First-order quantum phase transitions as condensations in the space of states,Journal of Physics A: Mathematical and Theoretical

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First-order quantum phase transitions as condensations in the space of states
Journal of Physics A: Mathematical and Theoretical ( IF 2.0 ) Pub Date : 2021-01-13 , DOI: 10.1088/1751-8121/aba144
Massimo Ostilli ₁ , Carlo Presilla _{2,

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Affiliation

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 $\mathbb{F}={\mathbb{F}}_{\text{cond}}\oplus {\mathbb{F}}_{\mathrm{norm}}$ where ${\mathbb{F}}_{\text{cond}}$ is the subspace spanned by the eigenstates of V with minimal eigenvalue and ${\mathbb{F}}_{\mathrm{norm}}={\mathbb{F}}_{\text{cond}}^{\perp }$ . If, in the thermodynamic limit, M _cond/M → 0, where M and M _cond are, respectively, the dimensions of $\mathbb{F}$ and ${\mathbb{F}}_{\text{cond}}$ , the above decomposition of $\mathbb{F}$ becomes effective, in the sense that the ground state energy per particle of the system, ϵ, coincides with the smaller between ϵ _cond and ϵ _norm, the ground state energies per particle of the system restricted to the subspaces ${\mathbb{F}}_{\text{cond}}$ and ${\mathbb{F}}_{\mathrm{norm}}$ , respectively: ϵ = min{ϵ _cond, ϵ _norm}. It may then happen that, as a function of the parameter g, the energies ϵ _cond and ϵ _norm cross at g = g _c. In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace ${\mathbb{F}}_{\text{cond}}$ ) and a normal phase (system spread over the large subspace ${\mathbb{F}}_{\mathrm{norm}}$ ). Since, in the thermodynamic limit, M _cond/M → 0, the confinement into ${\mathbb{F}}_{\text{cond}}$ is actually a condensation in which the system falls into a ground state orthogonal to that of the normal phase, something reminiscent of Anderson’s orthogonality catastrophe (Anderson 1967 Phys. Rev. Lett. 18 1049). The outlined mechanism is tested on a variety of benchmark lattice models, including spin systems, free fermions with non uniform fields, interacting fermions and interacting hard-core bosons.

中文翻译：

一阶量子相变为状态空间中的凝聚

我们证明了一大类一阶量子相变，即每个粒子的基态能量连续但其一阶导数具有跳跃不连续性的跃迁，可以描述为状态空间中的凝聚。给定一个具有哈密顿H = K + gV的系统，其中K和V是作用在状态空间上的两个非交换算子 $\ mathbb {F}$ ，我们总是可以写出 $\ mathbb {F} = {\ mathbb {F}} _ {\ text {cond}} \ oplus {\ mathbb {F}} _ {\ mathrm {norm}}$ 其中 ${\ mathbb {F}} _ {\ text {cond}}$ 是V的本征态所跨越的子空间，其特征值和为最小 ${\ mathbb {F}} _ {\ mathrm {norm}} = {\ mathbb {F}} __ \ text {cond}} ^ {\ perp}$ 。如果在热力学极限下，M _cond / M →0，其中M和中号 _COND分别是的尺寸 $\ mathbb {F}$ 和 ${\ mathbb {F}} _ {\ text {cond}}$ ，上述分解的 $\ mathbb {F}$ 生效，在这个意义上，每个系统中，粒子的基态能量ε，与一致的之间的较小ε _COND和ε _范数，基态能量分别限制在子空间 ${\ mathbb {F}} _ {\ text {cond}}$ 和中的每个系统粒子 ${\ mathbb {F}} _ {\ mathrm {norm}}$ ：ϵ = min { ϵ _cond，ϵ _norm }。然后可能发生，作为参数g的函数，能量ϵ _cond和ϵ _norm在g = g _c处交叉。在这种情况下，一阶量子相变发生在一个凝聚相（系统限于小子空间 ${\ mathbb {F}} _ {\ text {cond}}$ ）和一个正常相（系统散布在大子空间 ${\ mathbb {F}} _ {\ mathrm {norm}}$ ）之间。由于在热力学极限M _cond / M →0中，限制 ${\ mathbb {F}} _ {\ text {cond}}$ 实际上是一种冷凝，其中系统进入与正相正交的基态，这使人联想到安德森的正交性灾难（安德森1967年物理学报莱特 181049）。在各种基准晶格模型上测试了概述的机理，包括自旋系统，具有非均匀场的自由费米子，相互作用的费米子和相互作用的硬核玻色子。

更新日期：2021-01-13

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