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arXiv:2312.05566 · Physical review. B./Physical review. B

Topological interfaces of Luttinger liquids

Ananda Roy, Hubert Saleur

Published Apr 12, 2024• 9 citations
arXivHTMLPDF

Abstract

Topological interfaces of two-dimensional conformal field theories contain information about symmetries of the theory and exhibit striking spectral and entanglement characteristics. While lattice realizations of these interfaces have been proposed for unitary minimal models, the same has remained elusive for the paradigmatic Luttinger liquid, i.e., the free, compact boson model. Here, we show that a topological interface of two Luttinger liquids can be realized by coupling special one-dimensional superconductors. The gapless excitations in the latter carry charges that are specific integer multiples of the charge of Cooper pairs. The aforementioned integers are determined by the windings in the target space of the bosonic fields --- a crucial element required to give rise to nontrivial topological interfaces. The latter occur due to the perfect transmission of certain number of Cooper pairs across the interface. The topological interfaces arise naturally in Josephson junction arrays with the simplest case being realized by an array of experimentally demonstrated $0\ensuremath{-}\ensuremath{\pi}$ qubits, capacitors and ordinary Josephson junctions. Signatures of the topological interface are obtained through entanglement entropy computations. In particular, the subleading contribution to the so-called interface entropy is shown to differ from existing field theory predictions. The proposed lattice model provides an experimentally realizable alternative to spin and anyon chains for the analysis of several conjectured conformal fixed points which have so far eluded ab initio investigation.

DOI
10.1103/physrevb.109.l161107

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Topological interfaces of two-dimensional conformal field theories contain information about symmetries of the theory and exhibit striking spectral and entanglement characteristics. While lattice realizations of these interfaces have been proposed for unitary minimal models, the same has remained elusive for the paradigmatic Luttinger liquid, i.e., the free, compact boson model. Here, we show that a topological interface of two Luttinger liquids can be realized by coupling special one-dimensional superconductors. The gapless excitations in the latter carry charges that are specific integer multiples of the charge of Cooper pairs. The aforementioned integers are determined by the windings in the target space of the bosonic fields --- a crucial element required to give rise to nontrivial topological interfaces. The latter occur due to the perfect transmission of certain number of Cooper pairs across the interface. The topological interfaces arise naturally in Josephson junction arrays with the simplest case being realized by an array of experimentally demonstrated 0\ensuremath−\ensuremathπ0\ensuremath{-}\ensuremath{\pi}0\ensuremath−\ensuremathπ qubits, capacitors and ordinary Josephson junctions. Signatures of the topological interface are obtained through entanglement entropy computations. In particular, the subleading contribution to the so-called interface entropy is shown to differ from existing field theory predictions. The proposed lattice model provides an experimentally realizable alternative to spin and anyon chains for the analysis of several conjectured conformal fixed points which have so far eluded ab initio investigation.
We explore interfaces and junctions joining multiple two-dimensional conformal field theories, with the goal of calculating entanglement entropies in their presence and exploring their holographic duals. In chapter 1 we start with an overview of the three subjects, collecting various well-known results and reviewing some foundational works.In chapter 2 we calculate the holographic entanglement entropy in the presence of a conformal interface for a geometric configuration in which the entangling region lies on one side of the interface. For the supersymmetric Janus solution we find exact agreement between the holographic and conformal field theory calculation of the entanglement entropy.In chapter 3 we calculate the entanglement entropy for topological interfaces in rational conformal field theories for the case where the interface lies at the boundary of the entangling interval and for the case where it is located in the center of the entangling interval. We compare the results to each other and also to the left/right entropy of a related boundary conformal field theory. We also comment on the entanglement entropies for topological interfaces in Liouville theory.In chapter 4 we consider entanglement through permeable junctions of N free boson and free fermion conformal field theories. We constrain the form of the general boundary state and calculate the replicated partition functions with interface operators inserted, from which the entanglement entropy is calculated. We find the functional form of the universal and constant terms to be similar to the N=2 case, depending only of the total transmission of the junction and the unit volume of the zero mode lattice. For N>2 we see a subleading divergent term which does not depend on the parameters of the junction. For N=3 we consider some specific geometries and discuss various limits.In chapter 5 we investigate topological interfaces between three-dimensional Abelian Chern-Simons theories in the context of the AdS_3/CFT_2 correspondence. We show that it is possible to connect the topological interfaces in the bulk Chern-Simons theory to topological interfaces in the dual conformal field theory on the boundary. In addition to the [U(1)]^{2N} Chern-Simons theory on AdS_3, we show that it is possible to find boundary counter terms which lead to the N conserved currents in the dual two-dimensional conformal field theory.
We show that a doped spin-1/2 ladder with antiferromagnetic intra-chain and ferromagnetic inter-chain coupling is a symmetry protected topologically non-trivial Luttinger liquid. Turning on a large easy-plane spin anisotropy drives the system to a topologically-trivial Luttinger liquid. Both phases have full spin gaps and exhibit power-law superconducting pair correlation. The Cooper pair symmetry is singlet dxyd_{xy}dxy​ in the non-trivial phase and triplet Sz=0S_z=0Sz​=0 in the trivial phase. The topologically non-trivial Luttinger liquid exhibits gapless spin excitations in the presence of a boundary, and it has no non-interacting or mean-field theory analog even when the fluctuating phase in the charge sector is pinned. As a function of the strength of spin anisotropy there is a topological phase transition upon which the spin gap closes. We speculate these Luttinger liquids are relevant to the superconductivity in metalized integer spin ladders or chains.
Gapless Luttinger liquid is conventionally viewed as topologically trivial, unless it hosts degenerate ground states and or entanglement spectrum, which necessitates partial bulk degree of freedom to be gapped out. Here we predict an emergent gapless topological Luttinger liquid which is beyond the conventional scenarios and is characterized by the nontrivial many-body bulk spin texture, and propose feasible scheme for experimental observation. We consider a one-dimensional spin-orbit coupled Fermi-Hubbard model with fractional filling, whose low-energy physics is effectively described by a spinless Luttinger liquid and is trivial in the conventional characterization. We show that, as being tuned by the filling factor and interaction strength, the many-body ground state may exhibit nontrivial winding in its bulk spin texture in the projected momentum space, manifesting an emergent topological phase. A topological transition occurs when the projected spin-state at a high symmetry momentum becomes fully mixed one, resulting from competing processes of particle scattering to the lower and higher subbands, for which the spin texture at such momentum point is ill-defined, but the Luttinger liquid keeps gapless through the transition. Surprisingly, at relatively small filling the Luttinger liquid remains topologically nontrivial even at infinitely strong interaction. The results can be generalized to finite temperature which facilitates the real experimental detection. This work shows a novel gapless topological Luttinger liquid whose characterization is beyond the low-energy effective theory, and can be verified based on current experiments.
We show that a doped spin-1/2 ladder with antiferromagnetic intra-chain and ferromagnetic inter-chain coupling is a symmetry protected topologically non-trivial Luttinger liquid. Turning on a large easy-plane spin anisotropy drives the system to a topologically-trivial Luttinger liquid. Both phases have full spin gaps and exhibit power-law superconducting pair correlation. The Cooper pair symmetry is singlet dxy in the non-trivial phase and triplet Sz=0 in the trivial phase. The topologically non-trivial Luttinger liquid exhibits gapless spin excitations in the presence of a boundary, and it has no non-interacting or mean-field theory analog even when the fluctuating phase in the charge sector is pinned. As a function of the strength of spin anisotropy there is a topological phase transition upon which the spin gap closes. We speculate these Luttinger liquids are relevant to the superconductivity in metalized integer spin ladders or chains.
In this thesis, we study quantum phase transitions and topological phases in low dimensional fermionic systems. In the first part, we study quantum phase transitions and the nature of currents in one-dimensional systems, using field theoretic techniques like bosonization and renormalization group. This involves the study of currents in Luttinger liquids, and the fate of a persistent current in a 1D system. In the second part of the thesis, we study the different types of Majorana edge modes in a 1D p-wave topological superconductor. Further we extend our analysis to the effect of an additional s-wave pairing and a Zeeman field on the topological properties, and present a detailed phase diagram and symmetry classification for each of the cases. In the third part, we concentrate on the topological phases in two-dimensional systems. More specifically, we study the experimental realization of SU(3) topological phases in optical lattice experiments, which is characterized by the presence of gapless edge modes at the boundaries of the system. We discuss the specific characteristics required by a such a three component Hamiltonian to have a non-zero Chern number, and discuss a schematic lattice model for a possible experimental realization.
In this paper we consider entanglement entropies in two-dimensional conformal field theories in the presence of topological interfaces. Tracing over one side of the interface, the leading term of the entropy remains unchanged. The interface however adds a subleading contribution, which can be interpreted as a relative (Kullback-Leibler) entropy with respect to the situation with no defect inserted. Reinterpreting boundaries as topological interfaces of a chiral half of the full theory, we rederive the left/right entanglement entropy in analogy with the interface case. We discuss WZW models and toroidal bosonic theories as examples.