Finite-temperature properties of antiferromagnetic quantum spin chains with random long-range couplings

We study excited states and finite temperature properties of chains of randomly placed spins with long range interactions. To this end we extend the recently introduced strong disorder renormalization group method in real space, well suited to study bond disordered antiferromagnetic power law coupled quantum spin chains. For bond disordered, short range coupled spin chains, defined by keeping only interactions between adjacent spins, we show that the distribution of the absolute value of the couplings is the infinite randomness fixed point distribution also for the excited states. Moreover, we derive the distribution function of the sign of the couplings and find that the number of negative couplings increases with temperature <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>T</a:mi> <a:mo>.</a:mo> </a:mrow> </a:math> For power law long range interaction with power exponent <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>α</b:mi> </b:math> we derive and solve the Master equation for the distribution function of couplings for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>α</c:mi> <c:mo>&gt;</c:mo> <c:mn>2</c:mn> </c:mrow> </c:math> . The coupling amplitude is found to have the strong disorder distribution with finite width <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mn>2</d:mn> <d:mi>α</d:mi> <d:mo>,</d:mo> </d:mrow> </d:math> with small corrections. We derive resulting finite temperature properties of both short and power law long ranged spin systems for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>α</e:mi> <e:mo>&gt;</e:mo> <e:mn>2</e:mn> </e:mrow> </e:math> . The magnetic susceptibility is dominated by paramagnetic <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>S</f:mi> <f:mo>=</f:mo> <f:mn>1</f:mn> <f:mo>/</f:mo> <f:mn>2</f:mn> </f:mrow> </f:math> -spin, resulting in a Curie law susceptibility. The distribution function of pair lengths, and the spin correlation function are found be the same as in the ground state. We find that the entanglement entropy increases logarithmically with the partition length <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>n</g:mi> </g:math> and has the same functional form as for the ground state of critical quantum spin chains with temperature dependent effective central charge <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mover accent="true"> <h:mi>c</h:mi> <h:mo>¯</h:mo> </h:mover> <h:mrow> <h:mo>(</h:mo> <h:mi>T</h:mi> <h:mo>)</h:mo> </h:mrow> </h:mrow> </h:math> , which converges at high temperature to <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mfrac> <j:mn>1</j:mn> <j:mn>2</j:mn> </j:mfrac> <j:mo form="prefix">ln</j:mo> <j:mn>2</j:mn> </j:mrow> </j:math> , half the value it has in the ground state. We confirm that the entanglement entropy grows with time <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mi>t</l:mi> </l:math> after a quantum quench logarithmically as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>S</m:mi> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> <m:mo>∼</m:mo> <m:mo form="prefix">ln</m:mo> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> <m:mo>/</m:mo> <m:mo>(</m:mo> <m:mn>2</m:mn> <m:mi>α</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> .