Many-body systems of identical arbitrary-spin particles, with separable spin and spatial degrees of freedom, are considered. Their eigenstates can be classified by Young diagrams, corresponding to nontrivial permutation symmetries (beyond the conventional paradigm of symmetric-antisymmetric states). The present work obtains the following. (a) Selection rules for additional nonseparable (dependent on spins and coordinates) k-body interactions: the Young diagrams, associated with the initial and final states of a transition, can differ by relocation of no more than k boxes between their rows. (b) Correlation rules in which eigenstate-averaged local correlations of k particles vanish if k exceeds the number of columns (for bosons) or rows (for fermions) in the associated Young diagram. It also elucidates the physical meaning of the quantities conserved due to permutation symmetry-in 1929, Dirac identified those with characters of the symmetric group-relating them to experimentally observable correlations of several particles. The results provide a way to control the formation of entangled states belonging to multidimensional non-Abelian representations of the symmetric group. These states can find applications in quantum computation and metrology.Many-body systems of identical arbitrary-spin particles, with separable spin and spatial degrees of freedom, are considered. Their eigenstates can be classified by Young diagrams, corresponding to nontrivial permutation symmetries (beyond the conventional paradigm of symmetric-antisymmetric states). The present work obtains the following. (a) Selection rules for additional nonseparable (dependent on spins and coordinates) k-body interactions: the Young diagrams, associated with the initial and final states of a transition, can differ by relocation of no more than k boxes between their rows. (b) Correlation rules in which eigenstate-averaged local correlations of k particles vanish if k exceeds the number of columns (for bosons) or rows (for fermions) in the associated Young diagram. It also elucidates the physical meaning of the quantities conserved due to permutation symmetry-in 1929, Dirac identified those with characters of the symmetric group-relating them to experimentally observable correlations of several particles. The results provide a way to control the formation of entangled states belonging to multidimensional non-Abelian representations of the symmetric group. These states can find applications in quantum computation and metrology.

arXiv:1402.0776Physical Review LettersNov 14, 201416 citations

Phase Transitions in Definite Total Spin States of Two-Component Fermi Gases

Vladimir Yurovsky

Second-order phase transitions have no latent heat and are characterized by a change in symmetry. In addition to the conventional symmetric and antisymmetric states under permutations of bosons and fermions, mathematical group-representation theory allows for non-Abelian permutation symmetry. Such symmetry can be hidden in states with defined total spins of spinor gases, which can be formed in optical cavities. The present work shows that the symmetry reveals itself in spin-independent or coordinate-independent properties of these gases, namely as non-Abelian entropy in thermodynamic properties. In weakly interacting Fermi gases, two phases appear associated with fermionic and non-Abelian symmetry under permutations of particle states, respectively. The second-order transitions between the phases are characterized by discontinuities in specific heat. Unlike other phase transitions, the present ones are not caused by interactions and can appear even in ideal gases. Similar effects in Bose gases and strong interactions are discussed.Second-order phase transitions have no latent heat and are characterized by a change in symmetry. In addition to the conventional symmetric and antisymmetric states under permutations of bosons and fermions, mathematical group-representation theory allows for non-Abelian permutation symmetry. Such symmetry can be hidden in states with defined total spins of spinor gases, which can be formed in optical cavities. The present work shows that the symmetry reveals itself in spin-independent or coordinate-independent properties of these gases, namely as non-Abelian entropy in thermodynamic properties. In weakly interacting Fermi gases, two phases appear associated with fermionic and non-Abelian symmetry under permutations of particle states, respectively. The second-order transitions between the phases are characterized by discontinuities in specific heat. Unlike other phase transitions, the present ones are not caused by interactions and can appear even in ideal gases. Similar effects in Bose gases and strong interactions are discussed.

arXiv:1609.02180Physical Review LettersMay 19, 20174 citations

Dynamics and thermodynamics in spinor quantum gases

H. Schmaljohann, M. Erhard, Jochen Kronjäger +2 more

arXiv:cond-mat/0406485Applied Physics BOct 25, 200454 citations

Spin Waves in Spin-Polarized Quantum Gases

Eugene P. Bashkin

Physical Review LettersSep 23, 198512 citations

Permutation symmetry in large- N matrix quantum mechanics and partition algebras

GEORGE R. BARNES, Adrian Padellaro, Sanjaye Ramgoolam

We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size $N$ . We solve the general 11-parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras ${P}_{k}(N)$ . The integer $k$ is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalized by a representation theoretic basis for the permutation invariant subspace which we construct for $N\ensuremath{\ge}2k$. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realized in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realization of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in $\mathrm{AdS}/\mathrm{CFT}$ are shown to obey selection rules based on Clebsch-Gordan multiplicites (Kronecker coefficients) of symmetric groups.We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size $N$. We solve the general 11-parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras ${P}_{k}(N)$. The integer $k$ is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalized by a representation theoretic basis for the permutation invariant subspace which we construct for $N\ensuremath{\ge}2k$. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realized in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realization of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in $\mathrm{AdS}/\mathrm{CFT}$ are shown to obey selection rules based on Clebsch-Gordan multiplicites (Kronecker coefficients) of symmetric groups.

arXiv:2207.02166Physical review. D/Physical review. D.Nov 29, 20229 citations

We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size

N

. We solve the general 11-parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras

{P}_{k}(N)

. The integer

k

is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalized by a representation theoretic basis for the permutation invariant subspace which we construct for $N\ensuremath{\ge}2k$. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realized in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realization of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in

\mathrm{AdS}/\mathrm{CFT}

are shown to obey selection rules based on Clebsch-Gordan multiplicites (Kronecker coefficients) of symmetric groups.