Quantum quenches across continuous and first-order quantum transitions in one-dimensional quantum Ising models

We investigate the quantum dynamics generated by quantum quenches (QQs) of the Hamiltonian parameters in many-body systems, focusing on protocols that cross first-order and continuous quantum transitions, both in finite-size systems and in the thermodynamic limit. As a paradigmatic example, we consider the quantum Ising chain in the presence of homogeneous transverse ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>g</a:mi> </a:math> ) and longitudinal ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>h</b:mi> </b:math> ) magnetic fields. This model exhibits a continuous quantum transition (CQT) at <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>g</c:mi> <c:mo>=</c:mo> <c:msub> <c:mi>g</c:mi> <c:mi>c</c:mi> </c:msub> </c:mrow> </c:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>h</d:mi> <d:mo>=</d:mo> <d:mn>0</d:mn> </d:mrow> </d:math> , and first-order quantum transitions (FOQTs) driven by <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>h</e:mi> </e:math> along the line <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>h</f:mi> <f:mo>=</f:mo> <f:mn>0</f:mn> </f:mrow> </f:math> ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>g</g:mi> <g:mo>&lt;</g:mo> <g:msub> <g:mi>g</g:mi> <g:mi>c</g:mi> </g:msub> </g:mrow> </g:math> ). In the integrable limit <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mi>h</h:mi> <h:mo>=</h:mo> <h:mn>0</h:mn> </h:mrow> </h:math> , the system can be mapped onto a quadratic fermionic theory; however, any nonvanishing longitudinal field breaks integrability and the spectrum of the resulting Hamiltonian is generally expected to enter a chaotic regime. We analyze QQs in which the longitudinal field is suddenly changed from a negative value <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:msub> <i:mi>h</i:mi> <i:mi>i</i:mi> </i:msub> <i:mo>&lt;</i:mo> <i:mn>0</i:mn> </i:mrow> </i:math> to a positive value <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:msub> <j:mi>h</j:mi> <j:mi>f</j:mi> </j:msub> <j:mo>&gt;</j:mo> <j:mn>0</j:mn> </j:mrow> </j:math> . We focus on values of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msub> <k:mi>h</k:mi> <k:mi>f</k:mi> </k:msub> </k:math> such that the spectrum of the post-QQ Hamiltonian <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mover accent="true"> <l:mi>H</l:mi> <l:mo>̂</l:mo> </l:mover> <l:mrow> <l:mo>(</l:mo> <l:mi>g</l:mi> <l:mo>,</l:mo> <l:msub> <l:mi>h</l:mi> <l:mi>f</l:mi> </l:msub> <l:mo>)</l:mo> </l:mrow> </l:mrow> </l:math> lies in the chaotic regime, where thermalization may emerge at asymptotically long times. We study the out-of-equilibrium dynamics for different values of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mi>g</n:mi> </n:math> , finding qualitatively distinct behaviors for <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:mi>g</o:mi> <o:mo>&gt;</o:mo> <o:msub> <o:mi>g</o:mi> <o:mi>c</o:mi> </o:msub> </o:mrow> </o:math> (where the chain is in the disordered phase), for <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mi>g</p:mi> <p:mo>=</p:mo> <p:msub> <p:mi>g</p:mi> <p:mi>c</p:mi> </p:msub> </p:mrow> </p:math> (QQ across the CQT), and for <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:mrow> <q:mi>g</q:mi> <q:mo>&lt;</q:mo> <q:msub> <q:mi>g</q:mi> <q:mi>c</q:mi> </q:msub> </q:mrow> </q:math> (QQ across the FOQT line).