Stacked quantum Ising systems and quantum Ashkin-Teller model

We analyze the quantum states of an isolated composite system consisting of two stacked quantum Ising (SQI) subsystems, coupled by a local Hamiltonian term that preserves the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msub> <a:mi mathvariant="double-struck">Z</a:mi> <a:mn>2</a:mn> </a:msub> </a:math> symmetry of each subsystem. The coupling strength is controlled by an intercoupling parameter <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>w</c:mi> </c:math> , with <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>w</d:mi> <d:mo>=</d:mo> <d:mn>0</d:mn> </d:mrow> </d:math> corresponding to decoupled quantum Ising subsystems. We focus on the quantum correlations of one of the two SQI subsystems, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi mathvariant="script">S</e:mi> </e:math> , in the ground state of the global system and study their dependence on both the state of the weakly coupled complementary part <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi mathvariant="script">E</g:mi> </g:math> and the intercoupling strength. We concentrate on regimes in which <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi mathvariant="script">S</i:mi> </i:math> develops critical long-range correlations. The most interesting physical scenario arises when both SQI subsystems are critical. In particular, for identical SQI subsystems, the global system is equivalent to the quantum Ashkin-Teller model, characterized by an additional <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msub> <k:mi mathvariant="double-struck">Z</k:mi> <k:mn>2</k:mn> </k:msub> </k:math> interchange symmetry between the two subsystem operators. In this limit, one-dimensional SQI systems exhibit a peculiar critical line along which the length-scale critical exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ν</m:mi> </m:math> varies continuously with <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mi>w</n:mi> </n:math> , while two-dimensional systems develop quantum multicritical behaviors characterized by an effective enlargement of the symmetry of the critical modes, from the actual discrete <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:msub> <o:mi mathvariant="double-struck">Z</o:mi> <o:mn>2</o:mn> </o:msub> <o:mo>⊕</o:mo> <o:msub> <o:mi mathvariant="double-struck">Z</o:mi> <o:mn>2</o:mn> </o:msub> </o:mrow> </o:math> symmetry to an effective continuous O(2) symmetry.