2025/10/08 by Edward McCann, McCann, Edward
Physics and Astronomy · #Physics of Superconductivity and Magnetism #Quantum and electron transport phenomena #Quantum many-body systems
paper · pdf · doi:10.1103/65hp-72hs
We model <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>p</a:mi> </a:math> -state Fock parafermions on a lattice in one dimension (with occupation per orbital of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mn>0</b:mn> <b:mo>,</b:mo> <b:mn>1</b:mn> <b:mo>,</b:mo> <b:mo>...</b:mo> <b:mo>,</b:mo> <b:mi>p</b:mi> <b:mo>−</b:mo> <b:mn>1</b:mn> </b:mrow> </b:math> ). For <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>p</c:mi> </c:math> a composite number, they may be mapped to <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>q</d:mi> <d:mi>m</d:mi> </d:msub> </d:math> -state parafermions where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi>q</e:mi> <e:mi>m</e:mi> </e:msub> </e:math> are the prime factors of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>p</f:mi> </f:math> . For a Hamiltonian with a single-particle spectrum, the parafermions decompose into <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msub> <g:mi>q</g:mi> <g:mi>m</g:mi> </g:msub> </g:math> -state parafermions. When <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>p</h:mi> </h:math> is a power of two, the decomposition is into fermions. We use this to construct a parafermionic Hamiltonian for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mi>p</i:mi> <i:mo>=</i:mo> <i:mn>4</i:mn> </i:mrow> </i:math> with a single-particle spectrum using a fermionic tight-binding model which is bilinear in creation and annihilation operators. The single-particle levels may be determined by diagonalizing a square matrix whose order scales linearly with system size, and they are the same as those of the fermionic model. We show that the intermediate statistics of the thermodynamic distribution function for the occupation numbers (known as Gentile statistics) are consistent with the mapping to fermions, and we provide an example calculation of the internal energy and heat capacity for a simple linear chain.