Machine Learning Topological Phases with a Solid-state Quantum Simulator

TL;DR

This paper demonstrates that convolutional neural networks can be trained on experimental data from a solid-state quantum simulator to accurately identify three-dimensional chiral topological insulator phases, showcasing machine learning's potential in topological phase detection.

Contribution

It introduces a novel application of machine learning, specifically CNNs, for experimental identification of topological phases in solid-state quantum systems, advancing the field's methodology.

Findings

CNNs successfully identify topological phases from experimental data

Machine learning enhances detection of topological phenomena

Experimental validation of CNNs in quantum materials

Abstract

We report an experimental demonstration of a machine learning approach to identify exotic topological phases, with a focus on the three-dimensional chiral topological insulators. We show that the convolutional neural networks---a class of deep feed-forward artificial neural networks with widespread applications in machine learning---can be trained to successfully identify different topological phases protected by chiral symmetry from experimental raw data generated with a solid-state quantum simulator. Our results explicitly showcase the exceptional power of machine learning in the experimental detection of topological phases, which paves a way to study rich topological phenomena with the machine learning toolbox.

Equations

$$\mathcal{H}_{\mathbf{k}}=\left(\begin{array}[]{ccc}0&q_{1}(\mathbf{k})-iq_{2}(\mathbf{k})&0\\ q_{1}(\mathbf{k})+iq_{2}(\mathbf{k})&0&q_{3}(\mathbf{k})+iq_{0}(\mathbf{k})\\ 0&q_{3}(\mathbf{k})-iq_{0}(\mathbf{k})&0\end{array}\right)$$

$${\theta }^{(\lambda )}=\frac{1}{4\pi }{\int }_{\text{BZ}}{ϵ}^{\mu \nu \tau }{A}_{\mu }^{(\lambda )}{\partial }_{k^{\nu }}{A}_{\tau }^{(\lambda )}{d}^{3}k$$

Full text

The discovery of topological phases has transformed our modern understanding of quantum phases of matter von1986quantized ; Hasan2010Colloquium ; Qi2011Topological . Unlike conventional phases, topological phases do not fit into the paradigm of symmetry breaking Lifshitz2013Statistical . Instead, distinct topological phases exist within each symmetry class Kitaev2009Periodic ; Schnyder2008Classification ; Ryu2010Topological ; Chiu2016Classification . In the experiment, the identification of topological phases typically relies on measuring certain properties that have a topological origin. For example, to identify the three-dimensional (3D) $\mathbb{Z}_{2}$ topological insulators (TIs), one can measure their robust surface states (e.g., Dirac cones) Hsieh2008topological ; Chen2009Experimental . Yet, for other topological phases, it can be technologically challenging to provide a nonambiguous experimental identification.

Machine learning, or more broadly speaking artificial intelligence, may provide new promising approaches to solve such challenges. In fact, its ideas and techniques have recently been explored in condensed matter physics, giving rise to a fresh research frontier of machine learning phases of matter Carleo2017Solving ; Deng2017Machine ; Gao2017Efficient ; Deng2017Quantum ; Wang2016Discovering ; Carrasquilla2017Machine ; Nieuwenburg2017Learning ; Chng2017Machine ; Zhang2017Quantum ; Zhang2018Machine ; ZhangYi2017ML ; Broecker2017Machine ; Wetzel2017Unsupervised ; Hu2017Discovering . Particularly, theoretical proposals for using supervised learning to identify topological phases have been proposed and have attracted considerable attention across multiple communities Zhang2017Quantum ; Zhang2018Machine ; ZhangYi2017ML . A striking advantage of this approach is that the neural networks can be trained to detect topological phases directly from raw data of local observables, such as spin textures or local correlations, though the topological properties are intrinsically nonlocal. Despite all this exciting theoretical progress, the power of machine learning in studying quantum phases of matter, especially topological phases, has not yet been demonstrated in laboratory.

In this Letter, we carry out such an experiment and demonstrate that neural networks can be trained to successfully identify topological phases, even with a very small portion of the experimentally generated raw data. By using the electron spins in a diamond nitrogen-vacancy (NV) center, we implement a model Hamiltonian for 3D chiral TIs and obtain the corresponding single-particle density matrices in the momentum space through quantum state tomography Yuan2017Observation . Using the measured density matrices as input data, we find that a trained 3D convolutional neural network (CNN) can correctly identify distinct topological phases. We emphasize that this approach only requires a small portion of data samples, which would significantly reduce the cost in practical experiments DataNumber . Our results demonstrate the experimental feasibility of the machine learning approach to study topological phases, which opens the door for experimental detection of exotic quantum phases with the machine learning toolbox.

We focus our discussion on 3D chiral TIs, which have attracted extensive attention in recent years Hosur2010Chiral ; Neupert2012Noncommutative ; Shiozaki2013Electromagnetic ; Wang2014Probe ; Hasebe2014Chiral ; Wang2015Quantized . Unlike the well-known $\mathbb{Z}_{2}$ TIs, chiral TIs break both time-reversal and particle-hole symmetry, but preserve chiral symmetry and are characterized by a $\mathbb{Z}$ topological invariant Kitaev2009Periodic ; Schnyder2008Classification ; Ryu2010Topological ; Chiu2016Classification . Due to chiral symmetry, chiral TIs exhibit a zero-energy flat band when the total number of bands is odd. Strong interactions may lead to nontrivial many-body effects in 3D topological flat bands Weeks2012Flat ; Neupert2012Noncommutative ; Wang2014Probe and chiral TIs could provide an ideal setting to study possible fractionalization in 3D Maciejko2010Fractional ; Swingle2011Correlated ; Wang2014Classification . In our experiment, we detect the topological flat band via energy dispersion measurements. Topological phase transitions and the accompanied gap closings are also observed by changing a control parameter in the Hamiltonian.

We consider a minimal three-band Hamiltonian for 3D chiral TIs $H=\sum_{\mathbf{k}\in\text{BZ}}\Psi_{\mathbf{k}}^{\dagger}\mathcal{H}_{\mathbf{k}}\Psi_{\mathbf{k}}$, where $\Psi_{\mathbf{k}}^{\dagger}=(c_{\mathbf{k},1}^{\dagger},c_{\mathbf{k},0}^{\dagger},c_{\mathbf{k},-1}^{\dagger})$ with $c_{\mathbf{k},\mu}^{\dagger}$ creating a fermion at momentum $\mathbf{k}=(k_{x},k_{y},k_{z})$ in the orbital (spin) state $\mu=1,0,-1$ and the summation is over the Brillouin zone (BZ); the single-particle momentum-resolved Hamiltonian is Neupert2012Noncommutative ; Wang2014Probe :

[TABLE]

with momentum-dependent functions $q_{1}(\mathbf{k})=t\sin k_{x}$, $q_{2}(\mathbf{k})=t\sin k_{y}$, $q_{3}(\mathbf{k})=t\sin k_{z}$, and $q_{0}(\mathbf{k})=t(\cos k_{x}+\cos k_{y}+\cos k_{z}+h)$ where $t=1$ is set to be the energy unit and $h$ is a dimensionless control parameter. The Hamiltonian has a chiral symmetry $SH_{\mathbf{k}}S^{-1}=-H_{\mathbf{k}}$ specified by the unitary transformation $S\equiv\text{diag}(1,1,-1)$. The topological properties for each band can be characterized by a topological invariant $\theta^{(\lambda)}$ TopInv , which has measurable consequences in 3D TIs, such as the magneto-electric polarization Qi2008Topological ; Essin:2009ui ; Shiozaki2013Electromagnetic ; Wang2015Quantized . For the middle band, direct calculations show that $\theta^{(m)}/\pi=0,1$ and $-2$ for $|h|>3$, $1<|h|<3$, and $|h|<1$, respectively.

We experimentally realize this model Hamiltonian and probe its topological characteristics using a single NV center in a diamond sample at room temperature [Fig. 1(a)]. Our experimental setup is based on a home-built confocal microscope with an oil-immersed objective lens. The three spin (orbital) degrees of freedom can be encoded into the ground state $|1\rangle,|0\rangle,|-1\rangle$ corresponding to $m_{s}=1,0,-1$ electron spin states of the NV center [Fig. 1(b)]. A magnetic field is applied along the NV symmetry axis to polarize nearby spins and remove the degeneracy between the $m_{s}=1$ and $m_{s}=-1$ states. With no inter-particle interactions, the Hamiltonian $H$ is diagonal in the momentum space, so different momentum components in each band are decoupled. One can thus measure each momentum component separately in the experiment and probe the properties of the entire topological band by collecting individual measurements.

From Eq. (1), one observes that the Hamiltonian $\mathcal{H}_{\mathbf{k}}$ has no diagonal terms and no direct coupling between states $|1\rangle$ and $|-1\rangle$, so we can implement the Hamiltonian at each $\mathbf{k}$ by two on-resonance microwaves as shown in Fig. 1(b) SM . In our experiment, we do not have a physical analog for the lattice momentum $\mathbf{k}$. Instead, an adiabatic passage is used to realize the eigenstate of each band of the Hamiltonian $\mathcal{H}_{\mathbf{k}}$ at some parametric point $\mathbf{k}$. Fig. 1(c) shows such an experimental sequence. The spin state is first initialized to the state $|0\rangle$. To probe the properties of the lower band, we first adiabatically prepare the state $|\Psi^{(l)}_{0}\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle+i|{-1}\rangle\right)$, which is the ground state of the Hamiltonian $\mathcal{H}_{\mathbf{k}}$ at $\mathbf{k}=(0,0,0)$. Then ground states at other $\mathbf{k}$ points can be attained by adiabatically tuning the microwaves to the final Hamiltonian $\mathcal{H}_{\mathbf{k}}$ SM . For the middle band, the state $|\Psi^{(m)}_{0}\rangle=|1\rangle$ is first prepared, which is the middle eigenstate of the Hamiltonian at $\mathbf{k}=(0,0,0)$, before tracing a similar adiabatic passage to reach any other states of the flat band. Interestingly, the middle-band states are effectively the dark state of the Hamiltonian $\mathcal{H}_{\mathbf{k}}$ at all $\mathbf{k}$ points, which offers an alternative interpretation for the macroscopic flat band. Thus, the adiabatic procedure to reach any middle-band eigenstates resembles the stimulated Raman adiabatic passage (STIRAP). It is intriguing that the energy band composed of dark states can be topologically nontrivial, as we demonstrate explicitly later by learning the topological invariants and probing of phase transitions. The topological gap between different bands guarantees the feasibility of the above adiabatic procedure, and in order to remain as adiabatic as possible, we ramp the microwaves more slowly when the gap is smaller. This also enables us to probe the system close to topological phase transitions.

After the adiabatic passage, we perform quantum state tomography for the qutrit system via measurements in eight different sets of bases, and retrieve the full density matrix via a maximum likelihood estimation SM . Through these tomography measurements on a momentum grid, we obtain full information about the corresponding band. Figs. 1(d) and 1(e) show histograms of the retrieved state fidelity $F_{\mathbf{k}}=\langle\Psi_{\mathbf{k}}|\rho_{\mathbf{k}}|\Psi_{\mathbf{k}}\rangle$ for the middle band on a $10\times 10\times 10$ regular grid, where $\Psi_{\mathbf{k}}$ is the theoretically obtained eigenstate and $\rho_{\mathbf{k}}$ is the experimentally measured density matrix, which serves as the input data for the neural network.

In the experiment, we use a pre-trained CNN to predict the topological invariant. This is in the setting of supervised learning, where the essential idea is the same as how one trains artificial neural networks to identify pictures of dogs and cats. A CNN is first trained with numerically sampled data (density matrices) from $h\in[-5,5]$, avoiding the intervals $[1.8,2.2]$ and $[3.8,4.2]$ where experimental data lie. Correct values of the topological invariant $\theta^{(m)}/\pi$ are numerically computed and assigned to the corresponding training samples. Fig. 2(a) shows a schematic of the CNN we used, where the input data are the state density matrices on a $10\times 10\times 10$ grid, and the output reports the classification probabilities $P(\theta^{m}/\pi=0,1,-2)$ for each possible phase SM . Fig. 2(b) shows the training and validation accuracies with increasing iterations. Both the training and validation accuracies increase rapidly at the beginning of the training process and then saturate at a high value ($\approx 98\%$). Near the topological phase transition points, wrong classifications may occur due to large quantum fluctuations, as observed in the inset of Fig. 2(b).

After training, we use the CNN to predict the topological phases of our experimental data at $h=2$ and $h=4$. We find that, when using the entire set of measured density matrices on the $10\times 10\times 10$ grid, the trained CNN can identify the topological phase with a nearly unity probability. A more interesting scenario arises when part of the data is missing or not accessible. This is inspired by the fact that in the case of learning to recognize dogs and cats, the trained network can sometimes successfully recognize images that are partially destroyed. As shown in Fig. 2(c), it is remarkable that the trained CNN can successfully identify (with a probability $\gtrsim 90\%$) different topological phases even with less than $10\%$ of the experimental data.

We remark that the machine learning approach outshines conventional methods in several aspects. First, the neural network needs not to know the precise formula for different topological invariants. What is necessary is the labelling of the training data. Thus, this approach has a broader applicability in general. Especially, it carries over to the cases of poorly understood phases, where the precise order parameters or topological characterization are not known a priori. Second, in terms of data efficiency, CNN beats conventional methods that use discrete integration, which is not directly applicable if data are not supplied on a regular grid SM . In practice, this would reduce the experimental cost dramatically since much fewer data samples are required. Third, machine learning tools have the advantage of directly using measurement results in the form of mixed states, whereas conventional methods typically rely on topological invariants commonly defined in terms of pure states. In addition, we mention that CNN can be used to identify topological phase transition points, despite the fact that the training data are sampled far away from the transition boundaries SM .

As mentioned before, an interesting property of this three-band chiral TI is the presence of a three-dimensional macroscopic flat band. To probe this dispersionless band, we measure the energy of the middle band via state tomography. Left panels of Fig. 3(a) and (b) show the measured energy spectrum on a $10\times 10$ grid in the $(k_{x},k_{y})$ plane for the momentum layer $k_{z}=\pi$. For other layers of $k_{z}$, we also observe the flat middle band, confirming the 3D nature of the macroscopic flat band. For $h=2$, we can clearly see a band gap, while for $h=3$, the gap closes at $(k_{x},k_{y},k_{z})=(\pi,\pi,\pi)$, signaling the topological phase transition.

To further probe these topological phase transitions, we measure the energies of the lower band and the middle band along a path connecting the high symmetry points in the momentum space for various values of the parameter $h$. Fig. 3(c) and (d) show the measured energy spectrum along the smooth path $\Gamma$-$X$-$M$-$\Gamma$-$R$-$X$-$M$-$R$, where $\Gamma,X,M,R$ are high-symmetry momentum points as illustrated in Fig. 3(e). By changing the parameter from $h=0$ to $h=3.5$, we see the gap between the lower band and the middle band closes at the $M$ point for $h=1$ [Fig. 3(c)] and at the $R$ point for $h=3$ [Fig. 3(d)], consistent with our theoretical predictions. The gapless points are deliberately avoided due to the breakdown of adiabaticity, but the trend of gap closing can be clearly seen. The topological phase transition points can also be successfully predicted by the trained CNN SM .

In conclusion, we have experimentally demonstrated a machine learning approach to identify topological phases with a solid-state quantum simulator. We show that a CNN can be trained to successfully identify topological phases, even with a minimal number of data samples. This demonstrates a striking advantage of the machine learning approach, which could significantly reduce the experimental cost. Since the CNN approach does not require knowing the formula for topological invariants, it generalizes directly to other exotic phases that are less well-understood.

Although we have explicitly focused on interaction-free topological phases, extensions including the effect of interaction are possible. One approach is to embed the interaction-free Hamiltonian into the parameter space of multiple qutrits and introduce interactive couplings between the qutrits, similar to the method in Ref. Roushan2014Observation . This represents an exciting possibility to probe the effect of interactions in 3D topological flat bands in such an interacting spin system. The use of modern machine learning tools in this setting may offer new insights in understanding topological phases with strong interactions.