Limits...
Optimal free descriptions of many-body theories

View Article: PubMed Central - PubMed

ABSTRACT

Interacting bosons or fermions give rise to some of the most fascinating phases of matter, including high-temperature superconductivity, the fractional quantum Hall effect, quantum spin liquids and Mott insulators. Although these systems are promising for technological applications, they also present conceptual challenges, as they require approaches beyond mean-field and perturbation theory. Here we develop a general framework for identifying the free theory that is closest to a given interacting model in terms of their ground-state correlations. Moreover, we quantify the distance between them using the entanglement spectrum. When this interaction distance is small, the optimal free theory provides an effective description of the low-energy physics of the interacting model. Our construction of the optimal free model is non-perturbative in nature; thus, it offers a theoretical framework for investigating strongly correlated systems.

No MeSH data available.


Related in: MedlinePlus

Mapping the optimal free model to a transverse-field Ising model.(a) The FM model and (b) the AFM model both at system size L=16. Contours indicate the transverse-field , dashed lines for those in the symmetry broken phase and solid otherwise (including the critical value =1). The background colour plot gives the distance minD(σ,σf) (log scale) signifying the success of the mapping. In this way the interacting system is given a description in terms of a free-fermion model. The region near the hz=0 classical axis of the AFM is removed (grey region) because our calculations do not resolve all symmetries of the system.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC5382319&req=5

f3: Mapping the optimal free model to a transverse-field Ising model.(a) The FM model and (b) the AFM model both at system size L=16. Contours indicate the transverse-field , dashed lines for those in the symmetry broken phase and solid otherwise (including the critical value =1). The background colour plot gives the distance minD(σ,σf) (log scale) signifying the success of the mapping. In this way the interacting system is given a description in terms of a free-fermion model. The region near the hz=0 classical axis of the AFM is removed (grey region) because our calculations do not resolve all symmetries of the system.

Mentions: Finally, we are in position to identify the optimal free model that describes the interacting system given by an instance of (5). In particular, we identify the free Ising model H±(,0) with transverse field , whose ground state's entanglement spectrum matches σ's obtained from (1) for each point (hz,hx). This is simply obtained by minimizing D(σ(hz,hx), σf(,0)) over . As a result we observe that in the FM case, adding infinitesimal interactions to the free Ising model with hz<1 maps the model to a free Ising with hz>1 in a discontinuous way, as shown in Fig. 3a. When hz>1, the introduction of interactions maps the model to a neighbouring free model continuously. In the AFM case, the interactions are irrelevant. Indeed, Fig. 3b shows that the whole phase diagram maps trivially to the free model even very near criticality.


Optimal free descriptions of many-body theories
Mapping the optimal free model to a transverse-field Ising model.(a) The FM model and (b) the AFM model both at system size L=16. Contours indicate the transverse-field , dashed lines for those in the symmetry broken phase and solid otherwise (including the critical value =1). The background colour plot gives the distance minD(σ,σf) (log scale) signifying the success of the mapping. In this way the interacting system is given a description in terms of a free-fermion model. The region near the hz=0 classical axis of the AFM is removed (grey region) because our calculations do not resolve all symmetries of the system.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5382319&req=5

f3: Mapping the optimal free model to a transverse-field Ising model.(a) The FM model and (b) the AFM model both at system size L=16. Contours indicate the transverse-field , dashed lines for those in the symmetry broken phase and solid otherwise (including the critical value =1). The background colour plot gives the distance minD(σ,σf) (log scale) signifying the success of the mapping. In this way the interacting system is given a description in terms of a free-fermion model. The region near the hz=0 classical axis of the AFM is removed (grey region) because our calculations do not resolve all symmetries of the system.
Mentions: Finally, we are in position to identify the optimal free model that describes the interacting system given by an instance of (5). In particular, we identify the free Ising model H±(,0) with transverse field , whose ground state's entanglement spectrum matches σ's obtained from (1) for each point (hz,hx). This is simply obtained by minimizing D(σ(hz,hx), σf(,0)) over . As a result we observe that in the FM case, adding infinitesimal interactions to the free Ising model with hz<1 maps the model to a free Ising with hz>1 in a discontinuous way, as shown in Fig. 3a. When hz>1, the introduction of interactions maps the model to a neighbouring free model continuously. In the AFM case, the interactions are irrelevant. Indeed, Fig. 3b shows that the whole phase diagram maps trivially to the free model even very near criticality.

View Article: PubMed Central - PubMed

ABSTRACT

Interacting bosons or fermions give rise to some of the most fascinating phases of matter, including high-temperature superconductivity, the fractional quantum Hall effect, quantum spin liquids and Mott insulators. Although these systems are promising for technological applications, they also present conceptual challenges, as they require approaches beyond mean-field and perturbation theory. Here we develop a general framework for identifying the free theory that is closest to a given interacting model in terms of their ground-state correlations. Moreover, we quantify the distance between them using the entanglement spectrum. When this interaction distance is small, the optimal free theory provides an effective description of the low-energy physics of the interacting model. Our construction of the optimal free model is non-perturbative in nature; thus, it offers a theoretical framework for investigating strongly correlated systems.

No MeSH data available.


Related in: MedlinePlus