Anisotropy-driven topological quantum phase transition in magnetic impurities

Abstract A few years ago, a topological quantum phase transition (TQPT) has been found in Anderson and Kondo 2-channel spin-1 impurity models that include a hard-axis anisotropy term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>D</mml:mi> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> with D &gt; 0. The most remarkable manifestation of the TQPT is a jump in the spectral density of localized electrons, at the Fermi level, from very high to very low values as D is increased. If the two conduction channels are equivalent, the transition takes place at the critical anisotropy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∼</mml:mo> <mml:mn>2.5</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">K</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">K</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> is the Kondo temperature for D = 0. This jump might be important to develop a molecular transistor. The jump is due to a corresponding one in the Luttinger integral, which has a topological non-trivial value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . Here, we review the main results for the spectral density and highlight the significance of the theory for the interpretation of measurements conducted on magnetic atoms or molecules on metallic surfaces. In these experiments, where D is held constant, the energy scale <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">K</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> is manipulated by some parameters. The resulting variation gives rise to a differential conductance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>I</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> , measured by scanning-tunneling spectroscopy, which is consistent with a TQPT at an intermediate value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">K</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . For non-equivalent channels and non-zero magnetic field, the topological phase is lost but still a peculiar behaviour in the spectral density is obtained which agrees with experimental observations. We also show that the theory can be extended to integer spin S &gt; 1 and two-impurity systems. This is also probably true for half-integer spin and non-equivalent channels in some cases.