### 2016

L Lepori, D Vodola, G Pupillo, G Gori, A Trombettoni

Effective Theory and Breakdown of Conformal Symmetry in a Long-Range Quantum Chain Journal Article

Annals of Physics, 374 , pp. 35-66, 2016, ISSN: 0003-4916.

Abstract | Links | BibTeX | Tags: Conformal field theory, Integrable models, Long-range systems, Quantum entanglement, Quantum phase transitions, Renormalization group

@article{Lepori2016,

title = {Effective Theory and Breakdown of Conformal Symmetry in a Long-Range Quantum Chain},

author = {L Lepori and D Vodola and G Pupillo and G Gori and A Trombettoni},

doi = {10.1016/j.aop.2016.07.026},

issn = {0003-4916},

year = {2016},

date = {2016-11-01},

journal = {Annals of Physics},

volume = {374},

pages = {35-66},

abstract = {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 $alpha$. 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 SD, found in the short-range Ising universality class, and an ``anomalous'' CS breaking term SAN. While SD originates from low-energy excitations in the spectrum, SAN originates from the higher energy modes where singularities develop, due to the long-range nature of the model. At criticality SAN flows to zero for $alpha>$2, while for $alpha<$2 it dominates and determines the breakdown of the CS. Out of criticality SAN breaks, in the considered approximation, the effective Lorentz invariance (ELI) for every finite $alpha$. As $alpha$ increases such ELI breakdown becomes less and less pronounced and in the short-range limit $alpharightarrowinfty$ 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 $alpha<$1. Finally we show that at every finite $alpha$ the critical exponents, defined as for the short-range ($alpharightarrowinfty$) 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 $alpha$ 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 SAN (SD) is present for 1$$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.},

keywords = {Conformal field theory, Integrable models, Long-range systems, Quantum entanglement, Quantum phase transitions, Renormalization group},

pubstate = {published},

tppubtype = {article}

}

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 $alpha$. 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 SD, found in the short-range Ising universality class, and an ``anomalous'' CS breaking term SAN. While SD originates from low-energy excitations in the spectrum, SAN originates from the higher energy modes where singularities develop, due to the long-range nature of the model. At criticality SAN flows to zero for $alpha>$2, while for $alpha<$2 it dominates and determines the breakdown of the CS. Out of criticality SAN breaks, in the considered approximation, the effective Lorentz invariance (ELI) for every finite $alpha$. As $alpha$ increases such ELI breakdown becomes less and less pronounced and in the short-range limit $alpharightarrowinfty$ 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 $alpha<$1. Finally we show that at every finite $alpha$ the critical exponents, defined as for the short-range ($alpharightarrowinfty$) 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 $alpha$ 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 SAN (SD) is present for 1$<alpha<$2 (for $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.

# Publications

### 2016

Effective Theory and Breakdown of Conformal Symmetry in a Long-Range Quantum Chain Journal Article

Annals of Physics, 374 , pp. 35-66, 2016, ISSN: 0003-4916.