Effective theory and breakdown of conformal symmetry in a long-range quantum chain

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• 作者：L. Lepori^a ; ^b ; ^{llepori81@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; ^{luca.lepori@pd.infn.it" class="auth_mail" title="E-mail the corresponding author} ; D. Vodola^b ; G. Pupillo^b ; G. Gori^c ; A. Trombettoni^c ; ^d
• 关键词：Long-range systems ; Integrable models ; Quantum phase transitions ; Renormalization group ; Conformal field theory ; Quantum entanglement
• 刊名：Annals of Physics
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：374
• 期：Complete
• 页码：35-66
• 全文大小：797 K

文摘

We deal with the problem of studying the symmetries and the effective theories of long-range models around their critical points. A prominent issue is to determine whether they possess (or not) conformal symmetry (CS) at criticality and how the presence of CS depends on the range of the interactions. To have a model, both simple to treat and interesting, where to investigate these questions, we focus on the Kitaev chain with long-range pairings decaying with distance as power-law with exponent α$\alpha$. This is a quadratic solvable model, yet displaying non-trivial quantum phase transitions. Two critical lines are found, occurring respectively at a positive and a negative chemical potential. Focusing first on the critical line at positive chemical potential, by means of a renormalization group approach we derive its effective theory close to criticality. Our main result is that the effective action is the sum of two terms: a Dirac action $S_D$, found in the short-range Ising universality class, and an “anomalous” CS breaking term $S_{\mathrm{AN}}$. While $S_D$ originates from low-energy excitations in the spectrum, $S_{\mathrm{AN}}$ originates from the higher energy modes where singularities develop, due to the long-range nature of the model. At criticality $S_{\mathrm{AN}}$ flows to zero for 1fd0c6b7a2c40d0749f46e4e3723" title="Click to view the MathML source">α>2$\alpha >2$, while for α<2$\alpha <2$ it dominates and determines the breakdown of the CS. Out of criticality $S_{\mathrm{AN}}$ breaks, in the considered approximation, the effective Lorentz invariance (ELI) for every finite α$\alpha$. As α$\alpha$ increases such ELI breakdown becomes less and less pronounced and in the short-range limit 1f31f6eb7748ab038a30b9dbc0" title="Click to view the MathML source">α→∞$\alpha \to \infty$ the ELI is restored. In order to test the validity of the determined effective theory, we compared the two-fermion static correlation functions and the von Neumann entropy obtained from them with the ones calculated on the lattice, finding agreement. These results explain two observed features characteristic of long-range models, the hybrid decay of static correlation functions within gapped phases and the area-law violation for the von Neumann entropy. The proposed scenario is expected to hold in other long-range models displaying quasiparticle excitations in ballistic regime. From the effective theory one can also see that new phases emerge for α<1$\alpha <1$. Finally we show that at every finite α$\alpha$ the critical exponents, defined as for the short-range (1f31f6eb7748ab038a30b9dbc0" title="Click to view the MathML source">α→∞$\alpha \to \infty$) model, are not altered. This also shows that the long-range paired Kitaev chain provides an example of a long-range model in which the value of α$\alpha$ where the CS is broken does not coincide with the value at which the critical exponents start to differ from the ones of the corresponding short-range model. At variance, for the second critical line, having negative chemical potential, only $S_{\mathrm{AN}}$ ($S_D$) is present for 1<α<2$1<\alpha <2$ (for 1fd0c6b7a2c40d0749f46e4e3723" title="Click to view the MathML source">α>2$\alpha >2$). Close to this line, where the minimum of the spectrum coincides with the momentum where singularities develop, the critical exponents change where CS is broken.