### 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.
### 2012

Markus Müller, Sebastian Diehl, Guido Pupillo, Peter Zoller

Engineered Open Systems and Quantum Simulations with Atoms and Ions Incollection

Berman, Paul; Arimondo, Ennio; Lin, Chun (Ed.): Advances In Atomic, Molecular, and Optical Physics, 61 , pp. 1-80, Academic Press, 2012.

Abstract | Links | BibTeX | Tags: Atomic physics, Open quantum systems, Quantum phase transitions, Quantum simulation, Topological phases of matter, Trapped ions, Unconventional pairing mechanisms

@incollection{Muller2012,

title = {Engineered Open Systems and Quantum Simulations with Atoms and Ions},

author = {Markus Müller and Sebastian Diehl and Guido Pupillo and Peter Zoller},

editor = {Paul Berman and Ennio Arimondo and Chun Lin},

doi = {10.1016/B978-0-12-396482-3.00001-6},

year = {2012},

date = {2012-07-01},

booktitle = {Advances In Atomic, Molecular, and Optical Physics},

volume = {61},

pages = {1-80},

publisher = {Academic Press},

series = {Advances in Atomic, Molecular, and Optical Physics},

abstract = {The enormous experimental progress in atomic, molecular, and optical (AMO) physics during the last decades allows us nowadays to isolate single, a few or even many-body ensembles of microscopic particles, and to manipulate their quantum properties at a level of precision, which still seemed unthinkable some years ago. This versatile set of tools has enabled the development of the well-established concept of engineering of many-body Hamiltonians in various physical platforms. These available tools, however, can also be harnessed to extend the scenario of Hamiltonian engineering to a more general Liouvillian setting, which in addition to coherent dynamics also includes controlled dissipation in many-body quantum systems. Here, we review recent theoretical and experimental progress in different directions along these lines, with a particular focus on physical realizations with systems of atoms and ions. This comprises digital quantum simulations in a general open system setting, as well as engineering and understanding new classes of systems far away from thermodynamic equilibrium. In the context of digital quantum simulation, we first outline the basic concepts and illustrate them on the basis of a recent experiment with trapped ions. We also discuss theoretical work proposing an intrinsically scalable simulation architecture for spin models with high-order interactions such as Kitaev's toric code, based on Rydberg atoms stored in optical lattices. We then turn to the digital simulation of dissipative many-body dynamics, pointing out a route for the general quantum state preparation in complex spin models, and discuss a recent experiment demonstrating the basic building blocks of a full-fledged open-system quantum simulator. In view of creating novel classes of out-of-equilibrium systems, we focus on ultracold atoms. We point out how quantum mechanical long-range order can be established via engineered dissipation, and present genuine many-body aspects of this setting: in the context of bosons, we discuss dynamical phase transitions resulting from competing Hamiltonian and dissipative dynamics. In the context of fermions, we present a purely dissipative pairing mechanism, and show how this could pave the way for the quantum simulation of the Fermi–Hubbard model. We also propose and analyze the key properties of dissipatively targeted topological phases of matter.},

keywords = {Atomic physics, Open quantum systems, Quantum phase transitions, Quantum simulation, Topological phases of matter, Trapped ions, Unconventional pairing mechanisms},

pubstate = {published},

tppubtype = {incollection}

}

The enormous experimental progress in atomic, molecular, and optical (AMO) physics during the last decades allows us nowadays to isolate single, a few or even many-body ensembles of microscopic particles, and to manipulate their quantum properties at a level of precision, which still seemed unthinkable some years ago. This versatile set of tools has enabled the development of the well-established concept of engineering of many-body Hamiltonians in various physical platforms. These available tools, however, can also be harnessed to extend the scenario of Hamiltonian engineering to a more general Liouvillian setting, which in addition to coherent dynamics also includes controlled dissipation in many-body quantum systems. Here, we review recent theoretical and experimental progress in different directions along these lines, with a particular focus on physical realizations with systems of atoms and ions. This comprises digital quantum simulations in a general open system setting, as well as engineering and understanding new classes of systems far away from thermodynamic equilibrium. In the context of digital quantum simulation, we first outline the basic concepts and illustrate them on the basis of a recent experiment with trapped ions. We also discuss theoretical work proposing an intrinsically scalable simulation architecture for spin models with high-order interactions such as Kitaev's toric code, based on Rydberg atoms stored in optical lattices. We then turn to the digital simulation of dissipative many-body dynamics, pointing out a route for the general quantum state preparation in complex spin models, and discuss a recent experiment demonstrating the basic building blocks of a full-fledged open-system quantum simulator. In view of creating novel classes of out-of-equilibrium systems, we focus on ultracold atoms. We point out how quantum mechanical long-range order can be established via engineered dissipation, and present genuine many-body aspects of this setting: in the context of bosons, we discuss dynamical phase transitions resulting from competing Hamiltonian and dissipative dynamics. In the context of fermions, we present a purely dissipative pairing mechanism, and show how this could pave the way for the quantum simulation of the Fermi–Hubbard model. We also propose and analyze the key properties of dissipatively targeted topological phases of matter.

# 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.

### 2012

Engineered Open Systems and Quantum Simulations with Atoms and Ions Incollection

Berman, Paul; Arimondo, Ennio; Lin, Chun (Ed.): Advances In Atomic, Molecular, and Optical Physics, 61 , pp. 1-80, Academic Press, 2012.