Knizhnik-Zamolodchikov equations and integrable hyperbolic Landau-Zener models

We study the relationship between integrable Landau-Zener (LZ) models and Knizhnik-Zamolodchikov (KZ) equations. The latter are originally equations for the correlation functions of two-dimensional conformal field theories, but can also be interpreted as multi-time Schrödinger equations. The general LZ problem is to find probabilities of tunneling from eigenstates at t=t_\text{in} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mtext mathvariant="normal">in</mml:mtext> </mml:msub> </mml:mrow> </mml:math> to eigenstates at t\to+∞ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>→</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> for an N× N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>×</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> time-dependent Hamiltonian \hat H(t) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mover> <mml:mi>H</mml:mi> <mml:mo accent="true">̂</mml:mo> </mml:mover> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> . A number of such problems are exactly solvable in the sense that their tunneling probabilities are elementary functions of Hamiltonian parameters. Recently, it has been proposed that exactly solvable LZ models of this type map to KZ equations. Here we use this connection to identify and solve a class of integrable LZ models with hyperbolic time dependence, \hat H(t)=\hat A+\hat B/t <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mover> <mml:mi>H</mml:mi> <mml:mo accent="true">̂</mml:mo> </mml:mover> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>A</mml:mi> <mml:mo accent="true">̂</mml:mo> </mml:mover> <mml:mo>+</mml:mo> <mml:mover> <mml:mi>B</mml:mi> <mml:mo accent="true">̂</mml:mo> </mml:mover> <mml:mi>/</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> , for N=2, 3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , and 4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>4</mml:mn> </mml:math> , where \hat A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo accent="true">̂</mml:mo> </mml:mover> </mml:math> and \hat B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mover> <mml:mi>B</mml:mi> <mml:mo accent="true">̂</mml:mo> </mml:mover> </mml:math> are time-independent matrices.