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.
Abstract Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.
Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schr\"odinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining the minimum energy. We show how applying more complex constraints involving higher powers of momenta can potentially remedy such a problem. We also propose an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system.
We solve the Schrödinger equation for a particle in the circle with a quantum mechanical bootstrap method. The main idea is that in physical situations one needs solutions of the recursion relations for the Fourier coefficients of the measure that vanish exponentially fast. We show that this results in a (generalized) continued fraction representation of the solution. The solution is an analytical function of the couplings and the energy. We also show how one this method directly computes the density of states and checking that the answer is physical is reduced to checking an inequality for a single variable. We also show how to reconstruct the wave function and how to extract the quasimomentum of the particle as a function of the energy. We illustrate our solution in the simplest case of the Mathieu equation, where we show that the method can easily resolve exponentially small bands in the band spectrum.
Abstract We test the bootstrap approach for determining the spectrum of one dimensional Hamiltonians. In this paper we focus on problems that have a two parameter search space in the bootstrap approach: the double well and a periodic potential associated with the Mathieu equation. For the double well, we compare the bootstrapped energies with contributions from perturbative and non-perturbative results, finding good agreement. For the periodic potentials, we notice that the bootstrap approach gives the band structure of the periodic potential, but it has trouble finding the quasi-momentum of the system. To make further progress on determining the dispersion relation of the bands, new techniques are needed.
Fractional calculus has become an essential framework in geophysics, optics, and biological systems to capture long-range correlations and anomalous transport. In this article, we extend this framework to explore a particle in a periodic potential, where the Schrodinger equation is extended to its fractional form. This allows us to study how the Levy index
q governs the formation and inversion of energy bands, offering a pathway to engineer new physical behaviors by tuning
q in periodic quantum systems. We solve the fractional Schrodinger equation (FSE) for periodic rectangular potentials of varying height
V0, barrier thickness
L, and well width
W using an imaginary-time evolution algorithm, supplemented by Gaussian process regression. This analysis reveals a qualitative shift in the system's band structure at
q=2, separating into distinct regimes of dispersion for
q>2 and
q<2. For
q>2, the energy bands invert as symmetric minima emerge within the first Brillouin zone and shift from
k=0 toward
k=±π/a with increasing
q. These degenerate minima define a Bloch-momentum qubit, suggesting an analog to valley degrees of freedom in valleytronics. The
q at which inversion completes scales as
q∝V0−0.28±0.05,
q∝L−0.35±0.08, and
q∝W−0.49±0.06 when varying parameters individually, indicating a tunable sensitivity to potential geometry. In contrast, for
q<2, the ground band hardens around
k=0, with a dispersion following
C∣k∣q+E0 near
k=0. This suggests an effective mass of 0 for
1<q<2 at the band's lowest energy state. These results demonstrate that the Levy index serves as a tunable degree of freedom in quantum periodic systems, capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics through effective-mass control.
We study a matrix product state algorithm to approximate excited states of translationally invariant quantum spin systems with periodic boundary conditions. By means of a momentum eigenstate ansatz generalizing the one of \"Ostlund and Rommer [see S. \"Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995);S. Rommer and S. \"Ostlund, Phys. Rev. B 55, 2164 (1997)], we separate the Hilbert space of the system into subspaces with different momentum. This gives rise to a direct sum of effective Hamiltonians, each one corresponding to a different momentum, and we determine their spectrum by solving a generalized eigenvalue equation. Surprisingly, many branches of the dispersion relation are approximated to a very good precision. We benchmark the accuracy of the algorithm by comparison with the exact solutions and previous numerical results for the quantum Ising, the antiferromagnetic Heisenberg spin-
1/2, and the bilinear-biquadratic spin-1 models.