Loop currents in ladder cuprates: A dynamical mean field theory study

Xiancong Lu Department of Physics, Xiamen University, Xiamen 361005, China D. Sénéchal Département de physique and Institut quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1

Abstract

We investigate the possibility of spontaneous loop currents in the two-leg ladder cuprate Sr_14-xCa_xCu₂₄O₄₁ by applying cluster dynamical mean field theory (CDMFT) to a seven-band Hubbard model for that compound, with an exact diagonalization solver. We sample several values of the local interaction $U_{d}$ and of the Cu-O energy difference $E_{pd}$ , by applying an external field that induces loop currents. We find no instance of spontaneous loop currents once the external field is brought to zero.

I Introduction

One of the most interesting features of cuprate superconductors is the pseudogap phenomenon, which is widely believed to be a key to understanding the mechanism of high-temperature superconductivity (HTSC) Timusk and Statt (1999). However, the origin of the pseudogap is still a matter of debate and the possibility of a spontaneously broken symmetry at low temperature within that state has not been excluded Keimer et al. (2015). One important possibility is the loop currents (LC) phase, proposed by Varma Varma (1999, 2006); Shekhter and Varma (2009), in which equilibrium orbital currents are circulating along the O-Cu-O plaquette within each unit cell, thus breaking time-reversal symmetry while preserving translational symmetry. Varma’s proposal has stimulated many experimental searches for the signature of microscopic orbital magnetic moments. Polarized neutron diffraction (PND) experiments have lent support to the existence of an intra-unit cell (IUC) magnetic order on CuO₂ planes Fauqué et al. (2006); Mook et al. (2008); Mangin-Thro et al. (2017) or involving apical oxygens Li et al. (2008). By contrast, nuclear magnetic resonance (NMR) Strässle et al. (2011); Mounce et al. (2013); Wu et al. (2015) and muon spin rotation ( $\mu$ SR) MacDougall et al. (2008); Sonier et al. (2009); Huang et al. (2012); Pal et al. (2016) have not found evidence of magnetic order. Varma’s hypothesis has also been investigated theoretically, with numerical methods and models often used in the study of strongly correlated electrons, such as exact diagonalizations (ED) Greiter and Thomale (2007); Thomale and Greiter (2008); Kung et al. (2014), variational Monte Carlo (VMC) Weber et al. (2009, 2014), and the variational cluster approximation (VCA) Lu et al. (2012). For the three-band Hubbard model with realistic parameters for high- $T_{c}$ cuprates, the results of these different methods are consistent: the LC phase is not stabilized as a ground state in the thermodynamic limit.

The existence of LCs was also investigated theoretically in the two-leg ladder, which is simpler and interpolates between one- and two-dimensional systems. By using the highly accurate density-matrix renormalization group (DMRG) technique, evidence for the existence of a “staggered-flux” phase was found for the two-leg ladder with long-range interaction, both at, and away from, half-filling Marston et al. (2002); Schollwöck et al. (2003). Analytical studies using a bosonization/renormalization group (RG) method also found stable regions of LCs for weak interactions Orignac and Giamarchi (1997); Chudzinski et al. (2008). However, a DMRG study on two-leg CuO ladders has found negative evidence towards the LC phase Nishimoto et al. (2009, 2010).

Recently, using polarized neutron diffraction, Bounoua et al. Bounoua et al. (2019) discovered the existence of a new kind of short-range magnetism in the two-leg ladder cuprate Sr_14-xCa_xCu₂₄O₄₁(SCCO- $x$ ) for two Ca contents ( $x=5$ and $x=8$ ). The measured magnetic structure factor can be reproduced by assuming a set of counter-propagating LCs around each Cu atom. This raises the possibility of a LC phase in the ladder cuprate. In this paper, we try to verify this for $x=8$ using cluster dynamical mean field theory (CDMFT) applied on a multi-band Hubbard model.

Refer to caption — Figure 1: (Color online). Left panel: the Cu₂O₃ lattice. Cu atoms are in blue, oxygen atoms in red. The expected loop currents for SCCO-8 are shown by arrows and the associated fluxes of opposite signs are indicated by blue and red triangles. Unit cells are delimited in yellow and the lattice vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ are shown. Right panel: the seven orbitals in a given unit cell, with their labels as they appear in Eq. 2.

II Model and method

II.1 Hamiltonian

The structure of SCCO- $x$ consists of an alternating stack of 1D CuO₂ chains and quasi-1D Cu₂O₃ two-leg ladder layers. We will focus of the ladder layer only and use a simplified description in terms of seven orbitals per unit cell: Two Cu $d_{x^{2}-y^{2}}$ orbitals (in blue on Fig. 1), two O $p_{x}$ and three O $p_{y}$ orbitals, respectively in green and red on the right panel of Fig. 1. The hopping amplitudes will be chosen to be the same as the ones often used in the three-band model for the cuprates, except that two of the oxygen sites in the unit cell involve both $p_{x}$ and $p_{y}$ orbitals, owing to the slightly different geometry of the model compared to the cuprates. The noninteracting Hamiltonian has the form

H_{0}=\sum_{\mathbf{k},\sigma}\mathbf{t}_{\mathbf{k}}C_{\mathbf{k},\sigma}^{\dagger}C_{\mathbf{k},\sigma}

(1)

where $C_{\mathbf{k},\sigma}$ stands for an array of annihilation operators associated with the seven orbitals per unit cell, as labeled on Fig. 1, and where the momentum-dependent matrix $\mathbf{t}_{\mathbf{k}}$ is shown in Eq. (2) below. That matrix is Hermitian (the upper triangle is not shown). The hopping amplitude between Cu and O orbitals is $t_{pd}$ and the energy difference between O and Cu orbitals is $E_{pd}$ . We assume for simplicity that the diagonal hopping amplitude $t_{pp}$ between oxygens is the same for $p_{x}-p_{x}$ and $p_{x}-p_{y}$ bonds. We will set $t_{pp}=1$ and $t_{pd}=1.5$ throughout ( $t_{pp}$ sets the energy scale). Finally, we will specifically investigate the loop current structure illustrated on Fig. 1 which, according to Bounoua et al. (2019), is appropriate for $x=8$ , corresponding to a doping of $\sim 17\%$ . We will, however, cover a fairly wide doping range around that value.

\small\mathbf{t}_{\mathbf{k}}=-\begin{pmatrix}0&&&&&&\\ 0&0&&&&&\\ -t_{pd}&t_{pd}&-E_{pd}&&&&\\ t_{pd}e^{-i\mathbf{k}\cdot\mathbf{e}_{2}}&0&t_{pp}\left(1+e^{-i\mathbf{k}\cdot\mathbf{e}_{1}}\right)&-E_{pd}&&&\\ 0&t_{pd}\left(-1+e^{-i\mathbf{k}\cdot\mathbf{e}_{1}}\right)&t_{pp}\left(1-e^{-i\mathbf{k}\cdot\mathbf{e}_{1}}\right)&0&-E_{pd}&&\\ 0&-t_{pd}e^{i\mathbf{k}\cdot\mathbf{e}_{2}}&0&t_{pp}\left(1+e^{i\mathbf{k}\cdot\mathbf{e}_{1}}\right)e^{i\mathbf{k}\cdot\mathbf{e}_{2}}&t_{pp}\left(1-e^{i\mathbf{k}\cdot\mathbf{e}_{1}}\right)e^{i\mathbf{k}\cdot\mathbf{e}_{2}}&-E_{pd}&\\ t_{pd}\left(1-e^{i\mathbf{k}\cdot\mathbf{e}_{1}}\right)&0&t_{pp}\left(1-e^{i\mathbf{k}\cdot\mathbf{e}_{1}}\right)&t_{pp}\left(1-e^{i\mathbf{k}\cdot\mathbf{e}_{1}}\right)e^{i\mathbf{k}\cdot\mathbf{e}_{2}}&0&t_{pp}\left(1+e^{i\mathbf{k}\cdot\mathbf{e}_{1}}\right)e^{i\mathbf{k}\cdot\mathbf{e}_{2}}&-E_{pd}\end{pmatrix}

(2)

To this noninteracting Hamiltonian we add local Hubbard interactions on the Cu and O atoms, so that the complete Hamiltonian reads

H=H_{0}+U_{d}\sum_{i\in\mathrm{Cu}}n_{i\uparrow}^{d}n_{i\downarrow}^{d}+U_{p}\sum_{j\in\mathrm{O}}n_{j\uparrow}^{p}n_{j\downarrow}^{p}-\mu\hat{N}_{\rm tot}

(3)

where the sum over $i$ runs over Cu sites, the sum over $j$ runs over the five O orbitals in each unit cell, $\mu$ is the chemical potential and $\hat{N}$ the total number of electrons in all the orbitals considered. $U_{d}$ and $U_{p}$ are the Coulomb repulsion of two holes sitting on the same copper orbital or the same oxygen orbital, respectively. We neglect the Coulomb interaction between different orbitals.

Figure 2: (Color online) Structure of the hybridization between the four bath orbitals of the Anderson impurity model and the two Cu orbitals of the cluster (for simplicity, the oxygen orbitals are not shown, even though they are part of the impurity model).

II.2 Impurity model

In order to reveal loop currents possibly arising in model (3), we use cluster dynamical mean-field theory (CDMFT) Lichtenstein and Katsnelson (2000); Kotliar et al. (2001); Liebsch et al. (2008); Sénéchal (2015) with an exact diagonalization solver at zero temperature (or ED-CDMFT). In CDMFT, the infinite lattice is tiled into identical units, or clusters, each of which is then coupled to a bath of uncorrelated, auxiliary orbitals. The parameters describing this bath (energy levels, hybridization, etc.) are then found by imposing a self-consistency condition.

In this work the cluster consists of a single unit cell (as shown on the right panel of Fig. 1), which is coupled to a bath of four uncorrelated orbitals. The Cu orbitals being the most correlated (because $U_{d}$ is considerably larger than $U_{p}$ ), we choose a simplified bath parametrization in which the bath orbitals are hybridized with the Cu orbitals only, even though $U_{p}\neq 0$ , as shown on Fig. 2. The corresponding Anderson impurity model (AIM) Hamiltonian is

H_{\rm imp}=H_{c}+\sum_{i,r}\theta_{ir}\left(c_{i}^{\dagger}a_{r}^{\phantom{\dagger}}+\mbox{H.c.}\right)+\sum_{r}\epsilon_{r}a_{r}^{\dagger}a_{r}^{\phantom{\dagger}},

(4)

where $H_{c}$ is the Hamiltonian (3), but restricted to a single cluster; cluster orbitals are labeled by the index $i$ and uncorrelated (bath) orbitals by the index $r$ . $\theta_{ir}$ is a complex hybridization parameter between cluster orbital $i$ and bath orbital $r$ , and $\epsilon_{r}$ is the energy level of bath orbital $r$ . All these parameters are assumed to be spin independent, as we are not looking for magnetic ordering.

In ED-CDMFT, the bath parameters $\theta_{ir}$ and $\epsilon_{r}$ are determined by an approximate self-consistent procedure, as proposed initially in Caffarel and Krauth (1994), that goes as follows: (i) initial values $\{\epsilon_{r},\theta_{ir}\}$ are chosen on the first iteration. (ii) For each iteration, the AIM (4) is solved, i.e., the cluster Green function $\mathbf{G}_{c}(\omega)$ is computed using the Lanczos method. The latter can be expressed as

\mathbf{G}_{c}(\omega)^{-1}=\omega-\mathbf{t}_{c}-\bm{\Gamma}(\omega)-\bm{\Sigma}_{c}(\omega)

(5)

where $\mathbf{t}_{c}$ is the one-body matrix in the cluster part of the impurity Hamiltonian $H_{\rm imp}$ , $\bm{\Sigma}_{c}(\omega)$ is the associated self-energy, and $\bm{\Gamma}(\omega)$ is the bath hybridization matrix:

\Gamma_{ij}(\omega)=\sum_{r}\frac{\theta_{ir}\theta_{jr}^{*}}{\omega-\epsilon_{r}}

(6)

(iii) The bath parameters are updated, by minimizing the distance function:

d(\bm{\epsilon},\bm{\theta})=\sum_{i\omega_{n}}W(i\omega_{n})\left[\mathbf{G}_{c}(i\omega_{n})^{-1}-\bar{\mathbf{G}}(i\omega_{n})^{-1}\right]

(7)

where $\bar{\mathbf{G}}(\omega)$ , the projected Green function, is defined as

\bar{\mathbf{G}}(\omega)=\frac{1}{N}\sum_{\mathbf{k}}\mathbf{G}(\mathbf{k},\omega)\quad,\quad\mathbf{G}(\mathbf{k},\omega)=\frac{1}{\omega-\mathbf{t}_{\mathbf{k}}-\bm{\Sigma}_{c}(\omega)}\leavevmode\nobreak\ \leavevmode\nobreak\ .

(8)

In the above $\mathbf{t}_{\mathbf{k}}$ is the one-body Hamiltonian (2) and $N$ is the (nearly infinite) number of sites. The matrices in the above are $7\times 7$ , for each spin projection. Essentially, $\bar{\mathbf{G}}(\omega)$ is the local Green function obtained by carrying the self-energy $\bm{\Sigma}_{c}(\omega)$ to the whole lattice. Ideally, $\bar{\mathbf{G}}(\omega)$ should coincide with the impurity Green function $\mathbf{G}_{c}(\omega)$ , but the finite number of bath parameters does not allow for this correspondence at all frequencies, and so a distance function $d(\epsilon_{r},\theta_{ir})$ is defined, with emphasis on low frequencies along the imaginary axis. The weight function $W(i\omega_{n})$ is where the method has some arbitrariness; in this work $W(i\omega_{n})$ is taken to be a constant for all Matsubara frequencies lower than a cutoff $\omega_{c}=2t_{pp}$ , with a fictitious temperature $\beta^{-1}=t_{pp}/50$ . (iv) We go back to step (ii) and iterate until the bath parameters or the bath hybridization function $\bm{\Gamma}(\omega)$ stop varying within some preset tolerance.

III Results and discussion

In order to probe the possible existence of loop currents in Model (3), we first need to define an operator representing them. We selected the following current loop operator, defined within the unit cell and following the arrows shown on the right panel of Fig. 1:

\hat{I}=i\Big{(}c_{1}^{\dagger}c_{7}+c_{7}^{\dagger}c_{3}+c_{3}^{\dagger}c_{5}+c_{5}^{\dagger}c_{2}+c_{2}^{\dagger}c_{3}+c_{3}^{\dagger}c_{1}\Big{)}+\mathrm{H.c}

(9)

We then impose an external field $I$ proportional to this operator on the system, i.e., we replace Hamiltonian (3) by $H+I\hat{I}$ . This external field induces a nonzero expectation value $\langle\hat{I}\rangle$ on the impurity model. We then reduce this external field to zero through a sequence of values (see Fig. 3) and monitor the expectation value $\langle\hat{I}\rangle$ . If spontaneous currents were possible, a nonzero value of $\langle\hat{I}\rangle$ would persist down to $I=0$ , which would indicate a spontaneous breaking of time reversal symmetry (TRS). This is impossible if the hybridization $\theta_{ir}$ is purely real. One can always require the hybridization parameter $\theta_{1r}$ to be real, because of an arbitrariness in the phase of the bath annihilation operator $a_{r\sigma}$ . This being done, the phase of the other hybridization $\theta_{2r}$ is determined by the CDMFT procedure. The complex-valued character of $\theta_{2r}$ is a necessary (but not sufficient) condition for a broken TRS state.

We have carried out a series of CDMFT computations on Model (3) with band parameters $t_{pp}=1$ and $t_{pd}=1.5$ , $U_{p}=3$ , and several values of $E_{pd}$ (0, 2, 4, 7), $U_{d}$ (6, 8, 10, 14) and chemical potential. In all cases an external current field $I$ was applied sequentially ( $I=$ 0.2, 0.1, 0.05, 0.01 and 0.0) in order to maximize the chances of finding a spontaneous current. In all cases no such current was found: $\langle\hat{I}\rangle=0$ , within numerical error ( $10^{-6}$ ). Plots of $\langle\hat{I}\rangle$ vs the electron density $n_{c}$ on the impurity (corresponding to a few values of the chemical potential $\mu$ ) are shown on Fig. 3. In each panel the different curves correspond to different values of the external current field $I$ , down to $I=0$ for the null curve. A few sample values of $U_{d}$ and $E_{pd}$ were chosen for the figure.

On Fig. 1, another current loop may be defined, that straddles four different unit cells, meeting at its center (dotted line on the figure). An operator $\hat{I}^{\prime}$ exists for this current loop, except that it is defined on the lattice model only, not on the impurity. Nevertheless, it is possible to formally compute the average of such an operator, from the lattice Green function $\mathbf{G}(\mathbf{k},\omega)$ of Eq. (8). We have checked that this average too is identically zero in the limit of zero external field.

We have also checked that our conclusions are unchanged if we add a sizable second-neighbor O-O hopping term $t_{pp}^{\prime}=-1$ . This hopping was deemed important to detect loop currents in Ref. Weber et al. (2014). In our work, such a coupling does not affect the impurity model, but affects the CDMFT solution through the self-consistency solution.

Even though we are bound to limit ourselves to a sampling of model parameters, we are strongly inclined to conclude that spontaneous orbital currents do not occur in the model we used to describe Sr_14-xCa_xCu₂₄O₄₁. If the results of Ref. Bounoua et al. (2019) are truly the signature of loop currents, the source of the discrepancy has to be found either in the model itself, or in the simple CDMFT treatment we have put in place. We have chosen an impurity model that contains a pair of triangular loops within the cluster, so as not to rely only on the measurement of lattice-based operators (as opposed to impurity-based operators). Of course the bath system itself is limited in size, but this is necessary in order to keep the problem numerically manageable. Increasing the number of bath orbitals would in general lead to better accuracy, but would not, from experience, change the nature of the ground state. Quantum Monte Carlo studies are impossible here, because of the sign problem, which becomes a phase problem for complex-valued Hamiltonians. In short, we do not believe that incremental improvements in the DMFT treatment of this problem would lead to different conclusions.

Acknowledgements.

We thank A.-M. Tremblay for discussions and comments. Computing resources were provided by Compute Canada and Calcul Québec. X.L. is supported by the National Natural Science Foundation of China (Grant No. 11974293) and the Fundamental Research Funds for Central Universities (Grant No. 20720180015).

References

Timusk and Statt (1999) T. Timusk and B. Statt, Reports on Progress in Physics 62, 61 (1999).
Keimer et al. (2015) B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, Nature 518, 179 (2015).
Varma (1999) C. M. Varma, Phys. Rev. Lett. 83, 3538 (1999).
Varma (2006) C. M. Varma, Phys. Rev. B 73, 155113 (2006).
Shekhter and Varma (2009) A. Shekhter and C. M. Varma, Phys. Rev. B 80, 214501 (2009).
Fauqué et al. (2006) B. Fauqué, Y. Sidis, V. Hinkov, S. Pailhès, C. T. Lin, X. Chaud, and P. Bourges, Phys. Rev. Lett. 96, 197001 (2006).
Mook et al. (2008) H. A. Mook, Y. Sidis, B. Fauqué, V. Balédent, and P. Bourges, Phys. Rev. B 78, 020506 (2008).
Mangin-Thro et al. (2017) L. Mangin-Thro, Y. Li, Y. Sidis, and P. Bourges, Phys. Rev. Lett. 118, 097003 (2017).
Li et al. (2008) Y. Li, V. Balédent, N. Barišić, Y. Cho, B. Fauqué, Y. Sidis, G. Yu, X. Zhao, P. Bourges, and M. Greven, Nature 455, 372 (2008).
Strässle et al. (2011) S. Strässle, B. Graneli, M. Mali, J. Roos, and H. Keller, Phys. Rev. Lett. 106, 097003 (2011).
Mounce et al. (2013) A. M. Mounce, S. Oh, J. A. Lee, W. P. Halperin, A. P. Reyes, P. L. Kuhns, M. K. Chan, C. Dorow, L. Ji, D. Xia, X. Zhao, and M. Greven, Phys. Rev. Lett. 111, 187003 (2013).
Wu et al. (2015) T. Wu, H. Mayaffre, S. Krämer, M. Horvatić, C. Berthier, W. N. Hardy, R. Liang, D. A. Bonn, and M.-H. Julien, Nature Communications 6, 6438 (2015).
MacDougall et al. (2008) G. J. MacDougall, A. A. Aczel, J. P. Carlo, T. Ito, J. Rodriguez, P. L. Russo, Y. J. Uemura, S. Wakimoto, and G. M. Luke, Phys. Rev. Lett. 101, 017001 (2008).
Sonier et al. (2009) J. E. Sonier, V. Pacradouni, S. A. Sabok-Sayr, W. N. Hardy, D. A. Bonn, R. Liang, and H. A. Mook, Phys. Rev. Lett. 103, 167002 (2009).
Huang et al. (2012) W. Huang, V. Pacradouni, M. P. Kennett, S. Komiya, and J. E. Sonier, Phys. Rev. B 85, 104527 (2012).
Pal et al. (2016) A. Pal, K. Akintola, M. Potma, M. Ishikado, H. Eisaki, W. N. Hardy, D. A. Bonn, R. Liang, and J. E. Sonier, Phys. Rev. B 94, 134514 (2016).
Greiter and Thomale (2007) M. Greiter and R. Thomale, Phys. Rev. Lett. 99, 027005 (2007).
Thomale and Greiter (2008) R. Thomale and M. Greiter, Phys. Rev. B 77, 094511 (2008).
Kung et al. (2014) Y. F. Kung, C.-C. Chen, B. Moritz, S. Johnston, R. Thomale, and T. P. Devereaux, Phys. Rev. B 90, 224507 (2014).
Weber et al. (2009) C. Weber, A. Läuchli, F. Mila, and T. Giamarchi, Phys. Rev. Lett. 102, 017005 (2009).
Weber et al. (2014) C. Weber, T. Giamarchi, and C. M. Varma, Phys. Rev. Lett. 112, 117001 (2014).
Lu et al. (2012) X. Lu, L. Chioncel, and E. Arrigoni, Phys. Rev. B 85, 125117 (2012).
Marston et al. (2002) J. B. Marston, J. O. Fjærestad, and A. Sudbø, Phys. Rev. Lett. 89, 056404 (2002).
Schollwöck et al. (2003) U. Schollwöck, S. Chakravarty, J. O. Fjærestad, J. B. Marston, and M. Troyer, Phys. Rev. Lett. 90, 186401 (2003).
Orignac and Giamarchi (1997) E. Orignac and T. Giamarchi, Phys. Rev. B 56, 7167 (1997).
Chudzinski et al. (2008) P. Chudzinski, M. Gabay, and T. Giamarchi, Phys. Rev. B 78, 075124 (2008).
Nishimoto et al. (2009) S. Nishimoto, E. Jeckelmann, and D. J. Scalapino, Phys. Rev. B 79, 205115 (2009).
Nishimoto et al. (2010) S. Nishimoto, E. Jeckelmann, and D. Scalapino, Physica C: Superconductivity and its Applications 470, S53 (2010), proceedings of the 9th International Conference on Materials and Mechanisms of Superconductivity.
Bounoua et al. (2019) D. Bounoua, L. Mangin-Thro, J. Jeong, R. Saint-Martin, L. Pinsard-Gaudart, Y. Sidis, and P. Bourges, “Loop currents in two-leg ladders cuprates,” (2019), arXiv:1912.07757 [cond-mat.str-el] .
Lichtenstein and Katsnelson (2000) A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B 62, R9283 (2000).
Kotliar et al. (2001) G. Kotliar, S. Y. Savrasov, G. Pálsson, and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001).
Liebsch et al. (2008) A. Liebsch, H. Ishida, and J. Merino, Phys. Rev. B 78, 165123 (2008).
Sénéchal (2015) D. Sénéchal, in Many-Body Physics: From Kondo to Hubbard, Vol. 5, edited by E. Pavarini, E. Koch, and P. Coleman (Forschungszentrum Jülich, 2015) pp. 13.1–13.22.
Caffarel and Krauth (1994) M. Caffarel and W. Krauth, Phys. Rev. Lett. 72, 1545 (1994).