Quantum half-orphans in kagomé antiferromagnets

The low-temperature physics of frustrated quantum magnets is controlled by the interplay between the geometric frustration, quantum fluctuations, entropic effects, and quenched disorder. In many interesting cases Savary and Balents (2016), the geometric frustration renders the leading exchange interactions unable to drive magnetic ordering even at temperatures much lower than the scale of these interactions. Instead, in a semiclassical picture, they confine the system to a manifold of minimally frustrated configurations, which can admit a description in terms of emergent degrees of freedom as in the case of classical spin ice Castelnovo et al. (2008). The low temperature physics is then controlled by how these emergent degrees of freedom behave in the presence of subdominant interactions as well as quantum and thermal fluctuations. In the extreme quantum limit, frustrated $S=1/2$ magnets can provide an arena for the physics of quantum number fractionalization; the Majorana fermion excitations of Kitaev’s honeycomb model being a celebrated example Kitaev (2006).

Interestingly, quenched disorder, i.e. non-magnetic substitutional impurities or lattice imperfections that affect the bond strengths in their vicinity, can provide a powerful probe of these emergent excitations and their unusual quantum numbers. For instance, the emergent magnetic monopole excitations of classical spin ice can be nucleated in the ground state of a disordered sample. Similarly, vacancies in Kitaev’s honeycomb model nucleate a Majorana fermion-Z₂ flux complex in their vicinity. More generally, a wide variety of features can be probed or created by non-magnetic impurities in magnetic systems, depending on the nature of the underlying state. If the clean system is a spin liquid, then non-magnetic impurities can lead to spinon deconfinement Poilblanc et al. (2006) or even spin-charge separation if they are mobile Läuchli and Poilblanc (2004). More conventional ordering patterns can also be characterized by the response to non-magnetic impurities Alloul et al. (2009). An enhancement of local correlations is often observed near spinless impurities in antiferromagnets Martins et al. (1997). In valence bond crystals, that break lattice symmetries, or valence bond solids that are non-degenerate, non-magnetic impurities generically induce a localized spin texture around them Poilblanc et al. (2006). In systems with competing phases at low temperatures, non-magnetic impurities can furthermore reveal or seed competing orders Kaul et al. (2008). Quantum critical systems have also been shown Sachdev and Vojta (2003); Höglund et al. (2007) to host responses characteristic of their criticality to the inclusion of spinless defects. All these different physical behaviors and the corresponding local textures or responses have been detected in magnetic compounds using local probe techniques, such as nuclear magnetic resonance (NMR). Let us for instance mention the spin “polaron” observed in the two-dimensional doped material, SrCu₂(BO₃)₂ material Haravifard et al. (2006); Yoshida et al. (2015), the imaging of spin-1/2 edge excitations in spin 1 Haldane compound Y₂BaNi_1-xMg_xO₅ Tedoldi et al. (1999), the magnetic response around an impurity observed in a $S=1/2$ kagomé compound (herbertsmithite) Olariu et al. (2008) and the localized bound states generated by impurities in topological magnets Yin et al. (2020).

Coming back to frustrated magnets, the SrCr_9pGa_12-9pO₁₉ (SCGO) compound, in which Cr³⁺ $S=3/2$ moments lie on the vertices of the pyrochlore-slab (bilayer kagomé) lattice, provides another striking example of the interplay between impurities and the underlying physical state Obradors et al. (1988); Ramirez et al. (1990); Limot et al. (2002). Even in the best samples of SCGO, nonmagnetic Ga atoms disrupt the corner-sharing network of triangles and tetrahedra formed by the Cr ions; these Ga impurities can be modeled as static vacancies in the Heisenberg antiferromagnet on this lattice. At a classical level, the Heisenberg model on the pyrochlore slab lattice remains disordered down to $T=0$, providing an interesting example of a classical spin liquid with full SU(2) symmetry of interactions Moessner and Berlinsky (1999); Moessner and Chalker (1998).

Following an earlier phenomenological analysis Schiffer and Daruka (1997) of experimental results on SCGO, Henley Henley (2001) described the $T=0$ liquid state in terms of an emergent fluctuating polarization field and noted that a triangle with two vacancies led to a lone or “orphan” spin on that triangle which behaves as a source (charge) for this polarization field. The polarization field of this charge leads, in this description, to an oscillating spin texture with a power-law envelope. Even without a detailed computation of this texture, this effective theory predicts that the classical spin-$S$ Heisenberg model on the pyrochlore slab lattice will have a net spin polarization of $S/2$ in the direction of an infinitesimal external magnetic field at $T=0$.

Motivated by this factor of two, which is suggestive of spin fractionalization, Henley dubbed this combination of the lone spin and the resulting spin-texture a “half-orphan”, modifying the terminology of “orphan spins” introduced in the earlier phenomenological studies Schiffer and Daruka (1997). In closely related work Moessner and Berlinsky (1999) that studied the classical Heisenberg antiferromagnet on the pyrochlore and pyrochlore-slab lattices, Moessner and Berlinsky also recognized the special role of these half-orphan degrees of freedom in dominating the low temperature magnetic response, and used this insight to model the experimental results on SCGO within a single-unit approximation that used as input properties of individual simplices (triangles and tetrahedra) with various degrees of dilution.

This striking $T=0$ prediction of a saturation magnetization of $S/2$ for the system with two vacancies on the same triangle led immediately to the questions: Does the low-temperature susceptibility show signatures of this “spin fractionalization” ? In other words, does the impurity susceptibility of this orphan-texture complex correspond to that of a classical spin $S/2$ object? If yes, how is this response resolved spatially ? Motivated by these questions, Ref. Sen et al., 2011 used a hybrid large-N field-theory as well as direct Monte-Carlo simulations to study the low but nonzero temperature behaviour. Perhaps surprisingly, the answer turns out to be in the affirmative, with an interesting spatial structure that will be important in our present study: At low temperatures $T$ in the spin-liquid regime, each lone spin on a triangle with two vacancies “sees” an effective magnetic field that has magnitude $h/2$ (where $h$ is the applied external field) and responds to it as a free spin $S$ in this field at temperature $T$. Exactly half of this paramagnetic response is cancelled off by the net diamagnetic response of the surrounding spin texture for which this orphan spin serves as a source, giving rise to a net susceptibility that equals the susceptibility of a spin $S/2$ at temperature $T$.

In conjunction with subsequent work Sen et al. (2012) that also characterized the entropic interactions between these half-orphan degrees of freedom, this approach provides a fairly detailed picture of both the orphan-texture complex and its susceptibility, as well as interactions between orphans, including a theory for low temperature glassy behaviour in the multi-orphan case. Importantly, it provides a reasonably satisfactory fit to NMR data on SCGO Limot et al. (2002); Bono et al. (2004), that could not previously be accounted for by more conventional ideas. We provide more details on orphan physics in the classical case in the Appendix for completeness, along with classical Monte Carlo simulations for kagomé systems, as starting from now we will focus on kagomé planes (and not bilayers).

While one expects that the classical physics of half-orphans would be reflected in some form in a semiclassical treatment of the corresponding quantum magnet with large but finite spin $S$, this expectation has thus far not been shored up by an actual controlled calculation of $1/S$ corrections to the classical picture. Nor has the fairly impressive success of classical descriptions of $S=3/2$ frustrated magnets such as SCGO been quantitatively examined from this point of view.

Spin $S=1/2$ kagomé magnets such as Herbertsmithite Mendels and Bert (2011); Khuntia et al. (2020) also feature some degree of dilution by nonmagnetic vacancies. Since it is well known that the classical kagomé magnet has a fairly broad low-temperature spin liquid regime, extending upwards from a lower cutoff of $T^{*}\sim 10^{-3}JS^{2}$ (that marks a subtle crossover to an ordered regime below this temperature Chern and Moessner (2013); Zhitomirsky (2008); Huse and Rutenberg (1992)), one expects that half-orphans seeded by vacancy-pairs should be fairly well-defined at the classical level in this regime. These observations provide the central motivation for the present study, in which we ask: Do these emergent half-orphan degrees of freedom nucleated by pairs of vacancies on the same triangle survive quantum effects in the Heisenberg antiferromagnet on the kagomé lattice?

The presence of nonmagnetic impurities in some of the best-known experimental realizations of frustrated quantum magnets on the kagomé lattice has motivated several studies of vacancy effects, which we build on here with our work. For instance, a single non-magnetic impurity is known to induce local dimer order in the spin-1/2 case Dommange et al. (2003) and does not generate a localized spin in the spin-3/2 case either Läuchli et al. (2007), while a finite concentration could lead to valence bond glass in the spin-1/2 case Singh (2010). Using a combination of series expansion and variational wavefunction studies, Gregor and Motrunich Gregor and Motrunich (2008) have also modeled inhomogeneous Knight shifts in compounds such as Herbertsmithite Mendels and Bert (2011) by studying the response of the $S=1/2$ kagomé antiferromagnet to nonmagnetic impurities. Again, there is no evidence that single vacancies induce the kind of dramatic enhancement that pairs of vacancies do in the classical picture.

In this manuscript, we study the finite-temperature as well as ground-state signatures of orphan physics in the spin-$S$ quantum Heisenberg model on the kagomé lattice for a large set of spin values ranging from $S=1/2$ up to $S=6$, allowing us to discuss the quantum to classical crossover. One striking result of our study is that the ground state of the quantum system with two vacancies on the same triangle has nonzero spin quantum number $S_{GS}$. This tracks $[S/2]$, which is the total quantum spin value (allowed to exist in the system) which is the closest to $S/2$, i.e. we find $S_{GS}\simeq[S/2]$ down to $S=2$. This provides a dramatic quantum manifestation of half-orphan physics at $T=0$. In some way, this result is reminiscent of the $S=1/2$ end-spins in cut $S=1$ Haldane gap chains, which provide the clearest and experimentally relevant signature of the underlying symmetry-protected topological order in these systems. However, this is qualitatively different: The presence of quantum half-orphans in the ground state of samples with two vacancies on the same triangle is blind to the integer or half-integer nature of elementary spin $S$, and is perhaps best thought of as a semiclassical effect that survives to surprisingly low values of $S$. Indeed, there is no obvious underlying topological order at the heart of this effect.

In our finite temperature studies, we find that the local susceptibility offers a very clear signature of orphan physics down to $S=2$. We attribute this to presence of quantum half-orphans in the ground state of these systems. From other probes such as the magnetization curve, we find that orphan physics can be detected to even lower values of quantum spin (down to $S=1/2$), with specific signatures not present for other types of impurity patterns. We also present an analysis of the histogram of spin magnetization due to a distribution of impurities, akin to a NMR Knight shift experiment, specific to the experimentally relevant $S=1/2$ situation. Our results are obtained with a finite-temperature method working in Krylov bases generated by an appropriate number of randomly distributed initial states (see details below), as well as Lanczos exact diagonalization (ED) and Density Matrix Renormalization Group (DMRG) for the study of ground-state properties.

The layout of this manuscript is as follows: In Sec. II we first study the $T=0$ ground state and present our results on the ground-state polarization, as well as spin textures obtained through ED and DMRG computations. We begin Section III with a brief description of the finite-temperature random sampling technique used to simulate thermodynamics of the quantum system, and then present finite-temperature results for orphan physics, including the local susceptibility response, the magnetization curve, as well as experimental predictions for the NMR Knight shift. Finally, Sec. IV summarizes our results along with a discussion of some outstanding issues. For completeness, in the Appendix, we also present the results of finite-temperature Monte Carlo simulations of vacancy effects in the classical Heisenberg model on the kagomé lattice, and identify the temperature regime in which orphan physics is clearly visible in the classical case.

We begin with perhaps the simplest question that gets to the heart of the intriguing conundrum posed by Henley, Moessner and Berlinksy’s classical $T=0$ argument for a saturation magnetization of $S/2$ that reflects the presence of a half-orphan nucleated by two vacancies on a triangle: What is the ground state spin quantum number of a spin-$S$ quantum antiferromagnet on a kagomé lattice with two vacancies on one triangle of the lattice?

To answer this question for the kagomé magnet with Hamiltonian

we first compute the total spin of the ground-state for some of the lattices shown in Fig. 1. This is done using Lanczos diagonalization for kagomé samples with an orphan spin, up to $S=6$. This is represented in Fig. 2, where we find that starting from $S\geq 2$, the ground-state is no longer a singlet. We compare the total spin $S_{\rm GS}$ of the ground state to $S/2$ (dashed line in Fig. 2) and find an overall trend towards this classical prediction. In particular, the ground state spin $S_{\rm GS}$ tracks in most cases the (half-)integer closest to $S/2$ which is allowed in the sample. The results in Fig. 2 are mostly for the $12$-sites samples with a pair impurity, but we find the same for the few larger clusters which we are able to simulate. In particular, we find $S_{\rm GS}=1$ for spin-2 lattices of sizes 15 and 18 with an orphan impurity. We were able to check also a combination of spin and lattice size where the global total spin is half integer $\in\{1/2,3/2,5/2,...\}$. Note for instance the case of $S=7/2$ spins on a $15$-sites sample with a pair impurity (13 sites effective) where we find that $S_{GS}=3/2$, which is indeed the allowed value closest to the classical prediction.

These results should be contrasted with all other cases we have tested, such as the pure case and different arrangements of the impurities (for one or two non-magnetic impurities). In these more conventional configurations, we always find the ground state to be of the lowest possible spin (a global singlet $S_{\rm GS}=0$ when the total number of spins is even). This is consistent with what is expected for conventional antiferromagnets.

In addition to this, we also study the ground state texture for cases where it is not a global singlet. This can be done by calculating $\braket{S^{z}_{i}}$ for all sites on the lattice in the highest polarized state in the ground state multiplet. This is shown explicitly in Fig. 3 for $S=11/2$. Since the ground-state has total spin $S_{\rm GS}=2$ (see Fig. 2), we have measured the distribution of magnetization in the $S^{z}_{\rm total}=\sum_{i=1}^{N}S_{i}^{z}=2$ sector. There is clearly a larger effect at the orphan site which tapers off with increasing distance, but due to the relative small size of the lattice, other sites are affected as well.

In Fig.4, we present other magnetization profiles for much larger system sizes using DMRG simulations White (1992); Schollwöck (2011); Fishman et al. (2020). We have chosen a spin $S=2$ so that the ground-state is polarized and we measure the local $\braket{S^{z}_{i}}$ values in the $S^{z}_{\rm total}=1$ ground-state. Typically, we have kept up to $m=8000$ states (using only U(1) quantum number corresponding to the conservation of $S^{z}$) to achieve a discarded weight below $5.10^{-5}$. As often used in such simulations, we have chosen cylinder geometries, i.e. periodic boundary conditions in the short direction and open ones in the other. From these plots, it is clear that the spin texture is well localized around the orphan spin. Moreover, the numerical values found in these larger clusters are within $10\%$ of the ones obtained by ED on a much smaller 12-site cluster with 2 impurities.

In order to make contact with experiments, we now move to the finite-temperature regime to see if signatures of orphan spins remain and how to characterize them.

In this section, we study the thermodynamics of the quantum version of the Heisenberg model in Eq. 1, for various values of the spin $S$ ranging from $S=1/2$ up to $S=4$, on samples of the kagomé lattice containing one or several non-magnetic impurities, some of which are depicted in Fig. 1.

While a full computation of the thermodynamic properties does require the complete set of eigenvalues (to get the partition function), or even eigenstates (to compute observables), it has been known for a long-time that finite-temperature properties can be approximated using only a few well-chosen pure states in the correct energy window, as done in the finite-temperature Lanczos method Jaklič and Prelovšek (1994). This is rooted at the foundations of statistical mechanics since a pure state with energy $E$ of a large system has the same local properties as a thermodynamic mixed state at the related temperature. These ideas have been put on more rigorous grounds over the years, and are known as quantum typicality Hams and De Raedt (2000); Goldstein et al. (2006); Popescu et al. (2006); Reimann (2007).

In most of our results, we thus use a typicality scheme based on the exact application of the Hamiltonian to a random initial vector $|v_{0}\rangle$. Rather than using a power-method with repeated applications of the Hamiltonian Sugiura and Shimizu (2013, 2012) which has a slower convergence, we use the formulation in the Krylov basis ${\rm Span}\{|v_{0}\rangle,H|v_{0}\rangle,H^{2}|v_{0}\rangle\ldots,H^{m}|v_{0}\rangle\}$, and choose $m$ large enough (typically between 100 and 500) such that at least two of the lowest energy states are converged. We found that this criterion implies large-enough Krylov spaces to ensure convergence at the lower temperature. Several previous works show that these approaches are particularly successful to study thermodynamic properties of frustrated magnets on lattices larger than those available with full diagonalization methods Shimokawa and Kawamura (2016); Schnack et al. (2018); Wietek et al. (2019); Prelovšek and Kokalj (2020), and we refer to them for details about this method. Despite using exact (machine precision) application of $H$, all the data presented using this technique present error bars, which results from averaging over a finite number of initial random vectors $|v_{0}\rangle$. In our simulations, we use between $100$ and $800$ initial random vectors, depending on the system size, which is quite a high number (see discussion in Ref. Schnack et al. (2020) which averages over $100$ initial states). The error bars are computed using standard jackknife techniques to correctly take into account the important correlations between the numerator and denominators in all thermal expectations values Aichhorn et al. (2003); Schnack et al. (2020).

Specific aspects of our computations are: (i) the presence of impurities imply the absence of translation symmetries (for some impurity patterns, a reflection symmetry is still present), (ii) we measure expectation values of observables (such as the orphan magnetization $S^{z}_{\rm orphan}$) which do not commute with $H$, hence requiring to store all Krylov vectors Wietek et al. (2019). Even though we use total magnetization $S^{z}_{\rm total}$ conservation and spin inversion $S_{i}^{z}\rightarrow-S_{i}^{z}$ symmetry, the first point implies very large Hilbert spaces, especially for large spin $S$ values. The second point implies a much larger memory requirement than for most applications of the typicality method which deal with observables which commute with the Hamiltonian. We mitigate this by storing all Krylov vectors in parallel for the largest samples. Both computationally demanding points limit the typicality method to kagomé samples with a relatively small number of spins. We can nevertheless reach samples with up to $N=25$ spins for $S=1/2$ and up to $N=10$ spins for $S=4$.

In the following, we will present results for the total susceptibility of the sample $\chi_{\rm tot}=\langle(S^{z}_{\rm total})^{2}\rangle/T$, the local susceptibility, defined as (see e.g.  Gregor and Motrunich (2008)) $\chi_{loc}=\langle S^{z}_{\rm orphan}S^{z}_{\rm total}\rangle/T$, or the magnetization curve $S^{z}_{\rm orphan}(h)$. Here $S^{z}_{\rm orphan}$ is the magnetization at an orphan site.

In this work, we have investigated whether and how the physics of half-orphans, originally described using a classical theory, survives quantum effects and leads to clearly identifiable signatures in the response of quantum spin-$S$ Heisenberg antiferromagnets on the kagomé lattice when nearest-neighbours non-magnetic impurities are present. We found that orphan-specific signatures are evident in local susceptibility data in an intermediate temperature range. We also found that the classical predictions for the screening cloud around an orphan spin are justified in the quantum case. Indeed the resulting spin texture is a more robust signature than the local susceptibility of the orphan spin itself, and appears to provide a window to orphan physics even in the $S=1/2$ case. Another striking result, for samples with a single orphan, is the strong dependence of the ground state spin quantum number on the spin-$S$ of the local moment itself. In complete agreement with the classical expectation, we found that the $S\geq 2$ kagomé antiferromagnet with an orphan impurity complex has a ground state spin which is the nearest (half-)integer to $S/2$ allowed in most cases. Finally, we also presented experimental predictions for NMR Knight shifts on $S=1/2$ kagomé systems doped with non-magnetic impurities, which point to the possibility of observing the enhanced orphan-induced magnetic response through a small, but resolvable, secondary peak.

In the appropriate temperature regime corresponding to spin liquid behaviour, these classical results on half-orphan spins are also expected to apply mutatis mutandis to other corner-sharing lattices such as the pyrochlore. Our results, taken in conjunction with this expectation, therefore provide strong motivation for other related studies. For instance, one follow-up suggested by our work is to consider whether the nonzero ground-state spin quantum number that we find in samples with an orphan impurity complex is also present for the Heisenberg model with sufficiently large spin-$S$ on other corner-sharing lattices, for which the original $T=0$ arguments go through unchanged. We hope to report on this in the near future.

A noteworthy feature of the orphan-induced spin texture studied here is that it appears to be a very local feature that exists in an intermediate temperature window, and this feature seems largely independent of the nature of the ground state, whose character can vary with the spin value $S$. For example, the Heisenberg kagomé antiferromagnet with spin-1/2 has a ground state which is non-magnetic but the precise nature of this spin liquid (gapless vs gapped) is still debated Jiang et al. (2008); Evenbly and Vidal (2010); Yan et al. (2011); Depenbrock et al. (2012); Iqbal et al. (2013); Capponi et al. (2013b); He et al. (2017); Liao et al. (2017); in the spin-1 case, the ground state is also non-magnetic although whether it breaks lattice symmetries Changlani and Läuchli (2015); Liu et al. (2015) or not Nishimoto and Nakamura (2015) is still a matter of debate; for larger values of $S$, spin-wave analysis, coupled-cluster or series expansions methods point to an ordered phase Chubukov (1992); Götze et al. (2011); Oitmaa and Singh (2016) while a large-$N$ analysis, which generalizes the symmetry from SU(2) to a larger symmetry group, leads to a non-magnetic state Sachdev (1992).

It would be worth understanding how to incorporate the quantum orphan features found in our work in various effective field theories for these putative ground states Gregor and Motrunich (2008). Indeed, preliminary results suggest that some signatures of orphan physics are already present when samples with orphan impurity complex are studied using relatively crude extensions of variational treatments inspired by such effective field theories. It would also be interesting to complement our study of quantum half-orphans by series expansion methods; this may be particularly fruitful since the textures induced by the orphans are relatively compact in the temperature range of their existence.