Topological spin phases of trapped Rydberg excitons in Cu$_2$O

TL;DR

This paper theoretically investigates Rydberg excitons in one-dimensional Cu$_2$O traps, revealing a topological Haldane phase with tunable spin states and rich phase diagrams influenced by trap geometry and interaction anisotropy.

Contribution

It predicts the emergence of a topological Haldane phase in Rydberg excitons and explores how trap geometry and anisotropy affect the spin phase diagram.

Findings

Presence of topological Haldane phase with long-range string order

Effect of trap geometry on spin state phases

High tunability of Rydberg exciton spin states

Abstract

We study theoretically Rydberg excitons in one-dimensional chains of traps in Cu$_2$O coupled via the van der Waals interaction. The triplet of optically active $p$-shell states acts as an effective spin-$1$ and the interactions between the excitons are strongly spin-dependent. We predict that the system {has the} topological Haldane phase with the diluted antiferromagnetic order, long-range string correlations, and finite excitation gap. We also analyze the effect of the trap geometry and interactions anisotropy on the Rydberg exciton spin states and demonstrate that a rich spin phase diagram can be realized showing high tunability of the Rydberg exciton platform.

Equations

$${\mathcal{H}}_{\mathrm{bond}}({\mathbf{S}}_1,{\mathbf{S}}_2)$$

$$+c_3(S_1^z{S}_2^z{)}^2+c_4(S_1^x{S}_2^x+S_1^y{S}_2^y{)}^2$$

$$+c_5[S_1^z{S}_2^z(S_1^x{S}_2^x+S_1^y{S}_2^y)+H.c.]$$

$$+c_6(S_1^x{S}_2^y-S_1^y{S}_2^x{)}^2.$$

$$\begin{gathered} c_0=-5.58\mathcal{E},c_1=9.53\mathcal{E},c_2=-8.97\mathcal{E}, \\ {c}_3=1.27\mathcal{E},c_4=6.59\mathcal{E},c_5=-3.18\mathcal{E},c_6=5.04\mathcal{E}. \end{gathered}$$

$${\psi }_k\propto \frac{1}{\sqrt{N}}\sum _{j=1}^N{\mathrm{e}}^{\mathrm{i}kj}{S}_j^z{\psi }_0,$$

$${\psi }_k\propto \frac{1}{\sqrt{N}}\sum _{j=1}^N{\mathrm{e}}^{\mathrm{i}kj}[{\mu }_1{S}_j^z+{\mu }_2{S}_j^z(S_j^x\pm \mathrm{i}{S}_j^y)]{\psi }_0,$$

$${\psi }_{i_1{i}_2{i}_3\dots }\propto M_{{\alpha }_1{\alpha }_2}^{i_1}{M}_{{\alpha }_2{\alpha }_3}^{i_2}{M}_{{\alpha }_3{\alpha }_4}^{i_3}\dots ,$$

$$C_{\mathrm{N}\acute{\mathrm{e}}\mathrm{el}}(j)\equiv (-1{)}^j⟨S_0^z{S}_j^z⟩,$$

$$C_{\mathrm{string}}(j)=⟨S_0^z{\mathrm{e}}^{\mathrm{i}\pi (S_1^z+S_2^z+\dots S_{j-1}^z)}{S}_j^z⟩,$$

$$|\dots ,0\dots ,0,1,-1,0,\dots ,1,-1,1,0\dots ⟩,$$

Full text

Introduction. Rydberg states of matter attract high interest nowadays. Strongly excited atoms are macroscopic quantum objects highly susceptible to external fields that serve for bench-top studies of quantum effects at a large scale Gallagher (2005). Enhanced polarizability of Rydberg atoms results in efficient interactions making them a platform of choice for quantum simulations Saffman et al. (2010).

Exciton, the Coulomb interaction correlated electron-hole pair, emerges in semiconductors when an electron is promoted optically from the filled valence to an empty conduction band. It is a direct analogue of the hydrogen-like atom Rashba and Sturge (1982). Perfectness of the natural cuprous oxide crystals has not only made it possible to discover large-radius excitons in semiconductors Gross and Karrjew (1952) but has lead also to a recent breakthrough: demonstration of stable highly-excited excitonic Rydberg states with principal quantum number $n$ up to 25 Kazimierczuk et al. (2014). By contrast to Rydberg atoms excitons exist in the crystalline environment, which manifests itself not only in quantitative differences of the binding energy and exciton radii, but also in different selection rules for optical transitions Zielińska-Raczyńska et al. (2016); Heckötter et al. (2017); Semina (2018) making $p$-shell excitons with the orbital angular momentum $1$ active in single-photon processes, unusual fine structure of the energy spectrum Thewes et al. (2015), as well as broken symmetries Aßmann et al. (2016); Schweiner et al. (2017). All this provides flexibility to control the excitonic states by light.

Even more rich consequences of the non-zero orbital angular momentum are expected in exciton-exciton interactions in cuprous oxide Walther et al. (2018), which, just like for atoms, are of paramount importance for Rydberg excitons physics Kazimierczuk et al. (2014). By now, the interactions are scarcely studied, however, according to the recent theoretical predictions Walther et al. (2018), the coupling between the excitonic states crucially depends on the mutual orientations of their angular momenta.

Similarly to atomic physics, the interaction effects are expected to be strongest in ensembles of localized Rydberg excitons. Here we demonstrate that already in a one-dimensional chain of trapped Rydberg excitons in Cu2O the ground state corresponds to a topologically non-trivial spin order — the Haldane phase Haldane (1983a) — with diluted antiferromagnetic order and a gapped spectrum of elementary excitations. Such a topological phase is inherent to integer spin, in particular spin-$1$, chains with antiferromagnetic coupling. The hallmarks of the Haldane phase are the non-trivial edge states behaving akin spin-$1/2$ fermions as well as the presence of the hidden long-range “string” order despite the apparent lack of distant spin-spin correlations den Nijs and Rommelse (1989). Despite numerous theoretical proposals Xu et al. (2018), so far the Haldane phase has been only probed by the neutron scattering Lu et al. (1991) or heat conductivity measurements Sologubenko et al. (2008); Kawamata et al. (2007) in anisotropic magnetic materials.

Thus, the Rydberg excitons provide a highly desirable table-top solid state platform with direct optical access to individual excitons for both fundamental studies of hidden symmetries and topological orders and prospective quantum simulations Imambekov et al. (2003); Pollmann et al. (2012); Xu et al. (2018); Nayak et al. (2008); Chiu et al. (2016).

Chain of Rydberg excitons. We consider Rydberg excitons confined in a one-dimensional (1D) periodic array of traps, see Fig. 1. The traps for excitons could be created either optically utilizing the ac Stark shift of excitons in the structured light waves, similar to the optical lattices of cold atoms Bloch (2005) or in tailored semiconductor environment where the band gap energy can be controlled by applying the strain or external electrostatic potential Trauernicht et al. (1986). It is assumed that each trap is occupied by a single exciton, that is in a $p$-shell state with the angular momentum of the envelope function being equal to $1$. For simplicity, we neglect here exciton internal spin degrees of freedom, i.e., the spins of electrons and holes, and the spin-orbit interaction. We introduce the pseudovector angular momentum operator $\bm{S}_{j}=(S_{j}^{x},S_{j}^{y},S_{j}^{z})$ for $j$th trap. Hereafter, we use the term spin to denote $\bm{S}_{j}$ for brevity, $z$ is the chain axis. We assume that the traps are sufficiently well separated, thus the excitons in neighboring traps are coupled to each other by the van der Waals interaction only. Owing to strong, $R^{-6}$, decay of the potential with the distance between the traps $R$, it is sufficient to consider the nearest neighbor approximation. The Hamiltonian of the chain has the form ${\mathcal{H}}=\sum_{j=1}^{N-1}{\mathcal{H}}_{\rm bond}(\bm{S}_{j},\bm{S}_{j+1})\>,$ where $N$ is the number of traps equal to the number of excitons in the system, $\mathcal{H}_{\rm bond}$ is the neighbors coupling Hamiltonian. The system has a rotational symmetry around the $z$ axis. Thus, $\mathcal{H}_{\rm bond}$ is characterized by $7$ real constants Klümper et al. (1993) $c_{0},\ldots,c_{6}$ and can be presented as

[TABLE]

The microscopic calculation in Ref. Walther et al. (2018) confirms that all $7$ parameters are significant. Specifically, for the exciton with principal quantum numbers $n={12\ldots 25}$ one has Walther et al. (2018)

[TABLE]

where the common factor $\mathcal{E}$ is $n^{15}\times 10/(2\pi)~{}\rm mHz\times\mu m^{6}/R^{6}$. It is assumed that the trap size $d$ exceeds by far the exciton size but is much smaller than the distance between the traps $R$. In this situation the interaction Hamiltonian is independent of the trap size and geometry in the leading order in $d/R\ll 1$. The Hamiltonian is invariant to the change of parameters $c_{2}\to-c_{2},c_{5}\to-c_{5}$ which corresponds to reflection $z\to-z$ for every second spin. Importantly, the spin-spin coupling is mostly antiferromagnetic ($c_{1}>0$) and strongly anisotropic.

Since the Rydberg blockade prevents two particles from being close to each other Gallagher (2005); Saffman et al. (2010); Kazimierczuk et al. (2014), in what follows we neglect the exciton tunneling between the traps. The interplay of the interactions and tunneling can enrich the spin phases in bosonic systems Belemuk et al. (2017). The trap, however, can effect angular momentum state of the exciton giving rise to the anisotropic single particle contributions sup . We first disregard the anisotropy and discuss its effect in the end of the paper.

Ground and excited state energies. The chain Hamiltonian can not be diagonalized analytically for $N>3$. Instead, we first solved numerically for the lowest energy states in small finite chains ($N\leq 14$) with the periodic boundary conditions 111We considered only chains with even $N$, since the odd number is incompatible with antiferromagnetic order, resulting in slower numerical convergence..

The results of calculation are shown in Fig. 2. The ground state energy, shown in the inset, quickly converges to the value of $E_{0}\approx-3.51N\mathcal{E}$. The ground state is non-degenerate for periodic boundary conditions and has the total spin $\sum_{j}S_{j}^{z}=0$ . The system is gapped, with the excitation gap slowly decreasing with the size. The lowest excited state has zero total spin projection and the excitation energy $E_{g}\gtrsim 1.1\cal E$ (red open symbols in Fig. 2). The states with $\sum_{j}S_{j}^{z}=\pm 1$ have significantly larger excitation energies ${E_{g}^{\prime}}\gtrsim 6\cal E$ (blue filled symbols). In order to reveal the structure of the excited states we have compared the results of exact diagonalization with those obtained from the single-magnon variational approach Arovas et al. (1988); Bartel et al. (2003). Namely, we used the trial wavefunction

[TABLE]

for the first excited state, where $\psi_{0}$ is the numerically calculated ground state, and the wave vector has been set to $k=\pi$ sup . The variational energy is close to the exact one, which indicates that the excited state is well described by the ansatz Eq. (3). The large overlap (about $0.99$) between Eq. (3) and the lowest excited state has been also confirmed numerically. The lowest excited states with the $\pm 1$ total spin projection being degenerate due to the time-reversal symmetry were sought in the form

[TABLE]

with $k=0$ sup and $\mu_{1,2}$ being the variational coefficients. The energy of this trial state is close to the exact numerical result as well, cf. blue circles and diamonds in Fig. 2.

In addition to the exact diagonalization of small finite chains we have also employed the infinite time-evolving block decimation (iTEBD) approach  Vidal (2003, 2004, 2007). The advantage of this numerical technique is that it is capable to directly address infinite chains. It is based on the representation of the ground state $\psi_{0}$ in the matrix-product form Orús (2014)

[TABLE]

where $M$ is a certain 3-rd rank tensor, indices $i_{j}=0,+1,-1$ label the projections of the spin $j$ and $\alpha_{j}=1\ldots\chi$ are auxiliary indices. If the spins were independent, one could use $\chi=1$, so that the $\alpha$ indices are suppressed, $M_{\alpha_{j}\alpha_{j+1}}^{i}\to M^{i}$ and the state Eq. (5) would reduce to a simple product state. For $\chi>1$ the state Eq. (5) can describe quantum entanglement (i.e., correlations) between the spins. The iTEBD approach enables high-accuracy description of arbitrary gapped noncritical 1D systems with local Hamiltonians Vidal et al. (2003); Orús (2014) at relatively low computational cost. In our case the convergence of energy better than 1% has been already reached for the rank $\chi\lesssim 20$.

The iTEBD results for the infinite system are shown in Fig. 2 by dashed horizontal lines. The energies of both ground and excited states well agree with the corresponding results for the finite chains. The excited state energies in the infinite system were estimated using the same variational approximation Eqs. (3) and (4) with $\psi_{0}$ being now the ground state found from iTEBD. We have also calculated the dispersion of excitations $E_{g}(k)$ in the single magnon approximation sup . In agreement with the analysis of the finite system, the spin-0 magnon branch is indeed gapped with the minimum at the edge of the Brillouin zone, $k=\pi$, while the magnons with $\pm 1$ spin projection have the minimum energy at $k=0$.

Spin structure of the ground state. Now we proceed to the analysis of the spin-spin correlations in the ground state. The calculation has been performed for the infinite chain using the iTEBD technique. The results are shown in Fig. 3. We start with the analysis of the pair spin-spin correlations depending on the spin-spin distance $j$. The calculation demonstrates the presence of short-range antiferromagnetic Néel order,

[TABLE]

At large distances the spin correlations Eq. (6) vanish, as illustrated by the red dashed curve in Fig. 3 showing the exponential decay with the correlation length $\approx 7$. Hence, based only on the analysis of local spin-spin correlations one could conclude that the considered spin phase has no long-range order. However, this is not the case. Our main result is that the infinite chains of Rydberg excitons do possess a long-range string order, characterized by the non-vanishing correlator den Nijs and Rommelse (1989)

[TABLE]

blue circles in Fig. 3. The presence of the non-local order parameter $C_{\rm string}(j)$ can be interpreted as a result of spontaneous breaking of a certain $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry, hidden in the system Kennedy and Tasaki (1992a). Such string order in the absence of N eel order has been first revealed in the Afflect-Kennedy-Lieb-Tasaki (AKLT) model with ${\mathcal{H}_{\rm bond}}=\bm{S}_{1}\cdot\bm{S}_{2}+(\bm{S}_{1}\cdot\bm{S}_{2})^{2}/3$ Affleck et al. (1988); Kennedy and Tasaki (1992b). The AKLT model is exactly solvable by the matrix product state Eq. (5) with certain rank-2 matrices Kennedy and Tasaki (1992a). Similar anzatz has allowed us to obtain analytically the ground state energy with $E_{0}=-3.49N\mathcal{E}$ sup , which is very close to the numerically obtained value of $E_{0}=-3.51N\mathcal{E}$ shown in the inset of Fig. 2. The simultaneous presence of the string order and the absence of the Néel order, along with the presence of the non-degenerate gapped ground state are the clear fingerprints of the Haldane phase of spin-1 excitons Haldane (1983a, b). Physically, this indicates so-called diluted antiferromagnetism where the wavefunction of the ground state $\psi_{0}$ can be represented as a superposition of basic functions in the form

[TABLE]

where the significant part of excitons is in the $S^{z}=0$ state, and the spins $S^{z}=\pm 1$ can occur only in pairs, i.e., $+1,-1$ or $-1,+1$, while the pairs $+1,+1$ or $-1,-1$ are forbidden. Thus, the string correlator singles out “diluted antiferromagnetic order”, where neighboring spins are always opposite with the possible arbitrary number of zeroes in between. The fact that the pair $+1,-1$ or $-1,+1$ can occur at any arbitrary place of the chain is related to the formation of the topological Haldane phase in our system. Such diluted antiferromagnetic state is schematically illustrated in Fig. 1.

The shooting gun evidence for the Haldane phase is the presence of edge states in a finite chain with open boundary conditions behaving as spin-$1/2$ fermions den Nijs and Rommelse (1989); Pollmann et al. (2012). Our numerical calculations for $N\leqslant 14$ exciton chains have indeed confirmed that, in the case of open boundary conditions, the ground state corresponds to $4$ closely lying levels corresponding to the combinations of edge spins-$1/2$ slightly split due to the finite chain size sup .

Phase diagram. The Haldane phase is a robust generic feature of 1D spin-1 chains with antiferromagnetic nearest-neighbor interaction. For instance, it is known to exist in isotropic bilinear-biquadratic spin chains with ${\mathcal{H}}_{\rm bond}=-\cos\theta\bm{S}_{1}\cdot\bm{S}_{2}+\sin\theta(\bm{S}_{1}\cdot\bm{S}_{2})^{2}$ in a wide range of angles around $\theta=\pi$, including the spin-1 anisotropic Heisenberg model Schollwöck et al. (2008). In order to demonstrate, that the formation of the Haldane phase for the Rydberg excitons is not a coincidence, we have analyzed the structure of the ground state depending on the anisotropy of the trap shape and on the sign of the coupling, either ferromagnetic or antiferromagnetic. The trap anisotropy has been described by adding the single particle terms $\mathcal{H}_{1}=D\sum_{j}[(S_{j}^{z})^{2}-2/3]$ to the Hamiltonian sup . The coupling anisotropy was described by additional term $\delta c_{1}S_{1}^{z}S_{2}^{z}$ in the bond Hamiltonian Eq. (1). The structure of the ground state was determined from comparison of the spin-spin and string correlations at large distances. The results of this analysis are summarized in Fig. 4, and the correlation functions are given in the Supplementary Materials sup .

We start the discussion of the phase diagram Fig. 4 with the role of the coupling term $\delta c_{1}S_{1}^{z}S_{2}^{z}$. Clearly, for large positive $\delta c_{1}$ the spin-spin interaction becomes antiferromagnetic, $\langle S_{1}^{z}S_{2}^{z}\rangle=-1$, while for negative $\delta c_{1}$ the system is driven in the ferromagnetic phase with $\langle S_{1}^{z}S_{2}^{z}\rangle=1$. The anisotropic ferromagnetic order, indicated by pink area in Fig. 4, is characterized by the long-range spin-spin correlations with $\langle S_{1}^{z}S_{2}^{z}\rangle<1$. The Haldane phase is realized in the antiferromagnetic regime in the wide range of the trap anisotropy parameter $D$ provided that it is not large negative. The impact of the anisotropy has also a transparent interpretation: Large positive values of $D$ favor $S^{z}=0$ ground state first facilitating the formation of the diluted antiferromagnetic state and, ultimately, the non-magnetic state with $S^{z}_{j}\equiv 0$ at each site. Similarly to the spin-$1$ Heisenberg model even slight anisotropy $D<0$ switches the system into the antiferromagnetic state Chen et al. (2003). Large negative values of $D$ push down $S^{z}=\pm 1$ states rendering the exciton chain to the set of $1/2$-pseudospins and suppressing the Haldane phase.

Acknowledgements. We are grateful to M. Aßmann, M. Bayer, T. Pohl, M.A. Semina, and V. Walther for fruitful discussions.