Numerical signatures of ultra-local criticality in a one dimensional Kondo lattice model

Heavy fermion criticality has been a long-standing problem in condensed matter physics. Here we study a one-dimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo coupling J_K <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> at fixed doping x. At large positive J_K <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> , we confirm the expected conventional Luttinger liquid phase with 2k_F=\frac{1+x}{2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> (in units of 2\pi <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> ), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In the J_K ≤ 0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo>≤</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> side, our simulation finds the existence of a fractional Luttinger liquid (LL \star <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>⋆</mml:mo> </mml:math> ) phase with 2k_F=\frac{x}{2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> , accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL \star <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>⋆</mml:mo> </mml:math> ) phase in higher dimensions. The LL \star <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>⋆</mml:mo> </mml:math> phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE) liquid phase at small positive J_K <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> . Then we mainly focus on the “critical regime” between the LE phase and the LL phase. Approaching the critical point from the spin-gapped LE phase, we often find that the spin gap vanishes continuously, while the spin-spin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) of J_K <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> , the dynamical spin structure factor obtained through the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a finite range of momentum space above a small energy scale (around 0.035 J <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>0.035</mml:mn> <mml:mi>J</mml:mi> </mml:mrow> </mml:math> ) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultra-local criticality with an infinite dynamical exponent z=+ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:math> . The numerical discovery here may have important implications on our general theoretical understanding of the strange metals in heavy fermion systems. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference.