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.
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.
Matrix models appear as fundamental descriptions of M-theory and D-brane dynamics, and via the gauge/gravity duality their gauge-invariant, or singlet, sector describes the purely gravitational degrees of freedom in the holographic dual. We present a new exact algorithm for computing observables of bosonic U(N) matrix models in the gauge-invariant singlet sector. This sector is spanned by an orthogonal basis of Schur polynomials (for a single matrix) and restricted Schur polynomials (for multiple matrices), which diagonalizes the free Hamiltonian and provides a natural truncation of the Hilbert space by excitation number. Matrix elements of the interaction Hamiltonian, or any gauge-invariant observable, are evaluated through a group-theoretic reduction to cosets and double cosets of suitable subgroups of the symmetric group, together with character sums on the symmetric group. The resulting entries are closed-form polynomials in the gauge-group rank N, assembled from group-theoretic data that are precomputed once and can be reused for any N and any coupling constants. We validate the one-matrix implementation against the exact mapping to N non-interacting fermions, demonstrating rapid convergence of the low-lying spectrum with the cutoff. The multi-matrix extension is outlined; its main bottleneck is the computation of restricted characters of the symmetric group, for which no algorithm comparable to the Murnaghan--Nakayama rule is currently known. The framework gives direct access to finite-N, finite-coupling dynamics of gauge-invariant states and opens a new computational window on the non-planar regime of holographic matrix models.
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.
Recent developments have established exact bosonization and fermionization with a Z2 symmetry as dualities through gauging. In this work, we apply fermionic gauging, which realizes generalized Jordan-Wigner transformations, to the Bose-Hubbard (BH) model with a global U(1) symmetry and derive an exact dual description in terms of fermionic composites, built from bosons and fermions. In 1D, this duality generalizes the exact mapping between the extended hard-core BH model and the spinless Fermi-Hubbard model to include soft-core bosons. At low energies, the mapping reduces to the well-known equivalence between the sine-Gordon model and the Thirring model. The oscillation wave vector of the fermionic composite correlation function in the gapless phase is fixed by their density, providing a novel manifestation of Luttinger's theorem. We verify the exact duality using density matrix renormalization group (DMRG) calculations and demonstrate that the gapless phase and the phase transition are governed by the compact boson conformal field theory. Our construction naturally extends to generic bosonic systems and higher dimensions, opening new avenues for studying Bose-Fermi mixtures in optical lattices and other strongly correlated quantum systems.
I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin, or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (nonrelativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the nonrelativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.
We describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases.