First-order quantum phase transitions as condensations in the space of states

Abstract

We demonstrate that a large class of first-order quantum phase transitions, namely, transitions in which the ground state energy per particle is continuous but its first order derivative has a jump discontinuity, can be described as a condensation in the space of states. Given a system having Hamiltonian $H=K+gV$, where $K$ and $V$ are two non commuting operators acting on the space of states $\mathbb{F}$, we may always write $\mathbb{F}={\mathbb{F}}_{\mathrm{cond}}\oplus {\mathbb{F}}_{\mathrm{norm}}$ where ${\mathbb{F}}_{\mathrm{cond}}$ is the subspace spanned by the eigenstates of $V$ with minimal eigenvalue and ${\mathbb{F}}_{\mathrm{norm}}={\mathbb{F}}_{\mathrm{cond}}^{\perp }$. If, in the thermodynamic limit, $M_{\mathrm{cond}}/M\to 0$, where $M$ and $M_{\mathrm{cond}}$ are, respectively, the dimensions of $\mathbb{F}$ and ${\mathbb{F}}_{\mathrm{cond}}$, the above decomposition of $\mathbb{F}$ becomes effective, in the sense that the ground state energy per particle of the system, $\epsilon$, coincides with the smaller between ${\epsilon }_{\mathrm{cond}}$ and ${\epsilon }_{\mathrm{norm}}$, the ground state energies per particle of the system restricted to the subspaces ${\mathbb{F}}_{\mathrm{cond}}$ and ${\mathbb{F}}_{\mathrm{norm}}$, respectively: $\epsilon =\min \{{\epsilon }_{\mathrm{cond}},{\epsilon }_{\mathrm{norm}}\}$. It may then happen that, as a function of the parameter $g$, the energies ${\epsilon }_{\mathrm{cond}}$ and ${\epsilon }_{\mathrm{norm}}$ cross at $g=g_{\mathrm{c}}$. In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace ${\mathbb{F}}_{\mathrm{cond}}$) and a normal phase (system spread over the large subspace ${\mathbb{F}}_{\mathrm{norm}}$)....

21 pages, 12 figures

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• Finite temperature quantum condensations in the space of states: a new perspective for quantum annealing
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