Nonrelativistic conformality, contact correlation, and duality in one dimension

The field theory of nonrelativistic fermions interacting via contact interactions can be used to calculate the properties of few-body systems of cold atoms confined in harmonic traps. The state-operator correspondence of nonrelativistic conformal field theory (NRCFT) shows that the energy eigenvalues (in oscillator units) of $N$ harmonically trapped fermions can be calculated from the scaling dimensions of $N$ -fermion operators in the NRCFT. They are also in one-to-one correspondence with zero energy, scale-invariant solutions to the $N$ -body problem in free space. We show that these two mappings of the trapped fermion problem to free space problems are related by an automorphism of the $SL(2,R)$ algebra of the conformal symmetry of fermions at the unitary limit. This automorphism exchanges the internal Hamiltonian of the gas with the trapping potential and hence provides a method for deriving virial theorems for trapped Fermi gases at the unitary limit. We also show that the state-operator correspondence can be applied directly in three spatial dimensions by calculating the scaling dimensions of two- and three-fermion operators and finding agreement with known exact results for energy levels of two and three trapped fermions at the unitary limit.The field theory of nonrelativistic fermions interacting via contact interactions can be used to calculate the properties of few-body systems of cold atoms confined in harmonic traps. The state-operator correspondence of nonrelativistic conformal field theory (NRCFT) shows that the energy eigenvalues (in oscillator units) of $N$ harmonically trapped fermions can be calculated from the scaling dimensions of $N$-fermion operators in the NRCFT. They are also in one-to-one correspondence with zero energy, scale-invariant solutions to the $N$-body problem in free space. We show that these two mappings of the trapped fermion problem to free space problems are related by an automorphism of the $SL(2,R)$ algebra of the conformal symmetry of fermions at the unitary limit. This automorphism exchanges the internal Hamiltonian of the gas with the trapping potential and hence provides a method for deriving virial theorems for trapped Fermi gases at the unitary limit. We also show that the state-operator correspondence can be applied directly in three spatial dimensions by calculating the scaling dimensions of two- and three-fermion operators and finding agreement with known exact results for energy levels of two and three trapped fermions at the unitary limit.

arXiv:0712.0867Physical Review AJul 11, 200824 citations

Scattering by a contact potential in three and lower dimensions

Lin, Q

We consider the scattering of nonrelativistic particles in three dimensions by a contact potential $\\Omega\\hbar^2\\delta(r)/ 2\\mu r^\\alpha$ which is defined as the $a\\to 0$ limit of $\\Omega\\hbar^2\\delta(r-a)/2\\mu r^\\alpha$. It is surprising that it gives a nonvanishing cross section when $\\alpha=1$ and $\\Omega=-1$ . When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter $\\Omega$ that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.We consider the scattering of nonrelativistic particles in three dimensions by a contact potential $\\Omega\\hbar^2\\delta(r)/ 2\\mu r^\\alpha$ which is defined as the $a\\to 0$ limit of $\\Omega\\hbar^2\\delta(r-a)/2\\mu r^\\alpha$. It is surprising that it gives a nonvanishing cross section when $\\alpha=1$ and $\\Omega=-1$. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter $\\Omega$ that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.

Jan 1, 20015 citations

One-Dimensional Semirelativistic Bosons with Contact Interactions

Fatih Erman, O. Teoman Turgut

Jan 1, 20260 citations

Symmetry, Duality, and Anholonomy of Point Interactions in One Dimension

Taksu Cheon, Tamás Fülöp, Izumi Tsutsui

arXiv:quant-ph/0008123Annals of PhysicsNov 1, 200175 citations

Scattering by a contact potential in three and lower dimensions

Qiong-gui Lin

We consider the scattering of nonrelativistic particles in three dimensions by a contact potential Ωℏ 2 δ( r )/2µ r α , which is defined as the a →0 limit of Ωℏ 2 δ( r - a )/2µ r α . It is surprising that it gives a nonvanishing cross section when α = 1 and Ω = -1. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter Ω that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.We consider the scattering of nonrelativistic particles in three dimensions by a contact potential Ωℏ 2 δ( r )/2µ r α , which is defined as the a →0 limit of Ωℏ 2 δ( r - a )/2µ r α . It is surprising that it gives a nonvanishing cross section when α = 1 and Ω = -1. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter Ω that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.

arXiv:quant-ph/0103013Journal of Physics A Mathematical and GeneralFeb 26, 20014 citations

N

harmonically trapped fermions can be calculated from the scaling dimensions of

N

-fermion operators in the NRCFT. They are also in one-to-one correspondence with zero energy, scale-invariant solutions to the

N

-body problem in free space. We show that these two mappings of the trapped fermion problem to free space problems are related by an automorphism of the

SL(2,R)

algebra of the conformal symmetry of fermions at the unitary limit. This automorphism exchanges the internal Hamiltonian of the gas with the trapping potential and hence provides a method for deriving virial theorems for trapped Fermi gases at the unitary limit. We also show that the state-operator correspondence can be applied directly in three spatial dimensions by calculating the scaling dimensions of two- and three-fermion operators and finding agreement with known exact results for energy levels of two and three trapped fermions at the unitary limit.

We consider the scattering of nonrelativistic particles in three dimensions by a contact potential $\\Omega\\hbar^2\\delta(r)/ 2\\mu r^\\alpha$ which is defined as the

a\\to 0

limit of $\\Omega\\hbar^2\\delta(r-a)/2\\mu r^\\alpha$. It is surprising that it gives a nonvanishing cross section when

\\alpha=1

and

\\Omega=-1

. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter

\\Omega

that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.