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arXiv:hep-th/0404195

Families of Commuting Integrals of Motion with Certain Symmetries

Evgueniy Vitchev

Published Apr 25, 2004• 1 citation
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Abstract

Some results for commuting local integrals of motion for free bosonic theories are presented, which can be applied in construction of integrable boundary states. Usable program code for calculation of these integrals of motion is also introduced.
DOI
10.48550/arxiv.hep-th/0404195

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Some results for commuting local integrals of motion for free bosonic theories are presented, which can be applied in construction of integrable boundary states. We consider a particular model of two-dimensional quantum field theory on a disk, whose bulk dynamics is that of a two-component free massless Bose field X = (X, Y), and interaction occurs at the boundary, where the boundary values (XB, YB) are constrained to special curve—the “paperclip brane”. The interaction breaks conformal invariance, but we argue that it preserves integrability. We propose exact expression for the disk partition function (and more general overlap amplitudes ⟨P| B⟩ of the boundary state with all primary states) in terms of solutions of certain ordinary linear differential equations. Usable program code for calculation of commuting integrals of motion is also introduced.
Infinitely many commuting operators for the elliptic quantum group Uq,p(slN^)U_{q,p}(\hat{sl_N})Uq,p​(slN​^​)
We construct two classes of infinitely many commuting operators associated with the elliptic quantum group Uq,p(slN^)U_{q,p}(\hat{sl_N})Uq,p​(slN​^​). We call one of them the integral of motion Gm{\cal G}_mGm​, (m∈N)(m \in {\mathbb N})(m∈N) and the other the boundary transfer matrix TB(z)T_B(z)TB​(z), (z∈C)(z \in {\mathbb C})(z∈C). The integral of motion Gm{\cal G}_mGm​ is related to elliptic deformation of the NNN-th KdV theory. The boundary transfer matrix TB(z)T_B(z)TB​(z) is related to the boundary Uq,p(slN^)U_{q,p}(\hat{sl_N})Uq,p​(slN​^​) face model. We diagonalize the boundary transfer matrix TB(z)T_B(z)TB​(z) by using the free field realization of the elliptic quantum group, however diagonalization of the integral of motion Gm{\cal G}_mGm​ is open problem even for the simplest case Uq,p(sl2^)U_{q,p}(\hat{sl_2})Uq,p​(sl2​^​).
In this work, we present bilayer flat-band Hamiltonians, in which all bulk states are localized and specified by extensive local integrals of motion (LIOMs). The present systems are bilayer extension of the Creutz ladder, which is studied previously. In order to construct models, we employ building blocks, cube operators, which are linear combinations of fermions defined in each cube of the bilayer lattice. There are eight cubic operators, and the Hamiltonians are composed of the number operators of them, the LIOMs. A suitable arrangement of locations of the cube operators is needed to have exact projective Hamiltonians. The projective Hamiltonians belong to a topological classification class, the BDI class. With the open boundary condition, the constructed Hamiltonians have gapless edge modes, which commute with each other as well as the Hamiltonian. This result comes from a symmetry analogous to the one-dimensional chiral symmetry of the BDI class. These results indicate that the projective Hamiltonians describe a kind of symmetry-protected topological phase matter. Careful investigation of topological indexes, such as the Berry phase and string operator, is given. We also show that, by using the gapless edge modes, a generalized Sachdev-Ye-Kitaev model is constructed.