One-dimensional electronic systems can support exotic collective phases because of the enhanced role of electron correlations. We describe the experimental observation of a series of quantized conductance steps within strongly interacting electron waveguides formed at the lanthanum aluminate-strontium titanate (LaAlO 3 /SrTiO 3 ) interface. The waveguide conductance follows a characteristic sequence within Pascal's triangle: (1,3,6,10,15, …) ˙e2 /h, where e is the electron charge and h is the Planck constant. This behavior is consistent with the existence of a family of degenerate quantum liquids formed from bound states of n = 2, 3, 4, … electrons. Our experimental setup could provide a setting for solid-state analogs of a wide range of composite fermionic phases.

I n one-dimensional (1D) systems of interacting fermions (1)(2)(3)(4), correlations are enhanced relative to higher dimensions. A variety of theoretical approaches have been developed for understanding strongly correlated 1D systems, including Bethe ansatz and density matrix renormalization group (DMRG) (5). Experimentally, degenerate 1D gases of paired fermions have been explored in ultracold atom systems with attractive interactions (6). In the solid state, attractive interactions have been engineered in carbon nanotubes by means of a proximal excitonic pairing mechanism (7). Electron pairing without superconductivity, indicating strong attractive interactions, has been reported in low-dimensional SrTiO 3 nanostructures (8,9). However, bound states of three or more particles-analogs of baryon phases (10)-have been observed only in few-body bosonic systems (11).

SrTiO 3 -based electron waveguides can provide insight into strongly interacting fermionic systems. The total conductance through an electron waveguide is determined by the number of extended subbands (indexed by orbital, spin, and valley degrees of freedom) available at a given chemical potential m (12,13). Each subband contributes one quantum of conductance efoot_2 /h with transmission probability T(m) to the total conductance G = (e 2 /h) P i T i (m) (14). Quantized transport was first observed in III-V quantum point contacts (15,16) and subsequently in 1D systems (17)(18)(19). Quantized con-duction within 1D electron waveguides was recently demonstrated within LaAlO 3 /SrTiO 3 heterostructures (9). A unique aspect of this SrTiO 3 -based system is the existence of tunable electron-electron interactions (20) that lead to electron pairing and superconductivity (8).

Here, we investigated LaAlO 3 /SrTiO 3 -based 1D electron waveguides that are known to exhibit quantized ballistic transport as well as signatures of strong attractive electron-electron interactions and superconductivity (8,9,20). Fabrication details are described in (21). More than a dozen specific devices have been investigated. Parameters and properties for seven representative devices (devices 1 to 7) are given in table S1.

The conductance of these electron waveguides depends principally on the chemical potential m and the applied external magnetic field B (Fig. 1A). The chemical potential is adjusted with a local side gate V sg (9); for most experiments described here, the external magnetic field is oriented perpendicular to the LaAlO 3 /SrTiO 3 interface: B ¼ B z ẑ. Quantum point contacts formed in semiconductor heterostructures (15,16) exhibit conductance steps that typically follow a linear sequence: 2 × (1, 2, 3, 4, …) ⋅ e 2 /h, where the factor of 2 reflects the spin degeneracy. In an applied magnetic field, the electronic states are Zeeman-split, and they resolve into steps of (1, 2, 3, 4, …) ⋅ e 2 /h. In contrast, here we find that for certain values of magnetic field, the conductance steps for LaAlO 3 /SrTiO 3 electron waveguides follow the sequence (1, 3, 6, 10, …) ⋅ e 2 /h, or G n = n(n + 1)/2 ⋅ e 2 /h. As shown in Fig. 1B, this sequence of numbers is proportional to the third diagonal of Pascal's triangle (Fig. 1C, highlighted in red).

In order to better understand the origin of this sequence, it is helpful to examine the transconductance dG/dµ and plot it as an intensity map as a function of B and µ. Transconductance maps for devices 1 to 6 are plotted in Fig. 2. A peak in the transconductance demarcates the chemical potential at which a new subband emerges; these chemical potentials occur at the minima of each subband, and we refer to them as subband bottoms (SBBs). The peaks generally shift upward as the magnitude of the magnetic field is increased, sometimes bunching up and then again spreading apart. We observe many of the same features that were previously reported in 1D electron waveguides in LaAlO 3 /SrTiO 3 (9), such as electron pairing and re-entrant pairing, which indicate the existence of electron-electron interactions. Near a special value of the magnetic field, multiple subbands lock, and the total conductance as a function of chemical potential follows a Pascal series that is quantized in units of e 2 /h (see the labeled conductance plateaus in Fig. 2A).

Our approach to understanding the transport results described above begins with a single-particle description and incorporates interactions when the original description breaks down. Outside of the locked regions, the system is well described by a set of noninteracting channels, which places strong constraints on the theory of the locked regions. Any theory of the locked phases would need to explain the locking of the transconductance peaks as well as quantized conductance steps away from the locked regime.

Our single-particle description excludes interactions but takes into account the geometry of the electron waveguide that produces the underlying subband structure. The singleparticle picture has four components: confinement of electrons in the (i) vertical and (ii) lateral directions by the waveguide, and an external magnetic field that affects the electrons via the (iii) Zeeman and (iv) orbital effects. The intersection of more than two SBBs requires a special condition to be satisfied in the singleparticle model. The degeneracy requirement for obtaining the Pascal series (i.e., the crossing of 1, 2, 3, 4, … SBBs) is satisfied by a pair of ladders of equispaced levels. Indeed, a pair of ladders of equispaced levels is naturally produced by a waveguide with harmonic confinement in both vertical and lateral directions. In the presence of Zeeman interactions, the waveguide Hamiltonian can be written as 9), where m à x , m à y , and m à z are the effective masses along the x, y, and z directions; w y and w z are frequencies associated with parabolic transverse confinement in the lateral (y) direction and half-parabolic confinement in the vertical (z > 0) direction, respectively; g is the Landé factor; µ B is the Bohr magneton; and s = ±½ is the spin quantum number. Eigenenergies corresponding to the SBBs are given by

where the electron eigenstates |n z , n y , si are indexed by the orbital quantum numbers n z and n y and spin quantum number s, ħ is the Planck constant divided by 2p, and

p is the magnetic field-dependent frequency associated with parabolic confinement of the electron in the lateral direction (calculated from the bare frequency w y and the cyclotron frequency

). To obtain two equispaced ladders of states, we use the states associated with W for the first ladder and the states associated with w z , split by the Zeeman splitting, for the second ladder. The Pascal series is produced by the "Pascal condition": W = 4w z = 2gµ B B z /ħ. This condition requires fine-tuning of both the magnetic field B z and the geometry of the waveguide (w y /w z ). Meeting this condition results in crossings of increasing numbers of SBBs at a unique Pascal field B Pa . By fitting the SBB energies given by Eq. 2 to experimental data, we are able to generate a peak structure (Fig. 3A) that is in general agreement with and has the same sequence of peak crossings as the experimentally observed transconductance. (Estimates for the single-particle model parameters are listed in table S1.) By intentionally detuning the parameters away from the Pascal condition (e.g., Fig. 3B), the SBBs no longer intersect at well-defined magnetic field. Fits of the single-particle model to experimental data for devices 1 to 7 (Fig. 3C) show the expected correlation between w z and W(B Pa ), but we do observe deviations from the Pascal condition for all samples.

The experimental data deviate from the single-particle model in several important ways. At low magnetic fields, the predicted linear Zeeman splitting of subbands is not obeyed; instead, the two lowest subbands (|0, 0, ±½i) are paired below a critical magnetic field, B P (9). At higher magnetic fields, re-entrant pairing is observed as subbands intersect and lock over a range of magnetic field values near the Pascal field, B Pa . In our noninteracting model (Eq. 1), there is a unique Pascal field B Pa ; however, experimentally we find that the value of the Pascal field depends on the degeneracy n: B ðnþ1Þ Pa < B ðnÞ Pa . This shift of B Pa with the degeneracy may be caused by an anharmonic component to the confinement. Adding an anharmonic term to the single-particle model produces similar shifts of B Pa (21). Table S1 shows the pairing field B P and Pascal field B ð2Þ Pa for devices 1 to 7. Devices with similar geometries display a variety of pairing fields and Pascal fields. This is not unexpected, given a previous study (8) in which the pairing field was found to vary from device to device and could be as large as B P = 11 T. The cause for the differing strength of the pairing field is unknown but likely plays a role in the differing strengths of the locking for the Pascal degeneracies in this work. Fits of the transconductance data were made for the n = 2 and n = 3 peaks (or plateaus), and we found that the states are, in fact, locking together over a finite range of magnetic fields (fig. S1) (21). The Pascal series of conductance steps is observed for a variety of devices written with both short (50 nm) and long (1000 nm) electron waveguides, and at different angles f with respect to the (100) crystallographic axis of the sample (angles are listed in table S1). Devices with wires written at angles of 0°, 45°, or 90°show no discernable difference.

The Pascal condition assumes that the magnetic field is oriented out of plane. To investigate the effect of in-plane magnetic field components on the Pascal conductance series, we measure angle-dependent magnetotransport, with the magnetic field oriented at an angle q with respect to the sample normal, within the y-z plane, B ¼ Bðsin q ŷ þ cos q ẑÞ (Fig. 4A). In the out-of-plane orientation (q = 0°), characteristic Pascal behavior is observed, with subband locking taking place near 6 T (Fig. 4D, q = 0°). As q increases, the subband locking associated with the n = 3 plateau destabilizes, while another (non-Pascal series) subband locking forms in a different region of parameter space (Fig. 4D, q = 20°, indicated by white lines). At larger angles (Fig. 4D, q = 50°), a dense network of re-entrant pairing, disbanding, and re-pairing is observed (movie S1). The strength of the re-entrant pairing of the |0, 0, ↓i and |0, 1, ↑i subbands is strongly dependent on the angle q of the applied magnetic field (Fig. 4C). The lower (B À R ) and upper (B þ R ) magnetic fields over which these SBBs are locked together are indicated by red and blue circles in Fig. 4D. The magnetic field range (DB

) is shown as a function of angle (Fig. 4C). The strength of the re-entrant pairing, DB R , initially increases with angle, jumps discontinuously at q = 30°as the SBBs (which have been shifting closer) snap together, and then decreases again. At q = 0°, there is a non-Pascal series crossing (no locking) of like-spin states (|0, 0, ↓i, |0, 1, ↓i), highlighted by crossed lines, which evolves into an avoided crossing at q = 10°. This feature is explored in Fig. 4B, where we plot conductance curves at B = 3 T for different angles.

A theoretical analysis more sophisticated than the single-particle model discussed above is required to capture the effects of electronelectron interactions. In the absence of interactions, the single-particle model described by Eq. 1 has band crossings but cannot predict any locking behavior. Prior work has demonstrated the existence of attractive electron-electron interactions in LaAlO 3 /SrTiO 3 nanostructures (8,20). We therefore constructed an effective lattice model for the waveguide by extending the noninteracting model to include phenomenological, local, two-body interactions between electrons in different modes. This effective model was investigated using DMRG, a numerical method that produces highly accurate results for strongly interacting systems in one dimension (5,(22)(23)(24)(25)(26)(27). The DMRG phase 2 of 4 S3. The first set of calculations reveal a phase boundary line between a vacuum phase and an electron pair phase that is characterized by a gap to singleelectron excitations. We associate this line to the n = 2 conductance step (G = 3e 2 /h). Extending this calculation to three electron modes with attractive interactions (n = 3 plateau) reveals a transition line from the vacuum phase to a "trion phase," which we associate with the n = 3 conductance step (G = 6e 2 /h). The trion phase is a 1D degenerate quantum liquid of composite fermions, each made up of three electrons, in which all one-and two-particle excitations are gapped out but threeparticle excitations are gapless. [See (21) for details of our theoretical model and DMRG calculations.]

We considered other theoretical explanations. The addition of spin-orbit coupling to the noninteracting model modifies the subband structure, producing avoided crossings of the transconductance peaks. Anharmonicity of the confining potential, in the absence of interactions, bends the subband structure but also does not produce locking. We rule out impurity scattering effects because of the ballistic nature of the transport. Moreover, without inter-electron interactions [e.g., negative U center (28)], an impurity cannot produce locking phenomena. We are not aware of other mechanisms for locking but cannot rule them out. Finally, we note that any theory of the locking phenomenon would need to have a noninteracting limit that matches with experiments (e.g., predicts conductance quantization).

Pascal composite particles predicted by our model would have a charge ne, where n = 2, 3, 4, …, and spin quantum numbers not yet determined. As with fractional fermionic states, it seems likely that the expected charge could be verified from a shot-noise experiment (29). The particular Pascal sequence observed here experimentally is a consequence of the number of spatial dimensions. Hypothetically, a material with four dimensions (three transverse to a conducting channel) could exhibit a conductance sequence (1, 4, 10, 20, …) ⋅ e 2 /h, the next diagonal in the Pascal triangle. The Pascal sequence of bound fermions is reminiscent of the "quantum dot periodic table" used to categorize multi-electron states in semiconductor nanostructures (30); the difference here is that the Pascal liquids consist of composite particles that are free to move in one spatial dimension, held together by mutual attraction rather than by an external potential profile. Pascal composite particles with n > 2 can be regarded as a generalization of Cooper pair formation, analogous to the manner in which quarks combine to form baryonic and other forms of strongly interacting, degenerate quantum matter. Interactions among Pascal particles are in principle possible; for example, trions could, in principle, "pair" to form bosonic hexamers. Coupled arrays of 1D waveguides can be used to build 2D structures. This type of structure is predicted to show a wide variety of properties, such as sliding phases (31)(32)(33) and non-Abelian excitations (34). Our highly flexible oxide nanoelectronics platform is poised to support these exotic forms of quantum matter. Data are from waveguide device 7. (A) Schematic of the sample as it is rotated with respect to the direction of the magnetic field B. n is the vector normal to the plane of the sample, and q = 0°r epresents an out-of-plane magnetic field orientation. (B) Conductance curves as a function of angle at |B| = 3 T. As the magnetic field is rotated away from an out-of-plane angle, we see an avoided crossing open up, which can be seen in the q = 10°curve as the plateau that begins to form near 3e 2 /h. We can also see evidence that re-entrant pairing starts to occur at larger angles (q > 30°) when the conductance increases by a step of 2e 2 /h, from 1e 2 /h to 3e 2 /h. (C) Re-entrant pairing strength as a function of angle q. (D) Transconductance dG/dm as a function of magnetic field strength and chemical potential. The magnetic field is rotated from an outof-plane orientation (q = 0°) to q = 50°in 10°steps. The in-plane component of the magnetic field is roughly perpendicular to the waveguide channel. At small angles, the Pascal series can be seen in the transport with bunches of 1, 2, and 3 subbands, but this is broken as the angle is increased. The reentrant pairing strength is indicated by the points where the states first lock together (red circles) and break apart (blue circles). T = 20 mK.