether the ground state is gapped or gapless also remains controversial, though the bulk of the numerical calculations seem to suggest a nonzero spin gap. By contrast, there has been no systematic numerical study of the corresponding random model with $\Delta>0$.

We then study the ground-state properties of the model by means of the exact diagonalization (ED) Lanczos method. We treat finite-size clusters with the total number of spins $N$ up to $N\leq 36$ ($N$ is taken to be a multiple of 4 with $8\leq N\leq 36$), periodic boundary conditions applied in all directions. The clusters of $N=16$ and $N=32$ possess the cubic symmetry of the bulk pyrochlore lattice. The numbers of independent bond realizations (samples) $N_{s}$ used in the sample average are for the order parameter, the spin gap, and the static spin structure factor $N_{s}=100$ and 5 for $N=8$–$32$ and 36, respectively, whereas for the dynamical spin structure factor $N_{s}=100$ and 50 for $N=16$ and 32, respectively. Error bars are estimated from sample-to-sample fluctuations. The finite-temperature properties are computed by the Hams–de Raedt method HamsRaedt ; supple . The computation is performed for the size $N=32$, where the averaging is made over 10 initial vectors and 10 independent bond realizations. Error bars of physical quantities are estimated from the scattering over both samples and initial states by using the bootstrap method Bootstrap .

We first investigate the existence or nonexistence of the magnetic LRO by computing the spin freezing parameter $\bar{q}$ defined by

[TABLE]

where $[\cdots]_{J}$ denotes the average over the disorder or samples SGreview . This quantity can detect the static spin order of any type, even including the random one like the SG order. The computed $\bar{q}$ is plotted versus $N^{-1/3}$ in Fig. 1(a) for various values of randomness $\Delta$, in which the spin-wave form $\bar{q}_{N}\approx\bar{q}_{\infty}+cN^{-1/3}$ is borne in mind. As can be seen from the figure, $\bar{q}$ is extrapolated to zero, indicating the absence of spin LRO for any $\Delta$, even including the SG one.

In Fig. 1(b), we show the size dependence of the spin-gap energy $\Delta E$. For smaller $\Delta<\Delta_{c}\simeq 0.6$, $\Delta E$ tends to be extrapolated to a nonzero value suggestive of a gapped behavior, whereas, for larger $\Delta>\Delta_{c}$, it is extrapolated to zero within the error bar suggestive of a gapless behavior. The changeover observed between the gapped and gapless behaviors suggests the occurrence of a randomness-induced phase transition between the two distinct types of nonmagnetic states. The randomness-induced gapless nonmagnetic phase stabilized at $\Delta>\Delta_{c}$ is likely to be the random-singlet state as identified in Refs. Watanabe ; Kawamura ; Shimokawa ; Uematsu ; Uematsu2 . Although the precise value of $\Delta_{c}$ remains uncertain, we tentatively quote $\Delta_{c}=0.6^{+0.1}_{-0.3}$.

Such a transition can also be detected via the quantity $R$. All the samples we have studied possess either singlet or triplet ground states, and $R$ is the ratio of the number of samples with triplet ground states expected to be related to the fraction of orphan spins accompanied by the divergent-like low-temperature susceptibility.

As can be seen from Fig. 1(c), $R$ vanishes for $\Delta<\Delta_{c}\sim 0.6$, while it grows taking nonzero values beyond $\Delta_{c}$. This observation is consistent with a gapped-gapless transition observed in Fig. 1(b), which strengthens our conclusion of a phase transition occurring between the randomness-irrelevant gapped QSL state and the randomness-relevant gapless QSL state.

In order to probe the properties of magnetic excitations, we also compute the dynamical structure factor $S_{\bm{q}}(\omega)$ given by Gagliano ; Shimokawa

[TABLE]

where $E_{0}$ is the ground-state energy, and $\eta$ is a phenomenological damping factor taking a sufficiently small positive value. We employ the continued fraction method to compute $S_{\bm{q}}(\omega)$ Gagliano , putting $\eta=0.02$. The $\omega$-dependence of the computed $S_{\bm{q}}(\omega)$ in the random-singlet state is shown for the case of the maximal randomness of $\Delta=1$ in Fig. 2 at the (2,0,0) $\bm{q}$-point (given in units of $2\pi/a$ where $a$ is the linear size of the cubic unit cell), at which the corresponding static spin structure factor $S_{\bm{q}}$ has relatively high intensity, in comparison with the corresponding data for the regular model of $\Delta=0$. The information of $S_{\bm{q}}$ exhibiting broad features in the ${\bm{q}}$-space is given in Fig. S1 of Supplemental Material supple . As can be seen from Fig. 2, the computed $S_{\bm{q}}(\omega)$ exhibits broad features in $\omega$ without any clear peak, accompanied by a long tail extending to larger $\omega$, indicating the absence of characteristic energy scale. We note that essentially similar $\omega$-dependence is observed for other ${\bm{q}}$-points as well. Such a feature is also common to the random-singlet states in 2D Kawamura ; Shimokawa ; Uematsu ; Uematsu2 . In the small-$\omega$ region, $S_{\bm{q}}(\omega)$ diminishes somewhat toward $\omega\rightarrow 0$, though it still stays gapless as can be confirmed from its system-size dependence, in sharp contrast to the gapped behavior observed for $\Delta=0$.

We next move to the finite-$T$ properties. In Fig. 3, we show the temperature dependence of (a) the specific heat per spin, and of (b) the susceptibility per spin. As can be seen from Fig. 3(a), while the specific heat exhibits a double-peak structure for vanishing and weaker randomness $\Delta$, such a structure is gone in the random-singlet state at $\Delta\gtrsim 0.6$. We note that the double-peak structure generally means bunch of low-energy excitations with two distinct energy scales, which, however, is not the case in the random-singlet state. For $\Delta\gtrsim 0.6$, as generically observed in the random-singlet states in 2D Watanabe ; Kawamura ; Uematsu ; Uematsu2 , it exhibits a $T$-linear behavior at lower temperatures, $C\simeq\gamma T$, probably originated from the type of low-energy excitations postulated in Uematsu2 ; AndHalVar . Such a change of behavior is also consistent with the occurrence of a phase transition within the nonmagnetic state at $\Delta=\Delta_{c}$ argued above. As can be seen from Fig. 3(b), the susceptibility exhibits a nearly $T$-independent behavior insensitive to $\Delta$ for $0.2\lesssim T\lesssim 2$, while in the region of the random-singlet state of $\Delta\geq\Delta_{c}$ exhibits a gapless behavior with a Curie-like tail at still lower $T$, suggesting the existence of orphan spins Watanabe ; Kawamura ; Uematsu ; Uematsu2 .

One sees from our present results that the properties of the randomness-induced QSL state, the random-singlet state, of the 3D pyrochlore Heisenberg model is rather similar to those of the 2D models Watanabe ; Kawamura ; Shimokawa ; Uematsu ; Uematsu2 . This observation may suggest that the nature of the random-singlet state is insensitive not only to the details of the lattice structure and the origin of frustration Watanabe ; Kawamura ; Shimokawa ; Uematsu ; Uematsu2 , but even to whether the spatial dimensionality is either two or three. In other words, the random-singlet state seems to be a universal state of magnets. Of course, system size studied here are very small, and care needs to be taken in reaching a definitive conclusion.

While the random-singlet features are expected to be most eminent for stronger randomness, being clearly visible even for smaller systems and at higher temperatures, they might be realized also for somewhat weaker randomness, manifesting themselves only at longer length scales and at lower temperatures not directly accessible by the present ED calculation (recall the large uncertainty in our estimate of $\Delta_{c}$), even with possible experimental relevance expected Kimchi ; Kimchi2 . Such a random-singlet-like state for weaker randomness is an adiabatic continuation of the one for stronger randomness so long as it remains gapless, and is essentially the same state as that for stronger randomness.

We now wish to discuss experimental implications. Our results are compared favorably with the experimental results on Lu2Mo2O5N2 Pattfield , at least qualitatively, i.e., the $T$-linear specific heat, the gapless susceptibility accompanied by an intrinsic Curie-like tail, and the broad spectrum of $S_{\bm{q}}(\omega)$. Quantitatively, however, some deviation remains. If we try to estimate the coefficient of the $T$-linear specific heat with the $J$-value deduced from the experimental Curie-Weiss temperature, we get $\gamma\sim 80$ mJ/molK2 from our result, which deviates considerably from the experimental value $\gamma\sim 10$ mJ/molK2 Pattfield . One possible cause might be our oversimplified assumption of the randomness. The other possibility is that further neighbor interactions Iqbal , especially $J_{3}$, might affect the underlying energetics at the quantitative level. Unfortunately, the system size accessible by the ED method is too small to take account of such $J_{3}$-effect in a meaningful manner. Anyway, the random-singlet state is stabilized as long as the system possesses a certain amount of randomness, frustration and quantum fluctuations, staying quite robust at the qualitative level against other details of the system.

In summary, we studied both the $T=0$ and $T>0$ properties of the bond-random $s=1/2$ NN Heisenberg model on the 3D pyrochlore lattice, and found that the randomness-induced gapless QSL state, the random-singlet state, is stabilized if the strength of the randomness exceeds a critical value. Its properties turn out to be rather similar to those of the frustrated random Heisenberg magnets in 2D, highlighting the possible universal character of the random-singlet state. Further studies are desirable to fully substantiate such a picture.