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arXiv:0808.0099

Quantum oscillation in the Hofstadter Model

Wenhu Xu, Li‐Ping Yang, Mingpu Qin, Tao Xiang

Published Aug 1, 2008• 0 citations
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Abstract

We have developed a novel quantum transfer matrix method to study thermodynamic properties of the Hofstadter model on square lattices. This method overcomes the finite size effect and allows the magnetization and other thermodynamic quantities to be accurately and efficiently calculated. The interplay between the lattice potential and the applied magnetic field leads to substantial corrections to the conventional de Haas-van Alphen (dHvA) oscillation of magnetization. It yields many paramagnetic peaks or diamagnetic deeps above the background of dHvA oscillation. This correction bears the fractal feature of energy spectra and can be used to probe the hierarchical structure of Hofstadter butterfly by experimental measurements.

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We have developed a novel quantum transfer matrix method to study thermodynamic properties of the Hofstadter model on square lattices. This method overcomes the finite size effect and allows the magnetization and other thermodynamic quantities to be accurately and efficiently calculated. The interplay between the lattice potential and the applied magnetic field leads to substantial corrections to the conventional de Haas-van Alphen (dHvA) oscillation of magnetization. It yields many paramagnetic peaks or diamagnetic deeps above the background of dHvA oscillation. This correction bears the fractal feature of energy spectra and can be used to probe the hierarchical structure of Hofstadter butterfly by experimental measurements.
Motivated by recent realizations of two-dimensional (2D) superconducting-qubit lattices, we propose a protocol to simulate Hofstadter butterfly with synthetic gauge fields in superconducting circuits. Based on the existing 2D superconducting-qubit lattices, we construct a generalized Hofstadter model on zigzag lattices, which has a fractal energy spectrum similar to the original Hofstadter butterfly. By periodically modulating the resonant frequencies of qubits, we engineer a synthetic gauge field to mimic the generalized Hofstadter Hamiltonian. A spectroscopic method is used to demonstrate the Hofstadter butterfly from the time evolutions of experimental observables. We numerically simulate the dynamics of the system with realistic parameters, and the results show a butterfly spectrum clearly. Our proposal provides a promising way to realize the Hofstadter butterfly on the latest 2D superconducting-qubit lattices and will stimulate the quantum simulation of novel properties induced by magnetic fields in superconducting circuits.
In a lattice model subject to a perpendicular magnetic field, when the lattice constant is comparable to the magnetic length, one enters the “Hofstadter regime,” where continuum Landau levels become fractal magnetic Bloch bands. Strong mixing between bands alters the nature of the resulting quantum phases compared to the continuum limit; lattice potential, magnetic field, and Coulomb interaction must be treated on equal footing. Using determinant quantum Monte Carlo and density matrix renormalization group techniques, we study this regime numerically in the context of the Hubbard-Hofstadter model on a triangular lattice. In the field-filling phase diagram, we find a broad wedge-shaped region of ferromagnetic ground states for filling factor ν ≤ 1 , bounded below by filling factor ν = 1 and bounded above by half filling the lowest Hofstadter subband. We observe signatures of SU(2) quantum Hall ferromagnetism at filling factors ν = 1 and ν = 3 . The phases near ν = 1 are particle-hole asymmetric, and we observe a rapid decrease in ground-state spin polarization consistent with the formation of skyrmions only on the electron doped side. At large fields, above the ferromagnetic wedge, we observe a low-spin metallic region with spin correlations peaked at small momenta. We argue that the phenomenology of this region likely results from exchange interaction mixing fractal Hofstadter subbands. The phase diagram derived beyond the continuum limit points to a rich landscape to explore interaction effects in magnetic Bloch bands. Published by the American Physical Society 2024
ABSTRACT Quantum simulation offers an effective approach to explore novel phenomena in complex quantum systems, such as the famous Hofstadter butterfly that are inaccessible in conventional natural systems. We investigate variants of the Hofstadter butterfly arising in lattices subject to inhomogeneous magnetic fields. Using a superconducting‐circuit platform, we implement a zigzag spin chain and a two‐leg spin ladder, where spatially modulated interaction phases produce position‐dependent magnetic fluxes through the plaquettes. By periodically modulating the resonance frequency of superconducting qubits, we engineer the desired synthetic magnetic field. The resulting single‐excitation dynamics map onto a tight‐binding model of a charged particle moving in a non‐uniform magnetic field, enabling the exploration of fractal energy spectra beyond the uniform‐field case. We numerically simulate the system dynamics using experimentally realistic parameters and reveal variants of the Hofstadter butterfly through a spectroscopic method. The simulation results demonstrate a high degree of agreement with theoretical predictions. Our proposal provides a promising route for investigating exotic fractal energy spectra via quantum simulation.
The competition between the length scales associated with the periodicity of a lattice potential and the cyclotron radius of a uniform magnetic field is known to have dramatic effects on the single-particle properties of a quantum particle, e.g., the fractal spectrum is known as the Hofstadter butterfly. Having this intricate competition in mind, we consider a two-component Fermi gas on a square optical lattice with opposite synthetic magnetic fields for the components, and study its effects on the many-body BCS-pairing phenomenon. By a careful addressing of the distinct superfluid transitions from the semimetal, quantum spin-Hall insulator, or normal phases, we explore the low-temperature phase diagrams of the model, displaying lobe structures that are reminiscent of the well-known Mott-insulator transitions of the Bose-Hubbard model.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds nnn lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
Multifractal properties of the two-dimensional Ising model are studied by means of the $f(\ensuremath{\alpha})$ spectra of singularities of probability measures supported by energy spectra. These measures are determined through calculations of energy degeneracies for finite-size systems up to $12\ifmmode\times\else\texttimes\fi{}12$ spins in cases of square and triangular lattices and up to $22\ifmmode\times\else\texttimes\fi{}22$ spins in the case of the hexagonal lattice. The calculations are performed in an exact manner using the transfer-matrix method. It is argued that, in the thermodynamic limit, the scaling exponent ${\ensuremath{\alpha}}_{\mathrm{max}}$ associated with the most probable energy of the system takes at the critical temperature a minimum value and, consequently, it is argued that a given system reveals a phase transition at a finite temperature if ${\ensuremath{\alpha}}_{\mathrm{max}}$ possesses a minimum at the finite temperature. It is also shown that, in the thermodynamic limit, the spectrum $f(\ensuremath{\alpha})$ shrinks when the critical temperature is approached.