Tighter upper bounds on the critical temperature of two-dimensional superfluids and superconductors from the BCS to the Bose regime

Abstract We discuss standard and tighter upper bounds on the critical temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>c</mml:mtext> </mml:msub> </mml:mrow> </mml:math> of two-dimensional superfluids and superconductors versus particle density n or filling factor ν for continuum and lattice systems from the Bardeen–Cooper–Schrieffer (BCS) to the Bose regime. We consider only one-band Hamiltonians, where the transition from the normal to the superfluid (superconducting) phase is governed by the Berezinskii–Kosterlitz–Thouless (BKT) mechanism of vortex-antivortex binding, such that a direct relation between the superfluid density tensor and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>c</mml:mtext> </mml:msub> </mml:mrow> </mml:math> exists. The standard critical temperature upper bound <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mtext>c</mml:mtext> <mml:mrow> <mml:mrow> <mml:mi>up</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> is obtained from the Ferrell-Glover-Tinkham sum rule for the optical conductivity, which constrains the superfluid density tensor components. We demonstrate that it is imperative to consider at least the full effect of phase fluctuations of the order parameter for superfluidity (superconductivity) and use the renormalization group to obtain the phase-fluctuation critical temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mtext>c</mml:mtext> <mml:mrow> <mml:mstyle scriptlevel="0"/> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> , a much tighter bound to the critical temperature supremum than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mtext>c</mml:mtext> <mml:mrow> <mml:mrow> <mml:mi>up</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> over a wide range of densities or filling factors. We also discuss a fundamental difference between superfluids and superconductors in regards to the vortex core energy dependence on density. Going beyond phase fluctuations, we note that theories including modulus fluctuations of the order parameter or particle-hole fluctuations valid throughout the BCS-Bose evolution are still lacking, but the inclusion of these fluctuations can only produce a critical temperature that is lower than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mtext>c</mml:mtext> <mml:mrow> <mml:mstyle scriptlevel="0"/> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> and thus produce an even tighter bound to the critical temperature supremum. We conclude by indicating that if the measured critical temperature exceeds <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mtext>c</mml:mtext> <mml:mrow> <mml:mstyle scriptlevel="0"/> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> in experiments involving two-dimensional single-band systems, then a non-BKT mechanism must be invoked to describe the superfluid (superconducting) transition.