We review the density of states (DOS) and related quantities of quasi one-dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low-energy behavior of non-interacting fermions which are subject to a static random backscattering potential will be described by the strictly one-dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high-Tc superconductors. We develop a non-perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker-Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.We review the density of states (DOS) and related quantities of quasi one-dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low-energy behavior of non-interacting fermions which are subject to a static random backscattering potential will be described by the strictly one-dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high-Tc superconductors. We develop a non-perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker-Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.

arXiv:cond-mat/0102160Annalen der PhysikSep 1, 200113 citations

Fluctuation effects in disordered Peierls systems

Lorenz Bartosch

Abstract We review the density of states (DOS) and related quantities of quasi one‐dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low‐energy behavior of non‐interacting fermions which are subject to a static random backscattering potential will be described by the strictly one‐dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high‐Tc superconductors. We develop a non‐perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker‐Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.Abstract We review the density of states (DOS) and related quantities of quasi one‐dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low‐energy behavior of non‐interacting fermions which are subject to a static random backscattering potential will be described by the strictly one‐dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high‐Tc superconductors. We develop a non‐perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker‐Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.

Annalen der PhysikJul 30, 20010 citations

Singularities and Pseudogaps in the Density of States of the Fluctuating Gap Model

Lorenz Bartosch

We review the one-dimensional fluctuating gap model (FGM) which describes non-interacting fermions subject to a static random backscattering potential. The FGM applies to the low-energy behavior of quasi one-dimensional Peierls and spin systems and has recently also been used to explain the pseudogap phenomenon in high- $T_c$ superconductors. After an elementary introduction to the FGM, we develop methods which allow for a simultaneous calculation of the density of states (DOS) and the inverse localization length. First, we recover all known results in the limits of zero and infinite correlation length of the random potential. Then, we attack the problem of finite correlation lengths. While a complex order parameter, which describes incommensurate Peierls chains, leads to a suppression of the DOS, i.e. a pseudogap, the DOS exhibits a singularity at the Fermi energy if the order parameter is real and therefore refers to a commensurate system. We confirm these results by calculating the DOS and the inverse localization length for finite correlation lengths and Gaussian statistics of the backscattering potential with unprecedented accuracy numerically. Finally, we consider the case of classical phase fluctuations which apply to low temperatures where amplitude fluctuations are frozen out. In this physically important regime which is also governed by finite correlation lengths, we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.We review the one-dimensional fluctuating gap model (FGM) which describes non-interacting fermions subject to a static random backscattering potential. The FGM applies to the low-energy behavior of quasi one-dimensional Peierls and spin systems and has recently also been used to explain the pseudogap phenomenon in high-$T_c$ superconductors. After an elementary introduction to the FGM, we develop methods which allow for a simultaneous calculation of the density of states (DOS) and the inverse localization length. First, we recover all known results in the limits of zero and infinite correlation length of the random potential. Then, we attack the problem of finite correlation lengths. While a complex order parameter, which describes incommensurate Peierls chains, leads to a suppression of the DOS, i.e. a pseudogap, the DOS exhibits a singularity at the Fermi energy if the order parameter is real and therefore refers to a commensurate system. We confirm these results by calculating the DOS and the inverse localization length for finite correlation lengths and Gaussian statistics of the backscattering potential with unprecedented accuracy numerically. Finally, we consider the case of classical phase fluctuations which apply to low temperatures where amplitude fluctuations are frozen out. In this physically important regime which is also governed by finite correlation lengths, we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.

Jan 1, 20002 citations

Singularities and Pseudogaps in the Density of States of Peierls Chains

Lorenz Bartosch, Peter Kopietz

We develop a nonperturbative method to calculate the density of states (DOS) $\ensuremath{\rho}(\ensuremath{\omega})$ of the fluctuating gap model describing the low-energy physics of electrons on a disordered Peierls chain. For a real order parameter field we calculate $\ensuremath{\rho}(0)$ (i.e., the DOS at the Fermi energy) exactly as a functional of the disorder for a chain of finite length $L$ . Averaging $\ensuremath{\rho}(0)$ with respect to a Gaussian probability distribution of the fluctuating Peierls order parameter, we show that for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ the average $〈\ensuremath{\rho}(0)〉$ diverges for any finite value of the correlation length above the Peierls transition. Pseudogap behavior emerges only if the Peierls order parameter is finite and sufficiently large.We develop a nonperturbative method to calculate the density of states (DOS) $\ensuremath{\rho}(\ensuremath{\omega})$ of the fluctuating gap model describing the low-energy physics of electrons on a disordered Peierls chain. For a real order parameter field we calculate $\ensuremath{\rho}(0)$ (i.e., the DOS at the Fermi energy) exactly as a functional of the disorder for a chain of finite length $L$. Averaging $\ensuremath{\rho}(0)$ with respect to a Gaussian probability distribution of the fluctuating Peierls order parameter, we show that for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ the average $〈\ensuremath{\rho}(0)〉$ diverges for any finite value of the correlation length above the Peierls transition. Pseudogap behavior emerges only if the Peierls order parameter is finite and sufficiently large.

arXiv:cond-mat/9810362Physical Review LettersFeb 1, 199924 citations

Phase fluctuations in two-dimensional superconductors and pseudogap phenomenon

Xu-Cheng Wang, Yang Qi

We study the phase fluctuations in the normal state of generic two-dimensional superconducting systems with $s$ -wave pairing. The effect of phase fluctuations of the pairing fields can be dealt with perturbatively using disorder averaging, after we treat the local superconducting order parameter as a static disordered background. It is then confirmed that the phase fluctuations above the two-dimensional Berezinskii-Kosterlitz-Thouless transition lead to a significant broadening of the single-particle spectrum, giving birth to the pseudogap phenomenon. Quantitatively, the broadening of spectral weights near the BCS gap is characterized by the ratio of the superconducting coherence length and the spatial correlation length of the superconducting pairing order parameter. Our results are tested on the fermionic attractive- $U$ Hubbard model on the square lattice, using the unbiased determinant quantum Monte Carlo method and stochastic analytic continuation.We study the phase fluctuations in the normal state of generic two-dimensional superconducting systems with $s$-wave pairing. The effect of phase fluctuations of the pairing fields can be dealt with perturbatively using disorder averaging, after we treat the local superconducting order parameter as a static disordered background. It is then confirmed that the phase fluctuations above the two-dimensional Berezinskii-Kosterlitz-Thouless transition lead to a significant broadening of the single-particle spectrum, giving birth to the pseudogap phenomenon. Quantitatively, the broadening of spectral weights near the BCS gap is characterized by the ratio of the superconducting coherence length and the spatial correlation length of the superconducting pairing order parameter. Our results are tested on the fermionic attractive-$U$ Hubbard model on the square lattice, using the unbiased determinant quantum Monte Carlo method and stochastic analytic continuation.

arXiv:2212.05737Physical review. B./Physical review. BJun 1, 202316 citations

Random-mass Dirac fermions in doped spin-Peierls and spin-ladder systems: One-particle properties and boundary effects

M. Steiner, Michele Fabrizio, Alexander O. Gogolin

Quasi-one-dimensional spin-Peierls and spin-ladder systems are characterized by a gap in the spin-excitation spectrum, which can be modeled at low energies by that of Dirac fermions with a mass. In the presence of disorder, these systems can still be described by a Dirac fermion model, but with a random mass. Some peculiar properties, like the Dyson singularity in the density of states, are well known and attributed to creation of low-energy states due to the disorder. We take one step further and study single-particle correlations by means of Berezinskii's diagram technique. We find that, at low energy $\ensuremath{\epsilon}$, the single-particle Green function decays in real space like $G(x,\ensuremath{\epsilon})\ensuremath{\propto}{(1/x)}^{3/2}$. It follows that at these energies the correlations in the disordered system are strong --- even stronger than in the pure system without the gap. At distances $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}\ensuremath{\propto}{\mathrm{ln}}^{2}(1/\ensuremath{\epsilon})$, the Green function crosses over to a standard exponential decay. The scale ${L}_{\ensuremath{\epsilon}}$ is different from the Thouless localization length ${\ensuremath{\lambda}}_{\ensuremath{\epsilon}}\ensuremath{\propto}\mathrm{ln}(1/\ensuremath{\epsilon})$. Additionally, we study the effects of boundaries on the local density of states. We find that the latter is logarithmically (in the energy) enhanced close to the boundary. This enhancement decays into the bulk as $1/\sqrt{x}$ saturating at $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}.$ We also discuss some implications of these results for the spin systems and their relation to the investigations based on the real-space renormalization group approach.Quasi-one-dimensional spin-Peierls and spin-ladder systems are characterized by a gap in the spin-excitation spectrum, which can be modeled at low energies by that of Dirac fermions with a mass. In the presence of disorder, these systems can still be described by a Dirac fermion model, but with a random mass. Some peculiar properties, like the Dyson singularity in the density of states, are well known and attributed to creation of low-energy states due to the disorder. We take one step further and study single-particle correlations by means of Berezinskii's diagram technique. We find that, at low energy $\ensuremath{\epsilon}$, the single-particle Green function decays in real space like $G(x,\ensuremath{\epsilon})\ensuremath{\propto}{(1/x)}^{3/2}$. It follows that at these energies the correlations in the disordered system are strong --- even stronger than in the pure system without the gap. At distances $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}\ensuremath{\propto}{\mathrm{ln}}^{2}(1/\ensuremath{\epsilon})$, the Green function crosses over to a standard exponential decay. The scale ${L}_{\ensuremath{\epsilon}}$ is different from the Thouless localization length ${\ensuremath{\lambda}}_{\ensuremath{\epsilon}}\ensuremath{\propto}\mathrm{ln}(1/\ensuremath{\epsilon})$. Additionally, we study the effects of boundaries on the local density of states. We find that the latter is logarithmically (in the energy) enhanced close to the boundary. This enhancement decays into the bulk as $1/\sqrt{x}$ saturating at $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}.$ We also discuss some implications of these results for the spin systems and their relation to the investigations based on the real-space renormalization group approach.

arXiv:cond-mat/9706096Physical review. B, Condensed matterApr 1, 199845 citations

T_c

superconductors. After an elementary introduction to the FGM, we develop methods which allow for a simultaneous calculation of the density of states (DOS) and the inverse localization length. First, we recover all known results in the limits of zero and infinite correlation length of the random potential. Then, we attack the problem of finite correlation lengths. While a complex order parameter, which describes incommensurate Peierls chains, leads to a suppression of the DOS, i.e. a pseudogap, the DOS exhibits a singularity at the Fermi energy if the order parameter is real and therefore refers to a commensurate system. We confirm these results by calculating the DOS and the inverse localization length for finite correlation lengths and Gaussian statistics of the backscattering potential with unprecedented accuracy numerically. Finally, we consider the case of classical phase fluctuations which apply to low temperatures where amplitude fluctuations are frozen out. In this physically important regime which is also governed by finite correlation lengths, we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.

L

. Averaging $\ensuremath{\rho}(0)$ with respect to a Gaussian probability distribution of the fluctuating Peierls order parameter, we show that for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ the average $〈\ensuremath{\rho}(0)〉$ diverges for any finite value of the correlation length above the Peierls transition. Pseudogap behavior emerges only if the Peierls order parameter is finite and sufficiently large.

We study the phase fluctuations in the normal state of generic two-dimensional superconducting systems with

s

-wave pairing. The effect of phase fluctuations of the pairing fields can be dealt with perturbatively using disorder averaging, after we treat the local superconducting order parameter as a static disordered background. It is then confirmed that the phase fluctuations above the two-dimensional Berezinskii-Kosterlitz-Thouless transition lead to a significant broadening of the single-particle spectrum, giving birth to the pseudogap phenomenon. Quantitatively, the broadening of spectral weights near the BCS gap is characterized by the ratio of the superconducting coherence length and the spatial correlation length of the superconducting pairing order parameter. Our results are tested on the fermionic attractive-

U

Hubbard model on the square lattice, using the unbiased determinant quantum Monte Carlo method and stochastic analytic continuation.

1/\sqrt{x}

saturating at $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}.$ We also discuss some implications of these results for the spin systems and their relation to the investigations based on the real-space renormalization group approach.