Realizing and manipulating space-time inversion symmetric topological semimetal bands with superconducting quantum circuits

We have experimentally realized novel space-time inversion (P-T) invariant Z2-type topological semimetal-bands, via an analogy between the momentum space and a controllable parameter space in superconducting quantum circuits. By measuring the whole energy spectrum of system, we imaged clearly an exotic tunable gapless band structure of topological semimetals. Two topological quantum phase transitions from a topological semimetal to two kinds of insulators can be manipulated by continuously tuning the different parameters in the experimental setup, one of which captures the Z2 topology of the PT semimetal via merging a pair of nontrivial Z2 Dirac points. Remarkably, the topological robustness was demonstrated unambiguously, by adding a perturbation that breaks only the individual T and P symmetries but keeps the joint PT symmetry. In contrast, when another kind of PT -violated perturbation is introduced, a topologically trivial insulator gap is fully opened.

Symmetry and topology, as the two fundamentally important concepts in physics and mathematics, have not only manifested themselves in science, but also provided us profound understanding of arresting natural phenomena. Recently, topological gapless systems, such as Weyl semimetals [1][2][3][4][5] and a variety of Dirac semimetals [6][7][8][9][10] as well as Z 2 topological metals/semimetals [11][12][13], have significantly stimulated research interest. Analogous to that in gapped topological systems such as topological insulators and superconductors, the discrete symmetry that is rather robust against symmetry-preserved perturbations can enrich the topological physics of gapless systems as well. As is known, the discrete timereversal (T ), space-inversion (P ), and charge-conjugate (C) symmetries are fundamental and intriguing in nature. For examples, in high energy physics, any local quantum field theory must preserve the joint CP T symmetry, which is required by the unitarity and Lorentz invariance of the theory, and the source of CP violation still remains as one of seminal mysteries in the Standard Model. While in condensed matter systems, it is ubiquitous that P , T and C put the constraints on band structures and lead to new topological classifications of band theories [14][15][16]. Among various combinations of P , T and C, the joint P T symmetry actually inverses the space-time coordinates x µ → −x µ with µ = 0, 1, 2, 3 and x 0 = t, and therefore evidences itself to be fundamental and significant in physics. Very recently, a theory of P T -invariant topological gapless bands has rigorously been established [17], through revealing a profound connection between the P T symmetry and an elegant KO theory of algebraic topology [18]. On the other hand, it is noted that artificial superconducting quantum circuits possess high controllability [19][20][21][22][23], providing a power-ful and ideal tool to quantum-simulate and explore novel quantum systems [24][25][26][27][28], including topological ones.
In this Letter, we have realized experimentally the novel P T symmetry-protected topological semimetalbands that represent a gapless spectrum on a squarelattice, via an analogy between the momentum space and a controllable parameter space in superconducting quantum circuits. By measuring the whole energy spectrum of our system, we have imaged clearly an exotic tunable gapless band structure of topological semimetals, shown as nontrivial Z 2 -type Dirac points in momentum space. The two new distinct quantum phase transitions from a topological semimetal to two different insulators can be manipulated by continuously tuning the different parameters in the simulated effective Hamiltonian, particularly one of which exhibits the Z 2 topology in the P T semimetal via merging a pair of nontrivial Z 2 Dirac points. Furthermore, to demonstrate unambiguously the topological robustness of P T symmetry, a perturbation that breaks only the individual T and P symmetries is intentionally added, with the joint P T symmetry being still preserved. It is verified by experimental date that the Dirac points of the topological semimetal-bands still present under such perturbations, though the point positions and the band pattern are changed drastically. In a sharp contrast, when another kind of perturbation is added to break the P T symmetry in our experiment, the energy gap is fully opened and the Dirac points disappear completely, showing the essential role of P T symmetry underlying the topological robustness. All of these illustrate convincingly the topological protection of P T semimetals. Notably, the present work is the first experimental realization and manipulation of fundamental space-time inversion symmetric topological semimetal-bands (without individual T and P symmetries) in nature, which opens a window for simulating and manipulating topological quantum matter.
The physical manifestation of P T symmetry in band theories can simply be seen from the commutation relation as [Â, H] = 0, where H is the system Hamiltonian, and the joint P T symmetry is represented by an anti-unitary operatorÂ [29]. WhenÂ 2 = 1, the topological classification of band-crossing points in twodimensional band structures corresponds to the reduced KO group, KO(S 1 ) ∼ = Z 2 , which implies that there exist band-crossing points having nontrivial Z 2 topological charges in two dimensions [17]. Although the KO theory of algebraic topology seems to be rather abstruse for most physicists, the predicted topological band crossing points can be realized in a simple but representative dimensionless Hamiltonian, which is explicitly given by [17] with the P T being denoted byÂ = σ 3K , where σ j is the jth Pauli matrix, andK denotes the complex conjugate operation. When −1 < λ < 1, the model (1), which describes actually a topologically nontrivial spin(1/2)orbital quantum system in two dimension, has four bandcrossing points possessing the P T -protected Z 2 (ν Z2 = 1) topological charges. It is noted that although the model (1) has bothP = σ 3î andT =Kî symmetries withî being the inversion of the wave vector k, the topological stability of these band-crossing points merely requires the joint P T symmetry according to the P T invariant topological band theory, namely, the T /P -symmetry is allowed to be broken individually while the P T topological protection still remains. Experimental demonstration of this new kind of symmetry protected topological gapless band will significantly deepen our understanding of topological quantum matter. However, there are several big challenges that hinder the realization and investigation of the topological properties of this kind of Hamiltonians in real condensed matter systems. The first is how to synthesize the materials with a designated Hamiltonian. Secondly, even if one is fortune enough to have such kind of real materials, it seems extremely hard to tune the parameters continuously for studying fruitful topological properties including various topological quantum phase transitions. Moreover, it is quite difficult in experiments to directly image the whole momentum-dependent electronic energy spectrum of a bulk condensed matter system, noting that only a part of electronic spectra (or information of Fermi surfaces/points) may be inferred from the angle-resolved photoemission spectroscopy data (or quantum oscillation measurements). Therefore, it is imperative and important as well as significant to use artificial quantum systems like superconducting quantum circuits to simulate H(k) faithfully and to explore topological properties of  Figure 1: Experimental scheme for the realization of the lattice Hamiltonian. a, States |2 and |1 of a transmon are used as the energy levels of an artificial spin-1/2 particle, whose three components may be denoted by the three Pauli matri-cesσ1,2,3. |0 is chosen as an ancillary level to probe the eigenvalues of a Hamiltonian. Microwaves with various frequencies, phases, and amplitude are applied for the construction of a semimetal Hamiltonian and circuit QED readout, respectively [30]. b, The constructed Hamiltonian is implemented with modulation of microwave amplitude, frequency, and phase, mapping to the momentum space of a square lattice.
the system. Below we will realize the Hamiltonian of Eq.(1) in the parameter (analogous to the momentum) space via implementing a kind of fully controllable quantum superconducting circuits, such that the band structure can directly be measured over the whole first Brillouin zone (BZ) of square lattices, enabling us to demonstrate the unique topological nature of the corresponding semimetal-bands and to clearly visualize some crucial properties.
The superconducting quantum circuits used in our experiment consist of a superconducting transmon qubit embedded in a three dimensional aluminium cavity [31][32][33][34][35][36]. The transmon qubit, which is composed of a single Josephson junction and two pads (250 µm × 500 µm), is patterned using standard e-beam lithography, followed by double-angle evaporation of aluminium on a 500 µm thick silicon substrate. The thicknesses of the Al film are 30nm and 80nm, respectively. The chip is diced into 3 mm × 6.8 mm size to fit into the 3D rectangular aliminium cavity with the resonance frequency of TE101 mode 9.053 GHz. The whole sample package is cooled in a dilution refrigerator to a base temperature 30 mK. The dynamics of the system is identical to an artificial atom located in a cavity which has been extensively discussed as a circuit QED [31,33,37]. We designed the energy level of the transmon qubit to let the system work in the dispersive region. The quantum states of the transmon qubit can be controlled by microwaves. Inphase quadrature (IQ) mixers combined with 1 GHz arbitrary wave generator (AWG) are used to adjust the amplitude, frequency, and phase of microwave pulses. To read out qubit states, we use ordinary microwave heterodyne setup. The output microwave is pre-amplified by HEMT at 4 K stage in the dilution refrigerator and further amplified by two low noise amplifiers in room temperature. The microwave is then heterodyned into 50 MHz and collected by ADCs. The readout is performed with so called "high power readout" scheme [38]. By sending in a strong microwave on-resonance with the cavity, the transmitted amplitude of the microwave reflects the state of the transmon due to the non-linearity of the cavity QED system [39].
According to the circuit QED theory, the coupled transmon qubit and cavity exhibit anharmonic multiple enengy levels. In our experiments, we use the lowest three energy levels, as shown in Fig.1a, namely, |0 , |1 , and |2 . The two states |2 and |1 behave as an artificial spin-1/2 particle, whose three components may be denoted by the three Pauli matrices σ 1,2,3 which can couple with the microwave fields. |0 is chosen as an ancillary level to probe the energy spectrum of the simulated system. The transition frequencies between different energy levels are ω 10 /2π = 7.17155 GHz, ω 21 /2π = 6.8310 GHz, respectively, which are independently determined by saturation spectroscopies. The energy relaxation time of the qubit is T 1 ∼ 15µs, the dephasing time is T * 2 ∼ 4.3µs. When we apply microwave drive along x, y, and z directions, the effective Hamiltonian of the qubit in the rotating frame (Fig.1b) may be written as ( = 1 for brevity) where Ω 1 (Ω 2 ) corresponds to the frequency of Rabi oscillations along X (Y) axis on the Bloch sphere, which is continuously adjustable by changing the amplitude and phase of microwave applied to the system. Ω 3 = ω 21 − ω, is determined by the detuning between the system energy level spacing ω 21 and microwave frequency ω. By carefully designing the waveform of AWG, we can control the frequency, amplitude, and phase of microwave. In our experiment, we first calibrated the parameters Ω 1 , Ω 2 , and Measured energy spectrum of a typical spacetime inversion invariant topological semimetal. a, Threedimensional plot of the band structure of spectroscopy measurement. By tuning the driving amplitude, frequency, and phase gradually, we image the band structure of the system in the momentum space point by point. b, Magnitude of energy gap obtained from direct measurements of the energy spectrum of the system as function of kx and ky in the first BZ. Four nontrivial Z2-type Dirac points located inside the bright regions can be observed at (0, ±π/2), (π, ±π/2), in a full agreement with the theoretical prediction. Ω 3 using Rabi oscillations and Ramsey fringes, and then designed the microwave amplitude, frequency and phase to let Ω 1 = 0, Ω 2 (k x ) = Ω sin k x , Ω 3 (k y ) = λΩ + Ω cos k y , with Ω = 10 MHz being chosen as the energy unit.
Exploiting the analogy between the above parameter space of our system and the k-space of a lattice Hamiltonian system, we now have Eq.(1) exactly. It is worth to mention that λ plays a crucial role in the realization of the P T invariant topological phase transition. To examine the band structure, we first set λ = 0 and measured the entire energy spectrum of the system over the first BZ, as shown in Fig. 2. In our experiment, for a given (k x , k y ) ∈ [−π, π)×[−π, π), we actually measured the resonant peak of microwave absorption, and determined the frequency of the resonant peak as a function of k x and k y [30]. A key feature of the P T invariant topological semimetal, which is the existence of nontrivial Z 2 -type Dirac points yielded by crossing bands, is clearly seen in Fig.2b. These are the directly imaged Dirac cones in the experiments, indicating that we have successfully realized the topological semimetal that preserves the P T symmetry. In addition, the positions of the Dirac points (Fig.2a) locate at (π, ±π/2) and (0, ±π/2), agreeing well with the theoretical calculation of Eq.(1) with λ = 0.
Remarkably, the present fully tunable experimental setup can also be exploited to examine the P T -protected topological stability of the Z 2 nontrivial band crossing points from the following aspects. First we check the topological stability of these band crossing points of nontrivial Z 2 charges. From the topological band theory, each of them should be stably present under whatever perturbations that preserve the joint P T symmetry and do not mix one point with another, while the individ-Gap a b Figure 3: Symmetry-related topological features of the Dirac points for two different but representative kinds of perturbations. a, When H ′ 1 = ησ2 is added with η = 0.5 in unit of Ω, which breaks both T and P but preserves the P T symmetry, Dirac-like points still exist, though the gapless point positions are shifted (marked by the green square) and the band pattern is distorted drastically, showing the robust of the topological nature protected by the P T symmetry. Top and bottom panels correspond respectively to the cases of η = 0 and η = 0.5 on the plane of ky = π/2. The bright yellow and dashed green lines denote the experimental data and theoretical calculations from Eq.(1) with H ′ 1 being added, respectively. b, Whenever the P T symmetry is broken by adding the term H ′ 2 = εσ1 with a constant ε (= 0.5Ω), a gap is fully opened. Here λ = 0 for both (a) and (b).
ual P and T symmetries may be violated at the same time [17]. In this experiment, by introducing the perturbation H ′ 1 = ησ 2 (with η = 1/2 Ω being a constant) to the system. Now the parameter of σ 2 reads Ω 2 (k x ) = Ω(sin k x + 1/2), which breaks both P and T but preserves P T , it is observed that although the band structure is distorted dramatically, and the positions as well as neighborhood geometries of band-crossing points are changed significantly, these band-crossing points are persistently present in the first BZ without opening any gap, as seen clearly from Fig.3a, being perfectly consistent with the aforementioned facts of the topological band theory. On the other hand, however, when another kind of perturbation H ′ 2 (k) = εσ 1 (e.g., a constant ǫ ∼ 0.5Ω ) is introduced to the original Hamiltonian, H = Ω/2 σ 1 + Ω sin k x σ 2 + (λΩ + Ω cos k y )σ 3 , it is clear that the P T symmetry is violated, since such perturbations break P but preserves T . Accordingly the topological protection, which requires the P T symmetry, is discharged [17]. In agreement with the theoretical prediction, a trivial insulating gap is observed to be fully opened, as shown in Fig.3b [40].
We now turn to examine the Z 2 nature of the topological charge, utilizing the fully tunable advantage of our setup. The spectroscopic data are shown in Fig.4a for representative values of λ at each stage of the whole process of merging and annihilation of the Z 2 bandcrossing points. According to general principles of topological band theory, merging two ν Z2 = 1 band-crossing points nucleates a band crossing point of trivial topological charge (ν Z2 = 2 ≡ 0 mod 2), which can be gapped  Figure 4: Quantum phase transitions from a topological gapless semimetal to a gapped insulator as changing parameter λ. a, Spectroscopy at kx ≈ 0 for various λ. From right to left λ are 0, 0.5, 1 and 1.5, respectively. It is seen that when λ is increased from 0 to 1, then larger than 1, the number of Dirac-like points decreases from 4, to 2, then to 0, where the gap gradually is opened, demonstrating that a topological P T invariant semimetal phase transits to a normal insulator phase. b, Magnitude of minimum energy gap Eg in the first Brillouin zone as a function of λ, as predicted theoretically from Eq.(1).
out even though the P T symmetry is still preserved [17]. As shown in Fig.4b, we continuously increase the parameter λ from 0 to 2. Starting from λ = 0, where two band-crossing points are well separated at k y = π/2 and −π/2, respectively, in the one-dimensional subsystem with k x = 0, the two band-crossing points are gradually moving closer and closer to each other (with regard to their distances to the BZ boundaries) when λ is increased smoothly, then they are merged to be a new band-crossing point at the edge of the first BZ for λ = 1, which should be a topologically trivial point according to the topological band theory as mentioned above. Indeed, when λ is further increased to be bigger than 1, it is observed that the band crossing point of a trivial topological charge is gapped out, leading to a topologically trivial insulator that has even the P T symmetry [41], which verifies the aforementioned theoretical prediction.
To summarize, we have reported the first experimental realization and manipulation of fundamental space-time inversion invariant topological semimetal bands possessing neither T nor P symmetry. The non-trivial bulk topological band structures of P T symmetry have directly been imaged with superconducting quantum circuits. Moreover, two exotic topological quantum phase transitions have been observed for the first time. The present work is expected to stimulate a huge experimental and theoretical interest on various P T symmetric topological metals/semimetals, paving the way for quantum-simulating novel topological quantum materials.
Acknowledgments This work was partly supported by the the NKRDP of China (Grant No. 2016YFA0301802),