We explore different limits of exactly solvable vector and matrix fermionic quantum mechanical models with quartic interactions at finite temperature. The models preserve a $U(1)\ifmmode\times\else\texttimes\fi{}SU(N)\ifmmode\times\else\texttimes\fi{}SU(L)$ symmetry at the classical level and we analyze them through bosonization techniques introducing scalar (singlet) and matrix (nonsinglet) bosonic fields. The bosonic path integral representations in the vector limits $(N,1)$ and $(1,L)$ are matched to fermionic Fock space Hamiltonians expressed in terms of quadratic Casimirs and some additional terms involving the Cartan subalgebra, which makes explicit the symmetries preserved by scalar and matrix bosonizations at the quantum level. For the case of nonsinglet bosonization we find an equivalence between the vector model and the Polychronakos-Frahm spin model. Using this relation, we compute the free energy. Finally, we compute the eigenvalue distribution in the large $N$, $L$ limit with $\ensuremath{\alpha}=\frac{L}{N}$ fixed. The model displays a third order phase transition as we vary the temperature which, in the $\ensuremath{\alpha}\ensuremath{\gg}1$ limit, can be characterized analytically. We conclude finding the critical curve in the parameter space were the eigenvalue distribution transitions from single to double cut.
We explore different limits of exactly solvable vector and matrix fermionic quantum mechanical models with quartic interactions at finite temperature. The models preserve a U(1)\ifmmode×\else\texttimes\fiSU(N)\ifmmode×\else\texttimes\fiSU(L) symmetry at the classical level and we analyze them through bosonization techniques introducing scalar (singlet) and matrix (nonsinglet) bosonic fields. The bosonic path integral representations in the vector limits (N,1) and (1,L) are matched to fermionic Fock space Hamiltonians expressed in terms of quadratic Casimirs and some additional terms involving the Cartan subalgebra, which makes explicit the symmetries preserved by scalar and matrix bosonizations at the quantum level. For the case of nonsinglet bosonization we find an equivalence between the vector model and the Polychronakos-Frahm spin model. Using this relation, we compute the free energy. Finally, we compute the eigenvalue distribution in the large N, L limit with \ensuremathα=NL fixed. The model displays a third order phase transition as we vary the temperature which, in the \ensuremathα\ensuremath≫1 limit, can be characterized analytically. We conclude finding the critical curve in the parameter space were the eigenvalue distribution transitions from single to double cut.
Abstract We explore systems with a large number of fermionic degrees of freedom subject to non-local interactions. We study both vector and matrix-like models with quartic interactions. The exact thermal partition function is expressed in terms of an ordinary bosonic integral, which has an eigenvalue repulsion term in the matrix case. We calculate real time correlations at finite temperature and analyze the thermal phase structure. When possible, calculations are performed in both the original Hilbert space as well as the bosonic picture, and the exact map between the two is explained. At large N , there is a phase transition to a highly entropic high temperature phase from a low temperature low entropy phase. Thermal two-point functions decay in time in the high temperature phase.
We discuss two types of quantum mechanical models that couple large numbers of Majorana fermions and have orthogonal symmetry groups. In models of vector type, only one of the symmetry groups has a large rank. The large N limit is taken keeping $gN=\ensuremath{\lambda}$ fixed, where g multiplies the quartic Hamiltonian. We introduce a simple model with $O(N)\ifmmode\times\else\texttimes\fi{}SO(4)$ symmetry, whose energies are expressed in terms of the quadratic Casimirs of the symmetry groups. This model may be deformed so that the symmetry is $O(N)\ifmmode\times\else\texttimes\fi{}O(2{)}^{2}$, and the Hamiltonian reduces to that studied in [I. R. Klebanov et al., Phys. Rev. D 97, 106023 (2018)]. We find analytic expressions for the large N density of states and free energy. In both vector models, the large N density of states varies approximately as ${e}^{\ensuremath{-}|E|/\ensuremath{\lambda}}$ for a wide range of energies. This gives rise to critical behavior as the temperature approaches the Hagedorn temperature ${T}_{\mathrm{H}}=\ensuremath{\lambda}$. In the formal large N limit, the specific heat blows up as $({T}_{H}\ensuremath{-}T{)}^{\ensuremath{-}2}$, which implies that TH is the limiting temperature. However, at any finite N, it is possible to reach arbitrarily large temperatures. Thus, the finite N effects smooth out the Hagedorn transition. We also study models of matrix type, which have two O(N) symmetry groups with large rank. An example is provided by the Majorana matrix model with $O(N{)}^{2}\ifmmode\times\else\texttimes\fi{}O(2)$ symmetry, which was studied in Klebanov et al. In contrast with the vector models, the density of states is smooth and nearly Gaussian near the middle of the spectrum.
We explore systems with a large number of fermionic degrees of freedomsubject to non-local interactions. We study both vector and matrix-like modelswith quartic interactions. The exact thermal partition function is expressed interms of an ordinary bosonic integral, which has an eigenvalue repulsion term inthe matrix case. We calculate real time correlations at finite temperature andanalyze the thermal phase structure. When possible, calculations are performedin both the original Hilbert space as well as the bosonic picture, and the exactmap between the two is explained. At large N, there is a phase transition toa highly entropic high temperature phase from a low temperature low entropyphase. Thermal two-point functions decay in time in the high temperature phase.
We study the $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}O({N}_{3})$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks Ni are all equal, this model has a large N limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of group invariant states for any set of Ni. It is non-vanishing only when each Ni is even. For equal ranks the number of singlets exhibits rapid growth with N: it jumps from 36 in the O(4)3 model to 595 354 780 in the O(6)3 model. We derive bounds on the values of energy, which show that they scale at most as N3 in the large N limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order 1/N. For N3=1 the tensor model reduces to $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})$ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with $SU({N}_{1})\ifmmode\times\else\texttimes\fi{}SU({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$ symmetry. Finally, we study the N3=2 case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large N limits where the ground state energies are of order N2, while the energy gaps are of order 1.
A bstract We describe a new exact lattice bosonization of matrix quantum mechanics (equivalently of non-relativistic fermions) that is valid for arbitrary rank N of the matrix. It is based on the exact operator bosonization of non-relativistic fermions introduced earlier in [1]. The trace identities, which characterize finite rank matrices, are automatically incorporated in the bosonic theory. The finite number N of fermions is reflected in the finite number N of bosonic annihilation-creation operators, and equivalently to the finite number N of lattice points. The fermion Hamiltonian is exactly mappable to a bosonic Hamiltonian. At large N , the latter becomes local and corresponds to the lattice version of a relativistic boson Hamiltonian, with a lattice spacing of order 1/ N . The finite lattice spacing leads to a finite entanglement entropy (EE) of the bosonic theory, which reproduces the finite EE of the fermionic theory. Such a description is not available in the standard bosonization in terms of fermion density fluctuations on the Fermi surface, which does not have a built-in short distance cut-off (see, however, a recent description of the finiteness of the fermionic EE in a collective field theory formalism [2]). The bosonic lattice constructed in our work is equipped with a geometry that is determined by the matrix potential or equivalently by the shape of the Fermi surface. Our bosonization also works in the double scaled c = 1 matrix model; in particular the bosonic EE again turns out to be finite, with the short distance cut-off turning into g s l s , and reproduces the matrix result. Once again, this is to be contrasted with the conventional dual 2D string theory, where the bosonic EE is naturally identified with that of the “tachyon”, the massless string mode, where one may imagine the short-distance cut-off to be the string length l s . This appears to indicate our bosonization as a different dual description to the c = 1 matrix model appropriate for “local physics” like quantum entanglement, by contrast with the conventional duality to the eigenvalue density which works well for asymptotic observables like S-matrices. We briefly discuss possible relation of our bosonization to D0 branes.
Exact lattice bosonization of the double-scaled c=1 matrix model
In a recent paper by two of us, it has been shown that entanglement entropy (EE) of matrix quantum mechanics (or, equivalently, of a non-relativistic fermion model) has a natural interpretation in terms of an exact bosonization of the fermion theory in terms of a lattice boson with lattice spacing ∼1/N (where N is the rank of the matrix, equal to the number of fermions). The theory of this lattice boson is distinct from a latticized version of the "collective variable'' theory which is formulated in terms of the fermion density ρ(λ) ("collective variable''); it is, in fact, more closely related to the Wigner distribution of the fermions. In this paper, we develop this connection further for the c=1 double scaled matrix model. Even though N becomes infinite, the double scaled theory is described by a boson on a finite and non-uniform lattice spacing, of order ∼gs(x)ls (where gs(x)≡gscosech2(x/ls) is a position dependent string coupling). We explicitly derive the c=1 EE, as well as the one and two-point functions of fermion density from the lattice boson. The results are explicitly finite, unlike the collective variable theory which is shown to yield divergent answers. Our results suggest that while local observables such as the EE and ⟨ρ(λ)⟩, are characterized by a unique short distance scale gs(x)ls, the asymptotic observables such as the S-matrix at every genus are characterized by a short distance scale ls. We show how this arises and discuss a possible interpretation in terms of D0 brane physics.