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arXiv:cond-mat/0109424 · Physical review. B, Condensed matter

Kosterlitz-Thouless universality in a Fermionic system

Shailesh Chandrasekharan A, James C. Osborn

Published Jul 26, 2002• 18 citations
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Abstract

An extension of the attractive Hubbard model is constructed to study the critical behavior near a finite-temperature superconducting phase transition in two dimensions using the recently developed meron-cluster algorithm. Unlike previous calculations in the attractive Hubbard model which were limited to small lattices, the algorithm is used to study the critical behavior on lattices as large as $128\ifmmode\times\else\texttimes\fi{}128.$ These precise results show that a fermionic system can undergo a finite temperature phase transition whose critical behavior is well described by the predictions of Kosterlitz and Thouless almost three decades ago. In particular it is confirmed that the spatial winding number susceptibility obeys the well known predictions of finite size scaling for $T<{T}_{c}$ and up to logarithmic corrections the pair susceptibility scales as ${L}^{2\ensuremath{-}\ensuremath{\eta}}$ at large volumes with $0<~\ensuremath{\eta}<~0.25$ for $0<~T<~{T}_{c}.$

DOI
10.1103/physrevb.66.045113

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An extension of the attractive Hubbard model is constructed to study the critical behavior near a finite-temperature superconducting phase transition in two dimensions using the recently developed meron-cluster algorithm. Unlike previous calculations in the attractive Hubbard model which were limited to small lattices, the algorithm is used to study the critical behavior on lattices as large as 128\ifmmode×\else\texttimes\fi128.128\ifmmode\times\else\texttimes\fi{}128.128\ifmmode×\else\texttimes\fi128. These precise results show that a fermionic system can undergo a finite temperature phase transition whose critical behavior is well described by the predictions of Kosterlitz and Thouless almost three decades ago. In particular it is confirmed that the spatial winding number susceptibility obeys the well known predictions of finite size scaling for T<TcT<{T}_{c}T<Tc​ and up to logarithmic corrections the pair susceptibility scales as L2\ensuremath−\ensuremathη{L}^{2\ensuremath{-}\ensuremath{\eta}}L2\ensuremath−\ensuremathη at large volumes with 0< \ensuremathη< 0.250<~\ensuremath{\eta}<~0.250< \ensuremathη< 0.25 for 0< T< Tc.0<~T<~{T}_{c}.0< T< Tc​.
We study a strongly correlated fermionic model with attractive interactions in the presence of disorder in two spatial dimensions. Our model has been designed so that it can be solved using the recently discovered meron-cluster approach. Although the model is unconventional it has the same symmetries as the Hubbard model. Since the naive algorithm is inefficient, we develop an algorithm by combining the meron-cluster technique with the directed-loop update. This combination allows us to compute the pair susceptibility and the winding number susceptibility accurately. We find that the sss-wave superconductivity, present in the clean model, does not disappear until the disorder reaches a temperature dependent critical strength. The critical behavior as a function of disorder close to the phase transition belongs to the Berezinky-Kosterlitz-Thouless universality class as expected. The fermionic degrees of freedom, although present, do not appear to play an important role near the phase transition.
The meron-cluster algorithm was previously used to extensively study the physics associated with the spontaneous breaking of a discrete symmetry. We recently discovered that a larger class of models with spontaneous breaking of continuous symmetries can also be simulated using the meron-cluster algorithm. Here we study one of these new models which belongs to the attractive Hubbard model family. In particular we study the spontaneous breaking of the U(1) fermion number symmetry in two dimensions and find clear evidence for a Kosterlitz-Thouless transition to a superconducting phase. 1.
Susceptibilities in the static limit ω→0 are evaluated to show that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional Hubbard model for optimal band parameters. By using quantum Monte Carlo methods, we have examined the size dependence of the spin susceptibility χ( Q ) at half filling and the pair susceptibilities with d - and s -wave symmetries for the repulsive and attractive interactions, respectively. The numerical results for systems of up to 14×14 sites show that the s -wave pair susceptibility scales as χ pair ∼ L 2 for the attractive case, in agreement with previous quantum Monte Carlo studies. The scaling χ pair ∼ L 2 also holds for the d -wave pair susceptibility for the repulsive Hubbard model if we adjust the band parameter t '. Therefore, it is likely that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional repulsive Hubbard model.
Recent research shows that the partition function for a class of models involving fermions can be written as a statistical mechanics of clusters with positive definite weights. This new representation of the model allows one to construct novel algorithms. We illustrate this through models consisting of fermions with and without spin. A Hubbard type model with both attractive and repulsive interactions becomes tractable using the new approach. Precision results in the two dimensional attractive model confirm a superfluid phase transition in the Kosterlitz-Thouless universality class.
We examine the Kosterlitz-Thouless universality class and show that essential scaling at this type of phase transition is not self-consistent unless multiplicative logarithmic corrections are included. In the case of specific heat these logarithmic corrections are identified analytically. To identify those corresponding to the susceptibility we set up a numerical method involving the finite-size scaling of Lee-Yang zeroes. We also study the density of zeroes and introduce a new concept called index scaling. We apply the method to the XY model and the c]osely related step model in two dimensions. The critical parameters (including logarithmic corrections) of the step model are compatible with those of the XY model indicating that both models belong to the same universality class. This result then raises questions over how a vortex binding scenario can be the driving mechanism for the phase transition. Furthermore, the logarithmic corrections identified numerically by our methods of fitting are not in agreement with the renormalization group predictions of Kosterlitz and Thouless.
The two-dimensional spin-1/2 XY model is investigated via an extensive quantum Monte Carlo simulation on square lattices as large as 128\ifmmode\times\else\texttimes\fi{}128. The transverse susceptibility and correlation length show precise thermodynamic scaling behaviors of a phase transition of the Kosterlitz-Thouless type at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.353(3). The correlation functions near the transition point exhibit a universal scaling behavior. Various critical exponents are determined and our results demonstrate that the universality class of the two-dimensional quantum system with continuous symmetry is the same as its classical analog. The specific heat exhibits a finite peak around kT/J=0.45, decreases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​, and falls off as T2{\mathit{T}}^{2}T2 near T=0, validating the spin-wave treatment. The ground-state energy is -0.5543(1)J per spin.