Observing Fermion Pair Instability of the Sachdev-Ye-Kitaev Model on a Quantum Spin Simulator

The Sachdev-Ye-Kitaev (SYK) model has become increasingly of great interest in studying exotic non-fermi liquid states without quasiparticle excitations, holography duality to Einstein gravity, and quantum chaos. However, the unnatural form of its Hamiltonian, including its strong randomness and fully nonlocal fermion interaction, makes its experimental investigation an intractable challenge. A promising solution to overcome this challenge is quantum simulation, whose role will be more pronounced particularly in the future when more qubits can be handled. We have enough control to demonstrate a first step towards quantum simulation of this system. We observed the fermion paring instability of the non-Fermi liquid state and the chaotic-nonchaotic transition at simulated temperatures, as was predicted by previous theory. These results demonstrate the feasibility of experimentally simulating the SYK model. It opens a new experimental avenue towards investigating the key features of non-Fermi liquid states, as well as the quantum chaotic systems and the AdS/CFT duality, thanks to the rich physics of the SYK model.

The concept of non-fermi liquid (NFL) state represents a series of exotic metallic states observed in strongly correlated condensed matter systems [1][2][3][4][5][6][7][8][9][10][11], which behave fundamentally differently from the standard Fermi liquid states that are characterized by the long-lived quasiparticles near the Fermi level. The most well-known (yet poorly understood) NFL is the "strange metal" phase at the optimal doping of the cuprates high temperature superconductors, where the resistivity scales linearly with temperature for a very large range of temperature in the phase diagram [12][13][14][15]. Just like many other NFLs that occur at various itinerant quantum critical points [16][17][18][19][20][21][22], the strange metal phase is preempted by a dome of ordered phase with pair condensate of fermions (high T c superconductor) at low temperature. Thus the strange metal phase is more fundamental than the superconductor phase itself: it is the "parent state" of the high T c superconductor, just like the Fermi liquid is the parent state (or normal state) of ordinary BCS superconductor.
There is a series of toy model for NFL, the Sachdev-Ye-Kitaev (SYK) model and its generalizations [23][24][25][26][27][28][29][30], that seem to capture key universal features. It is known the fermion has a completely different scaling behavior from the noninteracting fermions in the infrared limit, thus it has no quasi-particles and by definition it is a NFL. Secondly, it was found that the SYK model has marginally relevant "pairing instability" just like the ordinary Fermi liquid state [31], which is again consistent with one of the universal features of the NFLs observed experimentally. Last but not least, recently a generalization based on the SYK model has shown linear-T resistivity for a large temperature scale, and the scaling behavior of the SYK model is the key for the linear-T resistivity [32]. All these developments suggest that some version of the SYK model and its generalizations may indeed have to do with the experimentally observed NFLs, especially the strange metal phase.
Besides its potential connection to the condensed matter systems, the SYK model, and its generalized version SYK q models with a q−fermion interaction, are of great interest to the quantum information and string theory community. This series of models have the maximal chaotic behavior for q > 2, namely their Lyapunov exponent saturates a proven bound [33], which grants them a holographic dual to the (1 + 1)d Einstein gravity with a bulk black hole. The common wisdom is that a theory that is dual to a black hole through the holographic duality must be maximally chaotic, and the SYK model provides such an example [25][26][27][34][35][36][37][38]. Thus an experimental realization of the SYK model can also be thought of as a realization of black hole through the holographic duality.
The goal of this work is to experimentally investigate the SYK model for the first time, especially the pairing instability and the chaotic-nonchaotic transition predicted recently [31]. We realized the (0+1)d generalized SYK model with N = 8 Majorana fermions using a fourqubit nuclear magnetic resonance (NMR) quantum simulator, and measured the boson correlation functions at different simulated temperatures and perturbations. The early-time and late-time decay behaviors of fermion-pair correlations reflect the fact that there exist two different phases of the generalized SYK model, i.e., maximally chaotic non-Fermi liquid phase and perturbatively weak chaotic fermion pair condensate phase. The results reveal their competition under different perturbations, and also the thermal behavior at different simulated temperatures.
The Hamiltonian of (0 + 1)d generalized SYK model we considered is given by where χ i,j,k,l are Majorana fermion operators with indices i, j, k, l = 1, · · · , N , and both J ijkl and C ij are antisymmetric random tensors drawn from a Gaussian distribution: J ijkl = 0, J 2 ijkl = 3!J 2 4 /N 3 and C ij = 0, C ij C kl = J 2 /N 2 (δ ik δ jl − δ il δ jk ). Note that J 4 has the dimension of energy, while J has the dimension of (energy) 1/2 . At µ = 0, the Hamiltonian describes the pure SYK model, whose ground state is a maximally chaotic non-Fermi liquid. As pointed out in Ref. [31], the SYK fixed point could be unstable towards fermion pair condensate and spontaneous symmetry breaking. For instance, a positive µ term in the Hamiltonian Eq. (1) is a (marginally) relevant perturbation that drives the spontaneous breaking of the time-reversal symmetry T : In the T -breaking phase, the following bosonic fermion pair operator develops a persistent correlation b(t)b(0) ∼ constant that does not decay in time t. We will use this long-time boson correlation as an experimental signature for the T -breaking phase. The ordering of b actually has a simple mean-field understanding, since the µ term can also be written as −µb 2 /2 which favors b = 0 when µ > 0. This can be viewed as a (0 + 1)d analog of the Cooper instability of a non-Fermi liquid at low temperature. In the presence of b = 0, the pairing term −iµ b C ij χ i χ j in the mean-field Hamiltonian is most relevant at lowenergy, which leads to the non-chaotic ground state in the infrared limit, plus perturbatively irrelevant interaction that causes weak chaos. On the other hand, if µ is negative (µ < 0), the spontaneous symmetry breaking will not be favored and the system will remain in the maximally chaotic non-Fermi liquid phase. In experiment, we use four spins to simulate N = 8 Majorana fermions. The Hamiltonian can be encoded into the spin-1/2 operators via the Jordan Wigner transformation: Here σ x,y,z stand for Pauli matrices. There are 70 J ijkl 's, 28 C ij 's and four types of spin interactions (i.e., 1-, 2-, 3-and 4-body interactions) in the case of N = 8.
The physical system we used has four nuclear spins (C 1 , C 2 , C 3 and C 4 ) in the sample of trans-crotonic acid dissolved in d6-acetone. Its molecular structure and relevant parameters are shown in Figs. 2(a) and 2(b), respectively. The natural Hamiltonian of this system in rotating frame iŝ where ω i represents the chemical shift of spin i and J ij the coupling constant between spins i and j. The experiment was carried out on a Bruker DRX-700 spectrometer at room temperature (T = 298 K). The experiment is divided into three steps: initialization, evolution and measurement, as illustrated in Fig.  2(c). Under high-temperature approximation, the natural system is originally in the thermal equilibrium state where I is the identity and ∼ 10 −5 is the polarization. During our quantum computation, the evolution preserves the unit operator I, so we omit it and rewrite ρ eq = 4 i=1 σ z i . Hereinafter we used the deviation density matrices as 'states' [39]. Starting from ρ eq , the system was prepared into the initial 'states': . To implement these initial states, we first build a quantum circuit with only singlequbit rotations and free evolutions of the natural Hamiltonian, which allow us to get the states (before the first z-direction gradient field) whose diagonal elements equal to the eigenvalues of ρ i . The sixteen rotation angles θ j 's for different initial states were searched out by numerical optimization algorithm and given in the Supplementary Information. The CNOT gates were applied to remove zero quantum coherence that cannot be averaged out by the z-direction gradient fields. The states after the third z-direction gradient field are thus the diagonal density matrices, i.e., ρ d i = V † ρ i V , where V is the basis transformation between computational basis and eigenvectors of ρ i . When performing the V transformation, we can obtain the initial states ρ i .
In the second step, the dynamic evolution of generalized SYK Hamiltonian can be simulated with a controllable NMR system efficiently, as pointed out originally by Feynman [40]. The spin Hamiltonian in Eq. (1) is Within some desired accuracy ε, its evolution operator can be decomposed into [41], The accuracy ε(∼ J 2 t 2 /n) can be made as small as desired by choosing sufficiently large n.
there is an increase in accuracy. We can use 1-and 2-body interactions in NMR systems to simulate all kbody spin interactions [42,43], which can be seen in Supplementary Information. For example, 3-and 4-body interactions were implemented in previous experiments [44,45]. Although the total number of gates for simulating the dynamic evolution grows polynomially with the number of Majorana fermions N [46], the experimental running time is still beyond T 2 . To overcome the decoherence effect, each evolution was implemented using one shaped pulse, whose parameters were optimized by the gradient ascent pulse engineering (GRAPE) method [47]. All pulses have theoretical fidelities over 0.99, and are designed to be robust against the inhomogeneity of the control field.
Finally, we measure the boson correlation function to probe the instability of the SYK non-fermion-liquid ground state towards spontaneous symmetry breaking at different values of β and µ. The boson correlation function is defined as To remove its initial value fluctuation from sample to sample (here a sample means that we randomly generate a Hamiltonian with J ijkl 's and C ij 's), we average the normalized correlation function over random samples, To avoid the unphysical phase interference among different samples, we take the amplitude of the normalized correlation function before the disorder average. In experiment, we took 8 random samples. Starting from initial states ρ Real i and ρ Imag i , the real and imaginary parts of b(t)b(0) β can be obtained by measuring the observable of b, namely, The observable of b includes 28 spin operators, all of which are readout by the following five pulses: Here R j x,y stands for the j-spin rotation by π/2 about x or y axis. The experimental results are shown in Fig. 3. We normalized the correlation function to correct for the effect of decoherence. With this correction, our experiment is in good agreement with what is expected from simulations.
Let us first look at the low temperature result (β = 20) in Fig. 3(c). The boson correlation is measured for three different values of µ. For both µ = 0 and µ = −5, the boson correlations decay in time quickly following the similar manner. While for µ = 5, the boson correlation decays much slower and saturates to a relatively large value. This difference indicates the long-time order of the boson b in the µ > 0 spontaneous T -breaking phase. In contrast, for µ < 0, there is no such instability towards symmetry breaking.
The fact that the boson correlation still saturates to a finite value in the non-Fermi liquid phase for µ ≤ 0 is due to the finite size effect in our small system. In the non-Fermi liquid phase (µ ≤ 0), the saturate value of boson correlation decays towards zero with the system size. In the symmetry breaking phase (µ > 0), the saturate value scales towards a finite value in the thermodynamic limit. The numerical simulation result of this scaling behavior is shown in Supplementary Information. So the different behaviors of the boson correlation in our finitesize experiment indeed reflects the different phases of the generalized SYK model.
As we raise the temperature to β = 1 in Fig. 3(b), the boson condensation is destroyed by the thermal fluctuation and the long-time correlation is suppressed. In the thermodynamic limit the transition temperature should scale as T c ∼ exp(− √ πJ 4 /2J 2 µ) [31], as illustrated in the phase diagram Fig. 1(a). At infinite temperature (β = 0) in Fig. 3(a) We observe the slightly different behaviors of the boson correlations between the µ = 0 and the µ = 0 cases: the correlation decays fastest and to the lowest saturation value at the point µ = 0. This is consistent with the fact that the pure SYK model is maximally chaotic which scrambles the order parameter most thoroughly.
In summary, we present the first proof-of-principle experimental realization of the SYK model, and its generalization. The measurements of fermion-pair correlation functions exhibit the unstability of the maximally chaotic non-Fermi liquid phase of the SYK model against certain types of four-fermion perturbations, which drives the system into a less chaotic fermion pair condensed phase with spontaneous T -breaking. These successful experi-mental demonstrations rely heavily on the full controllability of our NMR quantum simulator. Our experiment demonstrate the first steps towards quantum simulations that explore phenomena from strongly interacting systems which are difficult to investigate by other methods. The method used here can also be adapted in other platforms such as superconducting circuits and trapped ions, and in the future may provide a new path towards exploring the maximally chaotic property of the holographic black hole as well.
Supplementary Information for "Observing Fermion Pair Instability of the Sachdev-Ye-Kitaev Model on a Quantum Spin Simulator" 1. Experimental parameters The rotation angles for preparing different initial states are shown in Tables. S1 ∼ S15. The values of β = 0 are not listed below due to ρ Imag i (0, µ) = 0.