Dynamical reduction of the dimensionality of exchange interactions and the "spin-liquid" phase of $\kappa$-(BEDT-TTF)$_2X$

TL;DR

This paper demonstrates that the anisotropy in the effective spin model of $$-(BEDT-TTF)$_2X$ salts leads to a spin-liquid phase arising from coupled spin chains, reconciling experimental observations with theoretical models.

Contribution

It reveals how intra-dimer quantum interference reduces the effective dimensionality, explaining the spin-liquid phase as a remnant of single-chain physics.

Findings

Anisotropy of the effective spin model differs from the tight-binding model.

Frustrated interchain interactions suppress long-range magnetic order.

The spin-liquid phase is linked to Tomonaga-Luttinger physics of a single chain.

Abstract

We show that the anisotropy of the effective spin model for the dimer Mott insulator phase of $\kappa$-(BEDT-TTF)$_2X$ salts is dramatically different from that of the underlying tight-binding model. Intra-dimer quantum interference results in a model of coupled spin chains, where frustrated interchain interactions suppress long-range magnetic order. Thus, we argue, the "spin liquid" phase observed in some of these materials is a remnant of the Tomonaga-Luttinger physics of a single chain. This is consistent with previous experiments and resolves some outstanding puzzles. An erratum [Phys. Rev. Lett. 120, 199901 (2018).] is added as an appendix.

Equations

$${\mathcal{H}}_H$$

$$M_1$$

$$M_2$$

Full text

Layered organic charge transfer salts show a wide range of exotic physics due to strong electronic correlations and geometrical frustration RPP . This includes unconventional superconductivity, incoherent metallic transport, multiferroicity, and antiferromagnetism. However, the putative spin liquid states in $\kappa$-(BEDT-TTF)2Cu2(CN)3 Shimizu , $\kappa$-(BEDT-TTF)2Ag2(CN)3 Shimizu16 (henceforth, CuCN and AgCN respectively) and $\beta$’-EtMe3Sb[Pd(dmit)2]2 Itou are, perhaps, the least understood of these.

CuCN is usually discussed in terms of the nearly triangular Heisenberg model RPP ; Balents . Here we demonstrate that the theoretical arguments that lead to this model are fallacious. They fail to account for quantum interference within the (BEDT-TTF)2 dimer. We derive the correct low-energy model including these effects and show that it leads to an anisotropic triangular lattice in the quasi-one-dimensional (q1D) regime, $J_{1}>J_{2}$, Fig. 1c. Thus, the spin model for the Mott dimer insulating phases of the organic charge transfer salts are remarkably similar to that describing Cs2CuBr4 and Cs2CuCl4 Radu-model , where deconfined spinons have been observed Radu ; Balents . Our results provide natural explanations for several previously puzzling experiments on the organics.

Electronic structure calculations demonstrate that a single molecular orbital contributes to the low-energy process in the $\kappa$-(BEDT-TTF)2X salts RPP ; Kino ; RossReview ; Koretsune , and that the band structure is described by the tight-binding ‘monomer model’ sketched in Fig. 1a at three quarters filling. This model is dimerised: $t_{b1}\gg t_{b2},t_{p},t_{q}$. At ambient pressure CuCN, AgCN and $\kappa$-(BEDT-TTF)2Cu[N(CN)2]Cl (henceforth $\kappa$-Cl) display a Mott dimer insulating phase, where excitations away from exactly one hole per dimer are bound RPP .

Electronic correlations arise from the Coulombic repulsion between two holes on the same monomer, $U_{m}$, or dimer, $V_{m}$. Thus, the effective Hamiltonian for the $i$th dimer is ${\cal H}_{b1}^{(i)}=-t_{b1}\sum_{\sigma}(\hat{c}_{i1\sigma}^{\dagger}\hat{c}_{i2\sigma}+H.c.)+U_{m}\sum_{\mu}\hat{n}_{i\mu\uparrow}\hat{n}_{i\mu\downarrow}+V_{m}\hat{n}_{i1}\hat{n}_{i2}$, where $\hat{c}_{i\mu\sigma}^{(\dagger)}$ annihilates (creates) an electron with spin $\sigma$ on the $\mu$th monomer of the $i$th dimer, $\hat{n}_{i\mu\sigma}=\hat{c}_{i\mu\sigma}^{\dagger}\hat{c}_{i\mu\sigma}$, and $\hat{n}_{i\mu}=\sum_{\sigma}\hat{n}_{i\mu\sigma}$. Other Coulomb matrix elements can also be included, but do not qualitatively change our results and are neglected below. The hopping between dimers is given by ${\cal H}_{1}=-t_{b2}\sum_{\langle i,j\rangle\sigma}(\hat{T}_{21}+\hat{T}_{21}^{\dagger})$ and ${\cal H}_{2}=\sum_{[i,j]\sigma}[-t_{p}(\hat{T}_{21}+\hat{T}_{21}^{\dagger})-t_{q}(\hat{T}_{22}+\hat{T}_{22}^{\dagger})]$, where $\hat{T}_{\nu\mu}=\hat{c}_{i\nu\sigma}^{\dagger}\hat{c}_{j\mu\sigma}$, $\langle i,j\rangle$ implies a pair of dimers equivalent to tetramer 1 (Fig. 1a), and $[i,j]$ implies a pair of dimers such as tetramer 2.

Kino and Fukuyama (KF) showed that for large enough $U_{m}$ an insulating phase emerges Kino . They argued that this could be understood as a dimer Mott insulator: if one integrates out the bonding combination of molecular orbitals this leaves an effective half-filled model containing only the antibonding combination of molecular orbitals, $a_{i\sigma}^{\dagger}=\frac{1}{\sqrt{2}}(c_{i1\sigma}^{\dagger}-c_{i2\sigma}^{\dagger})$. The ‘dimer model’ is ${\cal H}_{d}=-t_{1}\sum_{\langle i,j\rangle\sigma}(\hat{a}_{i\sigma}^{\dagger}\hat{a}_{j\sigma}+H.c.)-t_{2}\sum_{[i,j]\sigma}(\hat{a}_{i\sigma}^{\dagger}\hat{a}_{j\sigma}+H.c.)+U_{d}\sum_{i}\hat{a}_{i\uparrow}^{\dagger}\hat{a}_{i\uparrow}\hat{a}_{i\downarrow}^{\dagger}\hat{a}_{i\downarrow}$, Fig. 1b, where, for $V_{m}=0$, $t_{1}/t_{b2}=t_{2}/(t_{p}+t_{q})=\sqrt{2}(\cos\theta-\sin\theta)/4$, and $\tan\theta=(U_{m}/4t_{b1})-\sqrt{1+(U_{m}/4t_{b1})^{2}}$ RossReview . KF estimated the effective interaction between two holes on the same dimer as $U_{d}=E_{0}(0)+E_{0}(2)-2E_{0}(1)=2t_{b1}+({U_{m}}/{2})[1-\sqrt{1+({4t_{b1}}/{U_{m}})}]\simeq 2t_{b1}$ for $U_{m}\gg 4t_{b1}$, where $E_{0}(N)$ is the ground state of the dimer with $N$ holes. The $V_{m}\neq 0$ case is discussed in ScrivenPRB .

In the Mott dimer phase KF’s dimer model reduces to a Heisenberg model, Fig. 1c:

[TABLE]

where $\hat{\mathbf{S}}_{i}$ is the spin operator on the $i$th dimer, and in the dimer model $J_{1}=4t_{1}^{2}/U_{d}$ and $J_{2}=4t_{2}^{2}/U_{d}$.

Two decades of research have been based on these ideas. Thus, it is surprising that no one appears to have asked whether the same parameters for the Heisenberg model, Eq. (1), are found from both the monomer and dimer models. We do. The answer is no.

To calculate $J_{1}$ we perform a canonical transformation Amie ; Jaime17 ; DiracQ taking ${\mathcal{H}}_{0}=\sum_{i}{\mathcal{H}}_{b1}^{(i)}$ as our unperturbed Hamiltonian with the perturbation given by ${\mathcal{H}}_{1}$. We retain terms ${{O}(t_{b2}^{2})}$ yielding the interaction described by the first term in Eq. (1).

The monomer model yields a larger $J_{1}$ than the dimer model, Fig. 2. This can be straightforwardly understood. The dimer with two electrons admits several low-lying excited states that allow for additional superexchange pathways, these are omitted from the dimer model. The exact energy differences between the lowest energy singlets and triplets for tetramer 1 (Figs. 1a, 2) are in excellent agreement with the perturbative treatment of the monomer model but are very different from the $J_{1}$ calculated from the dimer model.

$J_{2}$ is calculated from the analogous treatment of the perturbation ${\mathcal{H}}_{2}$. Here the predictions of the monomer model are strikingly different from the dimer model. $J_{2}$ is very rapidly suppressed by $U_{m}$ in the monomer model, indeed $J_{2}$ becomes ferromagnetic ($<0$) for only moderate $U_{m}$ at $V_{m}=0$, Fig. 2. Again a comparison with the exact low-energy states of tetramer 2 (Figs. 1a, 2) demonstrates excellent agreement with the monomer model and profound differences from the dimer model. In the monomer model, $J_{2}$ remains finite and negative whereas $J_{1}\rightarrow 0$ as $U_{m}\rightarrow\infty$.

Why is $J_{2}$ so different from $J_{1}$? The essential difference is that there are two hopping pathways in $\mathcal{H}_{2}$ and only one in $\mathcal{H}_{1}$. This allows destructive interference between the different exchange pathways that contribute to $J_{2}$, which are necessarily absent in the calculation of $J_{1}$. Furthermore, processes with amplitudes $\propto t_{p}t_{q}$ can take place without incurring an energetic penalty $\propto U_{m}$, Fig. 3. Thus, such processes remain active even as $U_{m}\rightarrow\infty$. Processes $\propto t_{p}t_{q}$ can favor ferromagnetic interactions. To understand this, it is helpful to consider two limiting cases:

(i) Molecular limit: $(U_{m}-V_{m})/t_{b1}\rightarrow\infty$. A detailed understanding can be gained from considering the matrix elements

[TABLE]

where $|\sigma_{i}\rangle=\frac{1}{\sqrt{2}}\hat{c}_{i1\sigma}^{\dagger}\hat{c}_{i2\sigma}^{\dagger}(\hat{c}_{i1\overline{\sigma}}^{\dagger}+\hat{c}_{i2\overline{\sigma}}^{\dagger})|0\rangle$, $|S_{i}\rangle=\frac{1}{\sqrt{2}}(\hat{c}_{i1\uparrow}^{\dagger}\hat{c}_{i2\downarrow}^{\dagger}-\hat{c}_{i1\downarrow}^{\dagger}\hat{c}_{i2\uparrow}^{\dagger})|0\rangle$, $|T_{i}\rangle=\frac{1}{\sqrt{2}}(\hat{c}_{i1\uparrow}^{\dagger}\hat{c}_{i2\downarrow}^{\dagger}+\hat{c}_{i1\downarrow}^{\dagger}\hat{c}_{i2\uparrow}^{\dagger})|0\rangle$ and $({\cal H}_{b1}^{(i)}+{\cal H}_{b1}^{(j)})|\Psi_{n}\rangle=\varepsilon_{n}|\Psi_{n}\rangle$. $|S_{i}\rangle$ and $|T_{i}\rangle$ become degenerate as $(U_{m}-V_{m})/t_{b1}\rightarrow\infty$. In the effective Heisenberg model $J_{1}=2t_{b1}^{2}M_{1}+\dots$ and $J_{2}=2t_{p}t_{q}M_{2}+2t_{p}^{2}M_{1}+\dots$, where the ellipses include other terms at the same order, discussed below.

$M_{1}$ vanishes because the intermediate singlet and triplet excited states interfere destructively, whereas $M_{2}$ remains finite because the interference is constructive. All other contributions to $J_{1}$ vanish due to similar interference effects, thus $J_{1}=0$. In contrast, the dimer model predicts that $J_{1}\propto t_{b2}^{2}/t_{b1}$ in this limit. All terms in $J_{2}$ proportional to $t_{p}^{2}$ and $t_{q}^{2}$ also vanish by the same arguments. Including all terms at this order yields $J_{2}=-t_{p}t_{q}/2t_{b1}$.

(ii) In the $U_{m}=V_{m}$ limit the Hartree-Fock approximation becomes exact. This makes it straightforward to calculate the effective Heisenberg interaction, $J_{\text{gen}}$, for the more general perturbation $\mathcal{H}_{\text{gen}}=-\sum_{ij\mu\nu\sigma}(t_{\mu\nu}\hat{T}_{\mu\nu}+H.c.)$. One finds that $J_{\text{gen}}=2(t_{11}-t_{12}-t_{21}+t_{22})^{2}/{U_{m}}$. Thus, $J_{1}=2t_{b1}^{2}/U_{m}$ and $J_{2}=2(t_{p}-t_{q})^{2}/U_{m}$. In this limit the interference is a single particle phenomenon arising from the different phases of the two sites in the antibonding orbital. Thus, the details of the interference here are quite different from the molecular limit. Nevertheless, one again finds that in the monomer model interference effects significantly suppress $J_{2}$ relative to expectations of the dimer model, where $J_{2}\propto(t_{p}+t_{q})^{2}$.

$J_{2}>0$ for all $U_{m}=V_{m}$. More generally, increasing $V_{m}$ suppresses ferromagnetic exchange and eventually drives it antiferromagnetic, Fig. 4. Large ferromagnetic $J_{2}$ is inconsistent with experiment. This suggests that $V_{m}/U_{m}$ is reasonably large, consistent with first principles estimates ScrivenPRB ; Laura1 .

To consider specific materials we take the hopping integrals from previous first principles calculations Koretsune . For reasonable parameters the monomer model yields $J_{2}<J_{1}$ in marked contrast to the dimer model, which gives $J_{2}>J_{1}$, Fig. 4. We will describe this as dynamical dimensionality reduction (DDR). We will see below that DDR, and the frustration inherent in the system, leads to a natural interpretation of the spin liquid phase in terms of coupled chains. Another important difference from the predictions of the dimer model is that the values of the interactions, $U_{m}$ and $V_{m}$, are vital for determining the parameters of the Heisenberg model. These are not well known at present and may differ between materials, but the best estimates suggest that $U_{m}\gg t_{b1}$ and $V_{m}\lesssim U_{m}$ ScrivenPRB ; Laura1 ; Laura2 ; Mori .

The similarities between the hopping integrals in the BEDT-TTF and Pd(dmit)2 salts suggest that similar physics is at play in the latter. Again the effective Heisenberg model is given by Eq. (1) and there are two significant inter-dimer hopping pathways that contribute to $J_{2}$, but a single pathway dominates $J_{1}$ ScrivenPRL .

In the 1D limit ($J_{2}\rightarrow 0$) a paramagnetic Tomonaga-Luttinger liquid (TLL) is expected at low temperatures. Eq. (1) with $J_{1}>J_{2}>0$ received extensive attention LSWT ; StarykhPRB ; StarykhPRL ; Sorella ; Bursill ; Singh ; White ; Bishop ; Zheng99 ; Kohno ; JOF06 ; JOF07 ; RVB ; Bocquet following the observation of a strong inelastic continuum, consistent with deconfined spinons, in neutron scattering experiments on Cs2CuCl4 Radu , where $J_{2}\simeq 0.34J_{1}$ Radu-model . Cs2CuCl4 displays spiral order at low temperatures. Nevertheless, the observed inelastic continuum is quantitatively reproduced by theories based on disjunctive TLLs Kohno .

Classically, model (1) has spiral order in the chain limit LSWT . Quantum fluctuations enhance the one-dimensionality of this state LSWT ; StarykhPRB ; Singh ; White ; Zheng99 ; JOF06 ; JOF07 ; RVB . Indeed Starykh et al. argued that the model is q1D for $J_{2}<0.7J_{1}$ StarykhPRB . Numerical studies are particularly challenging because of the incommensurate wavevector that characterizes the spiral phase Singh ; White and several other ground states are found to be energetically competitive StarykhPRL ; Bishop ; Zheng99 ; Sorella ; Bursill ; Singh ; White .

However, this question may be academic: theory suggests that small interactions decide which competing phase is realized StarykhPRB ; JaimeSB , as one expects on general grounds in frustrated systems. Series expansions Singh find that if the magnetization does not vanish as $J_{2}\rightarrow 0$ then it becomes small extremely rapidly, consistent with the Néel temperature, $T_{N}\sim\exp{[-(J_{1}/J_{2})^{2}]}$, predicted from treating the intrachain dynamics via TLL theory and the interchain coupling via the random phase approximation (TL+RPA) Bocquet .

Therefore, our prediction that $J_{1}>J_{2}$ naturally explains the absence of long-range magnetic order in CuCN and AgCN. Namely, that the q1D limit survives even for relatively large $J_{2}/J_{1}<1$ and thus the spin liquid is a remnant of the TLL found in an isolated chain. Even if the materials order eventually, the exponential suppression of $T_{N}$ can easily move this orders of magnitude below the lowest temperatures studied (10s of mK). Why then is $\kappa$-Cl antiferromagnetic? Two perturbations are formally relevant StarykhPRB : interlayer exchange, $J_{z}$; and the staggered interlayer component of the interchain Dzyaloshinskii-Moriya (DM) interaction, $D$, cf. Fig. 1c Kagawa ; SmithL ; SmithB ; Antal ; Winter (an inversion center precludes DM coupling within the chains).

In the TL+RPA theory Bocquet the dynamic susceptibility is given by $\chi^{+-}_{3D}(\omega,{\bm{k}})=\chi_{1D}^{+-}(\omega,k_{x})/[1-\tilde{J}(\bm{k})\chi_{1D}^{+-}(\omega,k_{x})]$, where $\chi_{1D}^{+-}(\omega,k_{x})$ is the susceptibility of a single chain perpendicular to $D$ and the Fourier transform foot-bi of the interchain interactions is $\tilde{J}(\bm{k})=-J_{z}\cos k_{z}\pm\sqrt{J_{2}^{2}+D^{2}}[\cos(k_{y}/2)+\cos(k_{x}-k_{y}/2)]$ . $T_{N}$ is the highest temperature with a zero-frequency pole in $\chi^{+-}_{3D}(\omega,{\bm{k}})$. This is straightforwardly calculated as described in Bocquet . The solutions, Fig. 5, clearly indicate that for reasonable parameters it is possible to achieve $T_{N}/J_{1}\sim 0.1-0.2$, consistent with observed critical temperature ($\sim 20$ K) in $\kappa$-Cl, given our calculation of $J_{1}$, Fig. 2. Furthermore, $J_{z}$, which is unfrustrated, affects $T_{N}$ far more strongly than $D$ or $J_{2}$, suggesting this could be the essential difference between $\kappa$-Cl and CuCN. This could be tested by applying uniaxial strain perpendicular to the layers, which one would expect to increase $J_{z}$. This should increase $T_{N}$ in $\kappa$-Cl and perhaps even drive CuCN or AgCN antiferromagnetic for sufficiently large strains, if the Mott transition does not intervene. Two intermonomer hopping integrals are relevant to interlayer hopping Jacko , so the interference effects that suppress $J_{2}$ also affect $J_{z}$. Thus, different materials may have radically different $J_{z}$.

It is important to ask how the DDR picture of the dimer Mott insulating organics compares with experiment.

Heat capacity varies linearly with temperature in a TLL Giamarchi as observed in CuCN Yam08 . Thermal conductivity of CuCN does not reveal a term that varies linearly with temperature Yam09 . It has widely been assumed, on the basis of q2D theories, that this is inconsistent with the heat capacity measurement. However, in a weakly disordered spin chain the magnetic contribution to the thermal conductivity $\kappa_{\textrm{mag}}\propto T^{2}$ Rosch , which is consistent with the measurements of CuCN provided the magnetic contribution dominates the low temperature behavior Yam09 . One also expects a dip in $\kappa_{\textrm{mag}}$ at $g\mu_{B}B\sim 4k_{B}T$ Rosch , which is also observed Yam09SI .

At low-frequencies one expects a power-law in the optical conductivity of a TLL Giamarchi . This is observed in both CuCN Dressel12 and AgCN Dressel17 .

The bulk susceptibility of CuCN shows a broad maximum around $\sim 70$ K Shimizu . This can be fit reasonably well by high temperature series expansions for the isotropic triangular lattice Shimizu ; Zheng . However, for 1D chains one also expects a broad maximum at $T=0.64J$ Affleck , which would lead to the estimate $J_{1}\sim 100$ K in CuCN.

The nuclear magnetic resonance (NMR) relaxation rate, $1/T_{1}$, in $\kappa$-Cl is well understood in terms of the incipient magnetic order Yusuf1 ; Yusuf2 . In contrast, $1/T_{1}$ in CuCN is a long-standing problem Lee ; Qi08 ; Qi09 ; Baskaran ; Liu ; Kyung ; Grover ; Galitski ; Kawamura ; Li . $1/T_{1}$ in CuCN decreases as the temperature is lowered until a minimum is reached at $\sim 6$ K. A broad peak is then observed around 1 K. In spin chains one expects a minimum in $1/T_{1}$ at $T\sim J_{1}/10$ concomitant with the crossover to a TLL Dupont . There are no calculations, to date, describing $1/T_{1}$ in the presence of interchain interaction at temperatures above the TLL regime. Therefore, the only available comparison is with experimental results for Cs2CuCl4 Vachon . Note that $J_{2}/J_{1}$ and the DM interaction are similar, but not identical, in the two materials, so the analogy is imperfect. However, one does not expect charge fluctuations to be especially important in the organics as no dramatic changes are observed under pressure until the first order metal-insulator transition. In Cs2CuCl4 one observes a broad peak around $\sim 2.5$ K, associated with the emergence of short-range order (SRO) Vachon ; Tokiwa , that is strongly reminiscent of the peak at $\sim 1$ K in CuCN. Microscopically, this SRO may be associated with the binding of spinons into triplons Kohno ; Balents driving a dimensional crossover and cutting off the logarithmic divergence in $1/T_{1}$ expected in a TLL.

Therefore a natural explanation of $1/T_{1}$ in CuCN is that one sees a high temperature regime, a crossover to a TLL regime at $T\sim 6$ K and the emergence of SRO/triplons at $T\sim 1$ K. These crossovers could also be responsible for the anomalies observed at the same characteristic temperatures in many other experiments Yam08 ; Yam09 ; Manna ; Poirier12 ; Poirier14 . A clear prediction of this interpretation is that the emergence of SRO should lead to the broadening of NMR spectra as the temperature is lowered Vachon . This is indeed observed in 13C NMR in CuCN; an observation that has eluded explanation in q2D theories Shimizu06 .

Antal et al. recently concluded that electron spin resonance (ESR) “in $\kappa$-Cl resembles the ESR in 1D Heisenberg chains with a Dzyaloshinskii-Moriya interaction” Antal just as our calculations suggest. Therefore, our prediction that the spin correlations in the insulating state are q1D is consistent with many experiments.

Metallic organics display coherent in-plane electronic transport at low temperatures. DDR applies only to the spin correlations and so is not inconsistent with this. However, charge transport becomes incoherent above 20-40 K Broun ; Jaime-Dressel . This suggests that the coherent interference processes, responsible for DDR, may be washed out when the temperature is raised. This would imply a strong temperature dependence in $J_{2}/J_{1}$ and lead to a dimensional crossover at a much lower temperature scale than one would expect from the low temperature $J_{2}/J_{1}$.

$J_{2}$ favors $d_{x^{2}-y^{2}}$ superconductivity (taking the $x$ and $y$ axes to lie along the $J_{2}$ bonds), whereas $J_{1}$ favors $s+d_{xy}$ pairing RVB ; group . Experimentally, the pairing symmetry in the organics remains controversial, but our results appear to favor $s+d_{xy}$ superconductivity, perhaps with accidental nodes.

Cs2CuCl4 displays a rich phase diagram as the strength and orientation of the magnetic field is varied StarykhPRB ; Tokiwa ; Vachon . Therefore, a more complete mapping of the physics of the organics in terms of field strength and direction, particularly those with antiferromagnetic order, and a detailed comparison with q1D theory, including the full details of the DM interaction, would provide a powerful test of the ideas described above. So could quantitative understanding of magnetic Raman scattering Nakamura14 ; Nakamura16 ; DrichkoPRB ; Drichko .

I Appendix: Erratum

In the above Letter we presented a general calculation of the superexchange interactions in dimer Mott insulators and compared these results to the title materials. We contrasted the ‘monomer model’, with one orbital per molecule, to the so-called dimer approximation where the bonding and antibonding combinations of the two orbitals within each dimer are first constructed and the orbitals that are filled in the non-interacting limit are neglected. We showed that intradimer interference effects can lead to qualitative differences between the full monomer model and the dimer approximation. Most dramatically, intradimer interference can cause a quasi-one-dimensional Heisenberg model to arise as an effective low-energy model of a quasi-two-dimensional tight-binding model. We argued that this physics is relevant to the $\kappa$-(BEDT-TTF)${}_{2}X$.

However, we have subsequently discovered an important error in our application of the general theory to the BEDT-TTF salts. We wrote the tight-binding part of the Hamiltonian in the form $-\sum_{ij\sigma}t_{ij}\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma}$, where $\hat{c}_{i\sigma}^{(\dagger)}$ annihilates an electron with spin $\sigma$ in the $i$th Wannier orbital. We will continue to use this convention throughout this erratum. However, Koretsune and Hotta Koretsune write such terms in the form $+\sum_{ij\sigma}t_{ij}\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma}$. We failed to account for this sign difference when using their first principles parameters for the $\kappa$-(BEDT-TTF)${}_{2}X$ salts. This has important consequences for these materials. As discussed below, the general mechanism described in our Letter whereby a one-dimensional superexchange interaction results from a two-dimensional tight-binding model remains correct. However, it does not appear to be relevant to the $\kappa$-(BEDT-TTF)${}_{2}X$ salts.

The value of $J_{1}$ is independent of the signs of the hopping integrals, but $J_{2}$ is not (cf. Fig. 1). As emphasized in our Letter, interference effects can dominate the value of $J_{2}$, particularly when electron-electron interactions are large. In our Letter we analyzed the superexchange interactions analytically, in two limits. We begin this erratum by clarifying how the superexchange interactions are changed when the signs of the hopping integrals are reversed (the numbering below corresponds to that on pages 2 and 3).

(i) In the molecular limit, $(U_{m}-V_{m})/|t_{b1}|\rightarrow\infty$, the analytic forms we reported above are correct regardless of the signs of the hopping integrals. In this limit the only non-vanishing superexchange interaction is $J_{2}=-t_{p}t_{q}/2t_{b1}$. Changing the signs of all three hopping integrals takes $J_{2}\rightarrow-J_{2}$. This is consistent with the differences between Nagaoka ferromagnetism and Haerter-Shastry antiferromagnetism Powell ; Haerter ; Sposetti . However, the signs of the hopping integrals that Koretsune and Hotta found via density functional theory (DFT) imply that in the molecular limit the $\kappa$-phase salts are described by a antiferromagnetic Heisenberg model on a square lattice as $J_{1}=0$ and $J_{2}>0$. Thus, in this frequently studied limit our conclusion that there are important differences between the superexchange interactions calculated from the monomer and dimer models remains valid (the latter yields a Heisenberg model on the anisotropic triangular lattice in this limit RossReview ).

(ii) For $U_{m}=V_{m}$ changing the sign of $t_{b1}$ swaps the bonding and antibonding orbitals relative to the definitions on page 1. That is, with $t_{b1}<0$ the bonding orbital is created by $\hat{b}_{i\sigma}^{\dagger}=(1/\sqrt{2})(\hat{c}_{i1\sigma}^{\dagger}-\hat{c}_{i2\sigma}^{\dagger})$ and the antibonding orbital is created by $\hat{a}_{i\sigma}^{\dagger}=(1/\sqrt{2})(\hat{c}_{i1\sigma}^{\dagger}+\hat{c}_{i2\sigma}^{\dagger})$. These signs propagate through and change our analytical results. The general superexchange interaction in this limit for $t_{b1}<0$ is $J_{\textrm{gen}}=2(t_{11}+t_{12}+t_{21}+t_{22})^{2}/U_{m}$, which is importantly different from the expression for $t_{b1}>0$ given on page 3. The expression for $J_{1}$ is unchanged, but for $t_{b1}<0$ we have $J_{2}=2(t_{p}+t_{q})^{2}/U_{m}$, which again differs by a sign from the expression for $t_{b1}>0$, given in our Letter. An important consequence of these corrections is that the ratio $J_{2}/J_{1}$ from the dimer approximation is correct when $U_{m}=V_{m}$: observe that the monomer and dimer lines in Fig. 7 coincide at $V_{m}/U_{m}=1$ for both materials.

Our error also necessitates the correction of two figures from our Letter. In Figs. 6 and 7 we re-plot Figs. 2 and 4 with the signs of all the hopping integrals reversed (so as to correctly represent the results of DFT calculations Koretsune ). It can be seen from Fig. 6 that both $J_{1}$ and $J_{2}$ remain antiferromagnetic ($>0$) for all values of $U_{m}$. This is in contrast to the case reported above, with the signs of all hopping integrals reversed, where $J_{2}$ becomes ferromagnetic ($<0$) for sufficiently large $U_{m}$. This behavior is expected for large $U_{m}$ as, to leading order in $1/U_{m}$, we have $J_{2}=-t_{p}t_{q}/t_{b1}$; changing the signs of all three hopping integrals must change the sign of $J_{2}$ in this limit. Fig. 7 shows that the superexchange interactions in both CuCN and $\kappa$-Cl remain quasi-two-dimensional. Indeed, as one expects $U_{m}>V_{m}$, these results suggest that the dimer approximation underestimates $J_{2}/J_{1}$ and hence that the magnetic interactions are closer to the square lattice than one would expect from the dimer approximation. This emphasizes that accurate estimates of the interaction parameters are important for determining the ratio $J_{2}/J_{1}$ and hence for understanding the spin liquid state in CuCN.

Finally, to understand the role of the signs of hopping integrals in molecular Mott insulators more generally, the following observations may be helpful. Considering the pattern of hopping integrals (Fig. 1) given that the sign of $t_{ij}$ is reversed by a $\pi$ gauge transformation on exactly one of the sites ($i$ or $j$), changing only the sign of $t_{b1}$ is sufficient to change the results between those shown here and those reported above. Secondly, as changing the signs of all hopping integrals is equivalent to a particle-hole transformation, if there is one electron per dimer (rather than three) the formulae given in the original Letter hold. This is the relevant filling for organic anion systems such as the Ni(dmit)2 salts Kato , which do appear to show quasi-one-dimensional magnetism.

Thus we conclude the following: (i) The interference mechanism for realizing a quasi-one-dimensional (q1d) Heisenberg model from a quasi-two-dimensional (q2d) tight-binding model is not relevant to the $\kappa$-phase organics, but is possible and may be realized in other materials. (ii) Inter-dimer interference effects are still likely to be relevant to the $\kappa$-phase organics, but they tend to drive the system towards the square lattice limit ($J_{1}/J_{2}\rightarrow 0$). (iii) The dimer approximation is only accurate when the Hartree-Fock approximation is reasonable.

We thank Stephen Winter and Roser Valentí for helpful discussions.