We study a quantum mechanical system whose spectrum coincides with that of bilinear operators of the Sachdev-Ye-Kitaev model. The standard positivity-based quantum mechanical bootstrap is degenerate with respect to the boundary data: it does not distinguish the boundary conditions that select the SYK spectrum, and hence is insufficient to determine the eigenvalues. Instead, by considering fractional powers of operators, we obtain constraint equations that determine the spectrum without imposing positivity. The resulting roots converge to exact eigenvalues as the truncation order increases. We call this the direct bootstrap.
We study a quantum mechanical system whose spectrum coincides with that of bilinear operators of the Sachdev-Ye-Kitaev model. The standard positivity-based quantum mechanical bootstrap is degenerate with respect to the boundary data: it does not distinguish the boundary conditions that select the SYK spectrum, and hence is insufficient to determine the eigenvalues. Instead, by considering fractional powers of operators, we obtain constraint equations that determine the spectrum without imposing positivity. The resulting roots converge to exact eigenvalues as the truncation order increases. We call this the direct bootstrap.
We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two q^-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the complex conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. In the limit of large number N of Majoranas and q^∝N, we employ diagrammatic and combinatorial techniques to show that the spectral density of the model can be approximated by the weight function of the Al-Salam-Chihara Q-Laguerre polynomials. For q^=2,3,4, the analytical spectral density, with Q=Q(q^,N) computed analytically, agrees well with numerical results obtained by exact diagonalization. The spectrum is positive with a hard edge at zero energy and, for odd q^, low-energy excitations grow as a stretched exponential, though the functional form is different from that of the supersymmetric SYK model. The microscopic spectral density and level statistics close to the hard edge are very well approximated by that of an ensemble of random matrices belonging to non-standard universality classes, which indicates quantum chaotic dynamics.
Periodic structures are ubiquitous in quantum many-body systems and quantum field theories, ranging from lattice models, to compact spaces, to topological phenomena. However, previous bootstrap studies encountered technical challenges even for one-body periodic problems, such as a failure in determining the accurate dispersion relations for Bloch bands. In this work, we develop a new bootstrap procedure to resolve these issues, which does not make use of positivity constraints. We mainly consider a quantum particle in a periodic cosine potential. The same procedure also applies to a particle on a circle, where the role of the Bloch momentum k is played by the boundary condition or the θ angle. We unify the natural set of operators and the translation operator by a new set of operators { e i n x e i a p p s } . To extract the Bloch momentum k , we further introduce a set of differential equations for ⟨ e i n x e i a p p s ⟩ in the translation parameter a . At some fixed a , the boundary conditions can be determined accurately by analytic bootstrap techniques and matching conditions. After solving the differential equations, we impose certain reality conditions to determine the accurate dispersion relations, as well as the k dependence of other physical quantities. We also investigate the case of noninteger s using the Weyl integral in fractional calculus.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model. These are quantum mechanical models involving N Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The N=1 model with a single supercharge has unbroken supersymmetry at large N, but nonperturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large N equation, and also by performing numerical computations for small values of N. We also compute the large N spectrum of ``singlet'' operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large N computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two $\stackrel{^}{q}$-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the Hermitian conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. The spectrum is positive with a hard edge at zero energy. We employ diagrammatic and combinatorial techniques to compute analytically the low-order moments of the Hamiltonian. In the limit of large number N of Majoranas, we have found striking similarities with the moments of the weight function of the Al-Salam-Chihara Q-Laguerre polynomials. For $\stackrel{^}{q}=3$, 4, the Q-Laguerre prediction, with $Q=Q(\stackrel{^}{q},N)$ also computed analytically, agrees well with exact diagonalization results for $30<N\ensuremath{\le}34$ while we observe some deviations for $\stackrel{^}{q}=2$. The most salient feature of the spectral density is that, for odd $\stackrel{^}{q}$, low-energy excitations grow as a stretched exponential, with a functional form different from that of the supersymmetric SYK model. For $\stackrel{^}{q}=4$, a detailed analysis of level statistics reveals quantum chaotic dynamics even for time scales substantially shorter than the Heisenberg time. More specifically, the spacing ratios in the bulk of the spectrum and the microscopic spectral density and the number variance close to the hard edge are very well approximated by that of an ensemble of random matrices that, depending on N, belong to the chiral or superconducting universality classes. In particular, we report the first realization of level statistics belonging to the chiral Gaussian Unitary Ensemble universality class, which completes the tenfold-way classification in the SYK model.
We study a description of the large N limit of the Sachdev–Ye–Kitaev (SYK) model in terms of quantum mechanics without quenched disorder. Instead of random couplings, we introduce massive scalar fields coupled to fermions, and study a small mass limit of the theory. We show that, under a certain condition, the correlation functions of fermions reproduce those of the SYK model with a temperature dependent coupling constant in the large N limit. We also discuss a supersymmetric generalization of our quantum mechanical model. As a byproduct, we develop an efficient way of estimating the large N behavior of correlators in the SYK model.
Several condensed-matter platforms have been proposed recently to realize the Sachdev-Ye-Kitaev (SYK) model in their low-energy limit. In these proposed realizations, the characteristic SYK behavior is expected to occur under certain assumptions about the underlying physical system that (i) render all bilinear terms small compared to four-fermion interactions and (ii) ensure that the coupling constants are approximately all-to-all and independent random variables. In this work we explore, both analytically and numerically, the family of models that arises when we relax these assumptions in ways motivated by real physical systems. By relaxing (i) and allowing large bilinear terms, we obtain a novel, exactly solvable cousin of the SYK model. It exhibits two distinct phases separated by a quantum phase transition characterized by a power-law, ${\ensuremath{\sim}|\ensuremath{\omega}|}^{\ensuremath{-}1/3}$, scaling of the low-energy spectral density, despite being a noninteracting model. By relaxing (ii), we obtain close relatives of the SYK model which exhibit interesting behaviors, including a chaotic non-Fermi liquid phase with continuously varying fermion scaling dimension, and a phase transition to a disordered Fermi liquid as a function of interaction range and disorder length scale.