Abstract The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.
A bstract We propose a method to challenge the calculation of genus- g , bulk n -point, b -boundary, c -crosscap correlation functions with x boundary operators
Fg,n,b,cx F g , n , b , c x in two-dimensional conformal field theories (CFT 2 ). We show that
Fg,n,b,cx F g , n , b , c x are infinite linear combinations of genus- g , bulk ( n + b + c )-point functions F g , ( n + b + c ) , and try to obtain the linear coefficients in this work. We show the existence of a single pole structure in the linear coefficients at degenerate limits. A practical method to obtain the infinite linear coefficients is the free field realizations of Ishibashi states. We review the results in Virasoro minimal models
M M ( p, p ′) and extend it to the N = 1 minimal models
SM SM ( p, p ′).
Universal thermal data in conformal field theory (CFT) offer a valuable means for characterizing and classifying criticality. With improved tensor network techniques, we investigate the universal thermodynamics on a nonorientable minimal surface, the crosscapped disk (or real projective plane,
RP2). Through a cut-and-sew process,
RP2 is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term
21lnk related to a nonorientable genus, with
k a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{c}{4}ln\ensuremath{\beta}$, with $\ensuremath{\beta}$ the manifold size and
c the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.
In addition to the celebrated Affleck-Ludwig entropy originating from the open boundaries of the path-integral manifold, recent research has shown that the entropy correction on nonorientable manifolds such as the Klein bottle is also a universal characterization of critical systems with an emergent conformal field theory (CFT). Here, the authors show that the Klein bottle entropy can be interpreted as a boundary effect by transforming the Klein bottle into an orientable manifold, whereby the Klein bottle entropy bears a resemblance to the Affleck-Ludwig entropy. The authors then propose a generic scheme to extract these universal boundary entropies from lattice models. Numerical results for the
q-state Potts model are compared with the CFT predictions.
We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in
2 dimensions, with electromagnetic duality for finite gauge theories in
3 dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.
The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the
Z2 symmetry defect ${D}_{\ensuremath{\epsilon}}$ and the Kramers-Wannier duality defect ${D}_{\ensuremath{\sigma}}$. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function
ZD of the critical Ising model in the presence of a topological conformal defect
D is expressed in terms of the scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$ and conformal spins ${s}_{\ensuremath{\alpha}}$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{D}$ can be extracted from the eigenvalue spectrum of a transfer matrix
MD for the partition function
ZD. In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions ${Z}_{{D}_{\ensuremath{\epsilon}}}$ and ${Z}_{{D}_{\ensuremath{\sigma}}}$ of the critical Ising model with either a symmetry defect ${D}_{\ensuremath{\epsilon}}$ or a duality defect ${D}_{\ensuremath{\sigma}}$. We also explain how to coarse grain the corresponding transfer matrices ${M}_{{D}_{\ensuremath{\epsilon}}}$ and ${M}_{{D}_{\ensuremath{\sigma}}}$, from which we can extract accurate numerical estimates of ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\epsilon}}}$ and ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\sigma}}}$. Two key ingredients of our approach are (i) coarse graining of the defect
D, which applies to any (i.e., not just topological) conformal defect and yields a set of associated scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, and (ii) construction and coarse graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins ${s}_{\ensuremath{\alpha}}$.