Long-range Ising and Kitaev models: phases, correlations and edge modes
We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2015-01-01
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| Series: | New Journal of Physics |
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| Online Access: | https://doi.org/10.1088/1367-2630/18/1/015001 |
| _version_ | 1797750798209253376 |
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| author | Davide Vodola Luca Lepori Elisa Ercolessi Guido Pupillo |
| author_facet | Davide Vodola Luca Lepori Elisa Ercolessi Guido Pupillo |
| author_sort | Davide Vodola |
| collection | DOAJ |
| description | We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance r as $1/{r}^{\alpha }$ , as well as a related class of fermionic Hamiltonians that generalize the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents α , based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for $\alpha \lesssim 1$ , while connected correlation functions can decay with a hybrid exponential and power-law behavior, with a power that is α -dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every α . Along the critical lines, for all models breaking of conformal symmetry is argued at low enough α . For the fermionic models we show that the edge modes, massless for $\alpha \gtrsim 1$ , can acquire a mass for $\alpha \lt 1$ . The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions. |
| first_indexed | 2024-03-12T16:39:02Z |
| format | Article |
| id | doaj.art-f608cb494c5b42169cceaad71532461a |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:39:02Z |
| publishDate | 2015-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-f608cb494c5b42169cceaad71532461a2023-08-08T14:38:27ZengIOP PublishingNew Journal of Physics1367-26302015-01-0118101500110.1088/1367-2630/18/1/015001Long-range Ising and Kitaev models: phases, correlations and edge modesDavide Vodola0Luca Lepori1Elisa Ercolessi2Guido Pupillo3icFRC, IPCMS (UMR 7504) and ISIS (UMR 7006), Université de Strasbourg and CNRS, F-67000 Strasbourg, FranceicFRC, IPCMS (UMR 7504) and ISIS (UMR 7006), Université de Strasbourg and CNRS, F-67000 Strasbourg, France; Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, I-35131 Padova, ItalyDipartimento di Fisica e Astronomia, Università di Bologna and INFN, Via Irnerio 46, I-40127 Bologna, ItalyicFRC, IPCMS (UMR 7504) and ISIS (UMR 7006), Université de Strasbourg and CNRS, F-67000 Strasbourg, FranceWe analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance r as $1/{r}^{\alpha }$ , as well as a related class of fermionic Hamiltonians that generalize the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents α , based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for $\alpha \lesssim 1$ , while connected correlation functions can decay with a hybrid exponential and power-law behavior, with a power that is α -dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every α . Along the critical lines, for all models breaking of conformal symmetry is argued at low enough α . For the fermionic models we show that the edge modes, massless for $\alpha \gtrsim 1$ , can acquire a mass for $\alpha \lt 1$ . The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.https://doi.org/10.1088/1367-2630/18/1/015001long-range interactionsspin chainsKitaev chainsarea law violationedge modesMajorana fermions |
| spellingShingle | Davide Vodola Luca Lepori Elisa Ercolessi Guido Pupillo Long-range Ising and Kitaev models: phases, correlations and edge modes New Journal of Physics long-range interactions spin chains Kitaev chains area law violation edge modes Majorana fermions |
| title | Long-range Ising and Kitaev models: phases, correlations and edge modes |
| title_full | Long-range Ising and Kitaev models: phases, correlations and edge modes |
| title_fullStr | Long-range Ising and Kitaev models: phases, correlations and edge modes |
| title_full_unstemmed | Long-range Ising and Kitaev models: phases, correlations and edge modes |
| title_short | Long-range Ising and Kitaev models: phases, correlations and edge modes |
| title_sort | long range ising and kitaev models phases correlations and edge modes |
| topic | long-range interactions spin chains Kitaev chains area law violation edge modes Majorana fermions |
| url | https://doi.org/10.1088/1367-2630/18/1/015001 |
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