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arXiv:cond-mat/0504259 · Physical Review B

Path integral of the two-dimensional Su-Schrieffer-Heeger model

Marco Zoli

Published May 24, 2005• 19 citations
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Abstract

The equilibrium thermodynamics of the two-dimensional Su-Schrieffer-Heeger model is derived by means of a path integral method that accounts for the variable range of the electronic hopping processes. While the lattice degrees of freedom are classical functions of time and are integrated out exactly, the electron particle paths are treated quantum mechanically. The free energy of the system and its temperature derivatives are computed by summing at any $T$ over the ensemble of relevant particle paths which mainly contribute to the total partition function. In the low $T$ regime, the heat capacity over $T$ ratio shows an upturn peculiar of a glassy-like behavior. This feature is more sizeable in the square lattice than in the linear chain as the overall hopping potential contribution to the total action is larger in higher dimensionality.

DOI
10.1103/physrevb.71.205111

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The equilibrium thermodynamics of the two-dimensional Su-Schrieffer-Heeger model is derived by means of a path integral method that accounts for the variable range of the electronic hopping processes. While the lattice degrees of freedom are classical functions of time and are integrated out exactly, the electron particle paths are treated quantum mechanically. The free energy of the system and its temperature derivatives are computed by summing at any TTT over the ensemble of relevant particle paths which mainly contribute to the total partition function. In the low TTT regime, the heat capacity over TTT ratio shows an upturn peculiar of a glassy-like behavior. This feature is more sizeable in the square lattice than in the linear chain as the overall hopping potential contribution to the total action is larger in higher dimensionality.
The path integral formalism is applied to derive the full partition function of a generalized Su-Schrieffer-Heeger Hamiltonian describing a particle motion in a bath of oscillators. The electronic correlations are computed versus temperature for some choices of oscillators energies. We study the perturbing effect of a time averaged particle path on the phonon subsystem deriving the relevant temperature dependent cumulant corrections to the harmonic partition function and free energy. The method has been applied to compute the total heat capacity up to room temeperature: a low temperature upturn in the heat capacity over temperature ratio points to a glassy like behavior ascribable to a time dependent electronic hopping with variable range in the linear chain.
The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kB{\mathit{k}}_{\mathit{B}}kB​T. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.
We investigate the two-dimensional optical Su-Schrieffer-Heeger (SSH) model, in which the electron hopping amplitude is modulated by the difference between neighboring phonon coordinates. Using sign-problem-free auxiliary-field quantum Monte Carlo simulations, complemented by mean-field analysis, we determine the long-range ordered phases as a function of the electron-phonon coupling and phonon frequency. By examining both adiabatic and antiadiabatic regimes, we reveal the emergence of staggered and armchair valence bond solid (VBS) phases, as well as the O(4) antiferromagnetic phase. In addition, finite-temperature simulations show that the VBS transition occurs at critical temperatures significantly higher than in models with local electron-phonon coupling, consistent with the presence of lighter polarons in the metallic regime. These findings establish the ground-state and finite-temperature phase diagrams of the optical SSH model, which emphasize its similarities and contrasts with other electron-phonon systems.
The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the qp and atoms are restricted to the sites of a one-dimensional lattice. A path integral formalism is developed within the context of the canonical ensemble in which the qp is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. For the case of a free particle, analytical expressions for the energy, its fluctuations, and the qp-qp correlation function are derived and compared with the Monte Carlo simulations. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp for a periodic interaction potential, forcing the qp to occupy extended states. We consider a striped potential in one dimension, where every other lattice site is occupied by an atom with potential ε, and every other lattice site is empty. This potential serves as a stress test for the path integral formalism because of its rapid site-to-site variation. An analytical solution was determined in this case by utilizing Bloch's theorem due to the periodicity of the potential. Comparisons of the potential energy, the total energy, the energy fluctuations, and the correlation function are made between the results of the Monte Carlo simulations and the analytical calculations.
The authors study here a Su-Schrieffer-Heeger model on a square lattice by auxiliary-field quantum Monte Carlo. A novel approach allows analytical integration over the phonons, considerably reducing autocorrelation times and allowing investigation of new regions of the phase diagram. By varying the strength of the direct hopping at the O(4) symmetric point, they discover novel phases in this textbook model. In particular, for small values of the direct hopping, they observe emergent orthogonal Dirac fermions, with an exotic quantum phase transition akin to deconfined quantum criticality.
The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the qp and the atoms are restricted to sites on a one-dimensional lattice. A path integral formalism developed within the context of the canonical ensemble is utilized, where the qp is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp in the context of a square well potential, forcing the qp to occupy bound states. We consider a one-dimensional square well potential where all atoms on the lattice are occupied with one atom with an on-site potential except for a contiguous set of sites of various lengths centered at the middle of the lattice. Comparison of the potential energy, the energy fluctuations, and the correlation function are made between the results of the Monte Carlo simulations and the numerical calculations.