text:
\section*{Introduction}

Topological phases of matter, while being gapped in the bulk, often display gapless edge modes. On the one hand the edge modes are controlled by bulk topological invariants, and on the other hand the critical edge theory governs the full topological content of the bulk. This phenomenon is known as the bulk-edge correspondence (see for instance Refs.~\cite{BulkEdgeInEntanglement_Regnault,BulkEdgeBreaksDown_Cano,HowUniversalEntaqnglementSpectrum_Sondhi} for general discussions about its validity).  In the context of the fractional quantum Hall (FQH) effect, the bulk-edge correspondence has been pushed one step further with the pioneering work of Moore and Read~\cite{MooreReadCFTCorrelator} who expressed a large class of FQH model WFs as two-dimensional Conformal Field Theory (CFT) correlators. Assuming generalized screening, one can then establish that the associated low-energy edge-modes are described by the same CFT \cite{FromCFTtoMPShamiltonian}, making the correspondence between the bulk and edge properties explicit.

Among the challenges that emerged during the last few years in the realm of topologically ordered phases, understanding the interface between two distinct intrinsic topologically ordered phases stands out as one of the most fascinating~\cite{SchoutensNassMR,BaisSMatrix,SlingerlandTopoSymBreaking,SlingerlandTopoEdgeBoundary,BarkeshliClassificationDefects,KapustinBoundaries,LudwigCutAndGlue,TopoPhaseTransitionStringNets,KaneFibonacciCoset}.  Predicting what happens at such an interface is notoriously difficult~\cite{LevinLagrangianSubgroup,TaylorHughesTransitionSPT,Gapped_interface}, and even the transitions between Abelian states have not yet been classified~\cite{TryClassificationInterface,KitaevTryClassificationInterface,FuchsTryClassificationInterface}. In this article we consider the prototypical example of an interface between two FQH Abelian states.

The FQH effect is the first observed~\cite{QantumHallDiscovery1980} quantum phase of matter with intrinsic topological order. Many features of this strongly-correlated, non-perturbative problem were unraveled by the study of model wavefunctions (WFs)~\cite{HanssonReviewCFTforFQHE,MooreReadCFTCorrelator}. In particular, the experimentally observed fractional $e/3$ charges~\cite{FractionalChargeExperimentalEtienne,FractionalChargeExperimentalSu} were first described as excitations of the seminal Laughlin WF at filling factor $\nu=1/3$~\cite{LaughlinVariationalWF}. Model WFs have been an invaluable tool in our understanding of the FQH effect. They provide a bridge between the microscopic models in terms of strongly-correlated electrons and the low-energy effective description in terms of topological quantum field theories. Indeed the Laughlin WF is the densest zero energy ground state of a microscopic relevant Hamiltonian, while at the same time it exhibits non-Abelian anyons and a topology-dependent ground state degeneracy consistent with that of a Chern-Simons theory~\cite{HaldaneHierarchiesPseudoPot,PseudoPotentialsTrugmanKivelson}.

Theoretical approaches to understand interfaces between topological phases mostly rely on the cut and glue approach~\cite{LudwigCutAndGlue}, in which both phases are solely described by their respective edge theories. The interface emerges from the coupling between the two edges~\cite{CoupledWireFirstKane} and predictions can be made about the nature of the interface theory~\cite{BaisSMatrix,TaylorHughesTransitionSPT,BarkeshliClassificationDefects}. While powerful, these effective field theory approaches suffer from a complete lack of connection with a more physical, microscopic description. In order to overcome this limitation we have recently proposed a family of Matrix Product State (MPS) model wavefunctions for the Laughlin-Halperin interface capable of describing the whole system including both bulks and the interface \cite{Interface_bosons}. We found that these MPSs faithfully describe the bulks intrinsic topological order while presenting the expected universal low-energy physics at the interface. However the validity of these model WFs at the microscopic level still has to be established.

In the present article,  we provide such a detailed microscopic analysis of these model WFs through extensive comparison with Exact Diagonalization (ED). We focus on the fermionic interface between the $\nu =1/3$ Laughlin state and the $\nu = 2/5$ Halperin (332) state. This interface is relevant for condensed matter experiments, and could be realized in graphene. There, the valley degeneracy leads to a spin singlet state at filling fraction $\nu=2/5$~\cite{JainGrapheneTwoFifthIsHalperin,ExperimentalTwoFifthGraphene} while the system at $\nu=1/3$ is spontaneously valley-polarized~\cite{ExperimentalTwoFifthGraphene,OneThirdPolarizedGraphene,OneThirdPolarizedGrapheneBis}. Thus, changing the density through a top gate provides a direct implementation of the Laughlin-Halperin interface.

The paper is organized as follows: we first introduce a microscopic model reproducing the physics of the transition between a Laughlin phase at filling factor 1/3 and a Halperin (332) phase at filling 2/5. We then detail the MPS construction of both the Halperin (332) state, derived in Ref.~\cite{MyFirstArticle}, and of the model state for such an interface that we have introduced in Ref.~\cite{Interface_bosons}. From there we explicitly show how to perform the identification of the interface gapless theory and apply the procedure to our system. We finally compare the ED results of the microscopic Hamiltonian with the model state and show how the construction of the ansatz may be used to identify the spin and charge excitations in the many-body spectrum. The main results of this work appear in this discussion, as it validates the model state not only on the universal features it holds but more importantly at a microscopic level.

\section*{Results}

\paragraph*{Microscopic Model ---} The Halperin $(m,m,m-1)$ at filling factor $\nu=\frac{2}{2m-1}$ is the natural spin singlet~\cite{AntisymHalperin,PhDPapicMulticomponentStates,RegnaultDemixtionHalperin} generalization of the celebrated, spin polarized, Laughlin state~\cite{LaughlinVariationalWF,LaughlinWFonCylinder} at filling factor $\nu=1/m$. It describes a FQH fluid with an internal two-level degree of freedom~\cite{HalperinOriginalPaper,HalperinSecondPaper} such as spin, valley degeneracy in graphene or layer index in bilayer systems. For the sake of conciseness, we will refer to the internal degree of freedom as \emph{spin} in the following. The most relevant case for condensed matter systems is $m=3$, namely the fermionic $(3,3,2)$ Halperin state. Numerical evidence~\cite{GoerbigRegnaultGraphene,JainGrapheneTwoFifthIsHalperin} suggests that the plateau at filling $\nu_H=2/5$ observed in graphene~\cite{ExperimentalTwoFifthGraphene} is in a valley-pseudospin unpolarized Halperin (332) state. Interestingly, graphene brought at filling $\nu_L=1/3$ spontaneously valley-polarizes~\cite{ExperimentalTwoFifthGraphene,OneThirdPolarizedGraphene,OneThirdPolarizedGrapheneBis} and is described by a Laughlin state. Top-gating different regions to change the density, it seems possible to engineer a setup where the Halperin 332 and Laughlin 1/3 topological orders develop on either sides of a sample. These phases have distinct intrinsic topological orders and hence cannot be adiabatically connected to one another. Thus, the creation of such an interface requires a gap closing and the emergence of a critical boundary or of critical points. In this section, we exhibit a microscopic model describing such an interface between the Halperin (332) and the polarized Laughlin 1/3 phases.

\begin{figure*}
	\centering
	\includegraphics[width=0.85\textwidth]{ED_TestMicroscopicModel.pdf}
	\caption{\textbf{Exact diagonalization results:} a) Energy spectrum of Eq.~\eqref{eq:HamiltonianFermionic} for $N_{\rm L}=3$ and $2N_{\rm H}=6$ particles in $N_{\rm orb}=23$ orbitals, $N_{\rm orb}^{\rm Lgh}=8$ are polarized, on a cylinder of perimeter $L=12\ell_B$. Low energy features detach from the continuum, hinting toward an interesting low energy physics coming from the interface. b) Spin resolved densities for the ground state of Eq.~\eqref{eq:HamiltonianFermionic} for $N_{\rm H}=4$ spin up and $N_{\rm H}+N_{\rm L}=8$ spin down particles in $N_{\rm orb}=30$ orbitals, $N_{\rm orb}^{\rm Lgh}=10$ of which are completely polarized. It is the only state detaching from the continuum and arises for a center of mass momentum $K_y=40.5$. For this system size, the ED calculation captures both the interface and bulk physics as can be seen from the plateaus of densities at both edges of the sample.}
	\label{fig:MicroscopicTestInterfaceED}
\end{figure*}

\begin{widetext}
We first recall the expressions of the Laughlin 1/3 state \begin{equation}
\Psi^{\rm Lgh} (z_1, \cdots , z_{N_e}) = e^{- \frac{1}{4\ell_B^2} \sum_{i} |z_i|^2} \prod_{1\leq i < j \leq N_e} \left(z_i - z_j \right)^3 \, , \label{eq:LaughlinOnethird}
\end{equation} and of the Halperin 332 state \begin{equation}
\Psi^{\rm Hlp} (z_1, \cdots , z_{N_e/2}, w_1, \cdots , w_{N_e/2}, ) = e^{- \frac{1}{4\ell_B^2} \sum_{i} |z_i|^2 + |w_i|^2} \prod_{i < j} \left(z_i - z_j \right)^3 \left(w_i - w_j \right)^3 \prod_{i , j} \left(z_i - w_j \right)^2 \, , \label{eq:Halperin332State}
\end{equation} where the position of the $i$-th spin down (resp. up) electrons is denoted by $z_i$ (resp. $w_i$) and $\ell_B$ the magnetic length of the system. Here, it is implicit that the total many-body state is the proper antisymmetrization of the spatial part Eq.~\eqref{eq:Halperin332State} associated with the spin component $(\downarrow\cdots \downarrow \uparrow  \cdots \uparrow )$ with respect to both electronic spin and position. The vanishing properties of these states ensure that they completely screen the interacting Hamiltonian~\cite{RootPartitionBernevig,SimonRezayiCooper_PseudoPotentialMultiparticle,BulkEdgeInEntanglement_Regnault}: \begin{equation} \mathcal{H}_{\rm int} = \int {\rm d}^{2}\vec{r} \sum_{\sigma,\sigma' \in \{ \uparrow,\downarrow \} } \!\!\!\!\!\!\! - : \rho_\sigma (\vec{r}) \nabla^2 \rho_{\sigma'} (\vec{r}) : + : \rho_\uparrow (\vec{r}) \rho_{\downarrow} (\vec{r}) : +   \mu_\uparrow \rho_\uparrow(\vec{r}) \, , \label{eq:HamiltonianFermionic} \end{equation} respectively for $\mu_\uparrow= 0$ and $\mu_\uparrow=\infty$. Here $\mu_\uparrow$ is a chemical potential for the particles with a spin up,  $\rho_\sigma$ denotes the density of particles with spin component $\sigma$, and $: \, . \, :$ stands the for normal ordering. Hence, creating an interface between these two topologically ordered phases can be achieved by making $\mu_\uparrow$ spatially dependent without tuning the interaction~\cite{SchoutensNassMR}.
\end{widetext} We make a few additional assumptions, allowing the numerical study of such an interface. First, we send the cyclotron energy to infinity, \textit{i.e.} large enough so that only the Lowest Landau Level (LLL) is populated. We will always assume periodic boundary conditions along the $y$-axis, thus mapping the system on a cylinder with perimeter $L$. Let $c_\sigma^\dagger(\vec{r})$ be the creation operator of an electron of spin $\sigma$ at position $\vec{r}=(x,y)$. The LLL is spanned by the one-body orbital WFs: \begin{equation}
\psi_n(\vec{r}) = \dfrac{e^{i k_n y}}{\sqrt{L\sqrt{\pi}}} e^{-\frac{(x-x_n)^2}{2\ell_B^2}} \, , \label{eq:LLLbasiscylinder}
\end{equation} where the momentum along the compact dimension $k_n=\left(\frac{2\pi}{L} \right) n$, with $n \in \mathbb{Z}+1/2$, labels the orbitals and determines the center of the Gaussian envelope \begin{equation}
x_n = k_n \ell_B^2 \label{eq:SupMatCenterOrbitals}
\end{equation} along the cylinder axis. The corresponding creation operator is $c_{n,\sigma}^\dagger = \int {\rm d}^2 \vec{r} \,  \psi_n (\vec{r}) c_\sigma^\dagger(\vec{r})$. Once projected to the LLL, the Hamiltonian of Eq.~\eqref{eq:HamiltonianFermionic} with a spatially dependent chemical potential $\mu_\uparrow(\vec{r})$ is made of an interaction $\mathcal{H}_{\rm int}$ term and a polarization term $\mathcal{H}_{\rm pol}$. After projection, the Halperin 332 state (resp. the Laughlin state) becomes the \textit{densest} zero-energy state for a uniform chemical potential $\mu_\uparrow= 0$ (resp. $\mu_\uparrow=\infty$)~\cite{HaldaneHierarchiesPseudoPot}. Indeed, $\mathcal{H}_{\rm int}$ reduces to the zero-th and first Haldane pseudo-potentials~\cite{PseudoPotentialsPapicThomale,PseudoPotentialsTrugmanKivelson}. We now choose $\mu_\uparrow(\vec{r})$ such that the quadratic part reads \begin{equation}
	\mathcal{H}_{\rm pol} = U \sum_{k<0} c_{k,\uparrow}^\dagger c_{k,\uparrow} \, , \label{eq:PolarizationLaughlin}
\end{equation} corresponding to a smooth ramp from zero to $U$ in real space over a typical distance $2\pi/L$. Moreover, we assume that $U\gg |\mathcal{H}_{\rm int}|$ while remaining smaller than the cyclotron energy. This allows us to project our Hilbert space onto a subspace where the occupation for all orbitals with $n<0$ is \emph{zero} for the spin up electrons. The polarized Hilbert subspace is spanned by the occupation basis: \begin{equation} \label{eq:OccupationBasis} \begin{split}
& \ket{\{n_k^\downarrow\}_{k}, \{n_k^\uparrow\}_{k>0}} = \\ & \qquad \quad   \ket{\cdots n_{-3/2}^\downarrow n_{-1/2}^\downarrow (n_{1/2}^\downarrow,n_{1/2}^\uparrow) (n_{3/2}^\downarrow,n_{3/2}^\uparrow) \cdots } \, .
\end{split}
\end{equation}

To summarize, our model is described by a purely interacting Hamiltonian $\mathcal{H}_{\rm int}$, consisting of the two first Haldane pseudo-potentials, projected to a polarized subspace of the many-body LLL Hilbert space in which no spin up occupies the $n<0$ orbitals. ED studies shows that some low energy features emerges from the continuum, as shown in Fig.~\ref{fig:MicroscopicTestInterfaceED}a. The largest reachable system size consists of 4 spin up and 8 spin down particles. For a suitable choice of orbital number, edge excitations acquire a large energy due to finite size effects and we can isolate a single low energy state detached from the continuum. The spin resolved densities of this vector are depicted in Fig.~\ref{fig:MicroscopicTestInterfaceED}b. They reach plateaus far from the transition, corresponding to the expected results for the Laughlin ($\rho_\uparrow=0$, $\rho_\downarrow=1/3$) and Halperin bulks ($\rho_\uparrow=1/5$, $\rho_\downarrow=1/5$). It shows that our model Eq.~\eqref{eq:HamiltonianFermionic} indeed captures the physics of the interface at a microscopic level. The density inhomogeneity persists at the interface and is a probe of the interface reconstruction due to interactions.

\paragraph*{Tensor Network Description of the Bulks ---} \label{SupMatSec:CFTreminder}  From the study of quantum entanglement in strongly correlated systems, a new class of variational WFs, namely the tensor networks states (TNS) has emerged in recent years. TNS efficiently encode physically relevant many-body states, relying on their rather low entanglement (for a review, see Ref.~\cite{ReviewTNRomanOrus}). For a large set of FQH model states, a MPS -- the prototype of TNS -- have been derived~\cite{ZaletelMongMPS,MPStrialPapic}. Moreover, the computational toolbox of tensor networks has been applied to large system size simulations of FQH systems~\cite{DMRGmicroscopicHamiltonian,DMRGmulticomponent,RegnaultConstructionMPS,RegnaultCorrelLength,MyFirstArticle,MPSNonAbelian,MPSquasiholes}.

We now first briefly recall the theoretical background of the exact MPS description for the Laughlin 1/$m$ and the spin singlet Halperin $(m,m,m-1)$ states~\cite{ZaletelMongMPS,RegnaultConstructionMPS,MyFirstArticle}. The exact MPS description of an FQH state which can be written as a CFT correlator consists of an electronic part and  a background part~\cite{ZaletelMongMPS,RegnaultConstructionMPS}. The former can be deduced from the mode expansion of the primaries appearing in the CFT correlators. The main result of Ref.~\cite{MyFirstArticle} is a method to determine those primaries and the exact MPS representation of two-components Abelian states from a factorization of the \textbf{K}-matrix  as $\mathbf{K}=\mathbf{Q}\mathbf{Q}^T$~\cite{WenZeeKmatrix,HanssonReviewCFTforFQHE}. We choose $\mathbf{Q}$ to be upper diagonal for reasons which will become clear later on: \begin{equation} \mathbf{K} = \begin{pmatrix} m & m-1 \\ m-1 & m  \end{pmatrix} \; , \quad \mathbf{Q} = \begin{pmatrix} \frac{2m-1}{R_\perp} & \frac{m-1}{R_{\rm L}} \\ 0 & \frac{m}{R_{\rm L}}  \end{pmatrix} \label{eq:KmatrixFactorization} \end{equation} where we have defined $R_\perp=\sqrt{m(2m-1)}$ and $R_{\rm L}=\sqrt{m}$. The underlying CFT is that of a two component boson ($\varphi^\perp$,$\varphi^{\rm L}$)~\cite{YellowBook}. Their respective U(1)-charges are integers $n_\perp$, $n_{\rm L}$ if measured in units of $R_\perp$ and $R_{\rm L}$ respectively, and satisfy the constraint $\frac{n_\perp + (m-1) n_{\rm L}}{m} \in \mathbb{Z}$. Compared to Ref.~\cite{MyFirstArticle}, Eq.~\eqref{eq:KmatrixFactorization} is just a different choice of orthonormal basis for the two-component boson, which is related to the usual spin-charge formulation~\cite{SpinChargeVoit,giamarchiQuantumOneD} by: \begin{subequations} \label{eq:BasisChangeBosonsSpinCharge}  \begin{align}
 \varphi^\perp &= \sqrt{\dfrac{1}{2m}} \varphi^c + \sqrt{\dfrac{2m-1}{2m}} \varphi^s \\   \varphi^{\rm L} &= \sqrt{\dfrac{2m-1}{2m}} \varphi^c - \sqrt{\dfrac{1}{2m}} \varphi^s \, ,
\end{align}	\end{subequations} where $\varphi^c$ and $\varphi^s$ are the bosonic fields corresponding to the charge and spin respectively. We define the spinful electronic operators as \begin{subequations} \label{eq:VertexOperators}  \begin{align}	\mathcal{W}^\uparrow (z) & = : \exp  \left( i \sqrt{\dfrac{2m-1}{m}}\varphi^\perp(z) +i \dfrac{m-1}{\sqrt{m}}\varphi^{\rm L}(z)  \right)  :\chi \\  \mathcal{W}^\downarrow (z) & = : \exp \left(i \sqrt{m}\varphi^{\rm L}(z)  \right)  :
\end{align} \end{subequations} where $\chi =(-1)^{\frac{n_\perp + (m-1) n_{\rm L}}{m} }$ acts as a Klein factor ensuring correct commutation relations between the electronic operators. The $j$-th Landau orbital on the cylinder is characterized by its occupation numbers $n^\uparrow$ and $n^\downarrow$. The electronic part $A^{(n^\uparrow,n^\downarrow)}[j]$ of the Halperin MPS matrices only depends on the mode expansion of the electronic operators of Eq.~\eqref{eq:VertexOperators}: \begin{equation}
A^{(n^\uparrow,n^\downarrow)}[j] = \frac{1}{\sqrt{n^\uparrow ! n^\downarrow!}} \left(\mathcal{W}_{-j}^\uparrow \right)^{n^\uparrow} \left(\mathcal{W}_{-j}^\downarrow \right)^{n^\downarrow} \label{eq:ElectronicPartMatrices}
\end{equation} It simply reduces to $A^{(0,n^\downarrow)}[j]$ for the (spin down) polarized Laughlin state. Our variational ansatz relies on the following crucial point: the Laughlin CFT Hilbert space made of a single boson $\varphi^{\rm L}$ is embedded into the Halperin CFT Hilbert space. Hence both sets of MPS matrices share the same auxiliary space which makes the gluing procedure straightforward in the Landau orbital basis, \textit{i.e.} a simple matrix multiplication. The basis change of Eq.~\eqref{eq:BasisChangeBosonsSpinCharge} makes this embedding transparent since we extract $\varphi^{\rm L}$ from the two component free boson. At the transition between the Halperin and the spin down polarized Laughlin bulks, the cut and glue approach~\cite{LudwigCutAndGlue} predicts with renormalization group arguments~\cite{CoupledWireFirstKane} that the degrees of freedom related to $\varphi^{\rm L}$ gaps out when the tunneling of spin down electrons across the transition is relevant, which is often assumed. The one dimensional effective field theory at the transition is then expected to be the one of a single bosonic $\varphi^\perp$ field.

To obtain an infinite MPS representation of the states, we combine the electronic operators as $\mathcal{V}(z)= \mathcal{W}^\uparrow (z) \ket{\uparrow}+\mathcal{W}^\downarrow (z) \ket{\downarrow}$. The Operator Product Expansion (OPE) of vertex operators~\cite{YellowBook} together with the \textbf{K}-matrix factorization Eq.~\eqref{eq:KmatrixFactorization} ensures that the $N_e$-points correlator \begin{equation}
\langle \mathcal{O}_{\rm bkg} \prod_{i} \mathcal{V}(z_i)  \rangle \label{eq:CFTcorrelator}
\end{equation} reproduces the Halperin $(m,m,m-1)$ (resp. Laughlin $1/m$) WF with $N_e$ particles through a careful choice of the background charge $\mathcal{O}_{\rm bkg}^{\rm H} (N_e)=\exp\left\{-iN_e\frac{2m-1}{2} \left(\frac{1}{R_\perp}\varphi_0^\perp + \frac{1}{R_{\rm L}}\varphi_0^{\rm L} \right)\right\}$ $\left({\rm resp. } \; \mathcal{O}_{\rm bkg}^{\rm L} (N_e)=\exp\left\{-iN_e\frac{m}{R_{\rm L}} \varphi_0^{\rm L}\right\}\right)$. The choice of the background charge reflects the facts that the Laughlin 1/$m$ state is an excitation of the denser Halperin $(m,m,m-1)$ state. Indeed, it may be understood as the introduction of a macroscopic number of $\varphi^\perp$ quasiholes (or a giant quasihole~\cite{SchoutensNassMR}) to fully polarize the Halperin Hall droplet into a Laughlin liquid.  Spreading these background charges equally between the orbitals provides a site-independent MPS representation for the Laughlin and Halperin states on the cylinder. Labeling these site-independent MPS representation $B_{\rm L}$ and $B_{\rm H}$, we have: \begin{subequations} \begin{align}
		B_{\rm L}^{(n^\downarrow)}  &=  A^{(0,n^\downarrow)}[0] U_{\rm L} & & U_{\rm L}=e^{-\left(\frac{2\pi}{L}\right)^2 L_0^{\rm L} - i\frac{\varphi_0^{\rm L}}{R_{\rm L}}} \, , \label{eq:iMPSlaughlin} \\
		B_{\rm H}^{(n^\uparrow,n^\downarrow)}  &= A^{(n^\uparrow,n^\downarrow)}[0] U_{\rm H} & & U_{\rm H}=e^{-\left(\frac{2\pi}{L}\right)^2 L_0^\perp - i\frac{\varphi_0^\perp}{R_\perp}} U_{\rm L} \, , \label{eq:iMPShalperin}
\end{align}\end{subequations} where $L_0^\perp$ (resp. $L_0^{\rm L}$) is the Virasoro zero-th mode corresponding to the standard free boson action for $\varphi^\perp$ (resp. $\varphi^{\rm L}$). Because there is no spin up component in the electronic part of the Laughlin MPS matrices, we may add a shift in the $\varphi^\perp$ U(1)-charge to $U_{\rm L}$ at each orbital. Doing so allows to always fulfill the compactification constraint $\frac{n_\perp + (m-1) n_{\rm L}}{m} \in \mathbb{Z}$. Since the Laughlin transfer matrix should only be considered over $m$ orbitals to preserve the topological sectors~\cite{RegnaultConstructionMPS}, we simply impose the shift over any $m$ consecutive orbitals to be zero.

\begin{figure*}
	\centering
	\includegraphics[width=\textwidth]{ConstructionInterfaceModelWF.pdf}
	\caption{ \textbf{Construction of an interface model wavefunction:} a) Schematic representation of the MPS ansatz $\ket{\Psi_{\rm H-L}}$ for the Laughlin 1/3 - Halperin 332 interface on a cylinder of perimeter $L$. The Halperin iMPS matrices $B_{\rm H}$ (red) are glued to the Laughlin iMPS matrices $B_{\rm L} \otimes \mathbb{I}$ (blue) in the Landau orbital space. Due to the embedding of one auxiliary space into the other, the quantum numbers of $\ket{\mu_\perp}$ (see Eq.~\eqref{eq:ModelWFProduct}) are left unchanged by the Laughlin matrices all the way to the interface. It constitutes a direct access controlling the states of the interface chiral gapless mode, graphically sketched here with a double arrow. b) Spin resolved densities of the MPS ansatz state along the cylinder axis obtained at $P_{\rm max}=11$ for $L=25\ell_B$. They smoothly interpolate between the Laughlin ($2\pi\rho_\downarrow=1/3$ and $\rho_\uparrow=0$) and the Halperin ($2\pi\rho_\downarrow=2\pi\rho_\uparrow=1/5$) theoretical values. The density is a robust quantities for which it is safe to consider large perimeter with our truncation level. c) The EE $S_\mathcal{A}(L,x)$ follows an area law  (Eq.~\eqref{eq:AreaLawAndCorrection}) for the rotationally invariant bipartition $\mathcal{A}-\mathcal{B}$ depicted on top of the graph. The constant correction is numerically extracted by finite differences $S_\mathcal{A}(L,x)-L \partial_L S_\mathcal{A}(L,x)$ and plotted for $L=13\ell_B$ as a function of the position along the cylinder axis $x$. It smoothly interpolates between its respective Laughlin and Halperin bulk values and we see no universal signature of the critical mode at the interface. Away from the interface, i.e. $x<-7\ell_B$ on the Halperin side and $x>3\ell_B$ on the Laughlin side, the extracted $\gamma(x)$ agree with the theoretical expectation within 3\% accuracy. Thus, our MPS model WF describes the interface between two distinct topological orders. Note that the extraction and convergence of the subleading quantity $\gamma(x)$ (see Eq.~\eqref{eq:AreaLawAndCorrection}) requires large truncation parameters. The spikes appearing on both sides of the transitions are artifacts of the computations of the RSES (see Ref.~\cite{MyFirstArticle} for details). They corresponds to the points where a patch of three (resp. five) orbitals are added on the Laughlin (resp. Halperin) side of the finite size region which translate the bipartition $\mathcal{A}-\mathcal{B}$ to orbital space. They disappear with increasing $P_{\rm max}$, as shown by the points computed at $P_{\rm max}=14$ (black dots). }
	\label{fig:AnstazConstruction}
\end{figure*}

\begin{widetext}

\paragraph*{Model Wavefunction for the Interface ---} \label{SupMatSec:ModelWF} We introduce a MPS model WF to describe the low energy features of the previously described microscopic model. It has non-vanishing coefficients only over the polarized Hilbert subspace discussed previously. To construct the MPS ansatz, we use $B_{\rm H}$ (resp. $B_{\rm L}$) matrices Eq.~\eqref{eq:iMPShalperin} (resp.~\eqref{eq:iMPSlaughlin}) for unpolarized (resp. polarized) orbitals. Thus, the MPS ansatz expanded on the many-body states Eq.~\eqref{eq:OccupationBasis} reads
\begin{equation}
\ket{\Psi_{\rm H-L}} = \sum_{\{n_k^\downarrow\}_{k}, \{n_k^\uparrow\}_{k>0}}
\bra{\eta}
{\cdots B_{\rm L}^{n_{-3/2}^\downarrow} B_{\rm L}^{n_{-1/2}^\downarrow} B_{\rm H}^{(n_{1/2}^\uparrow,n_{1/2}^\downarrow)} B_{\rm H}^{(n_{1/2}^\uparrow,n_{1/2}^\downarrow)} \cdots }
\ket{\mu_\perp}
\ket{\{n_k^\downarrow\}_{k}, \{n_k^\uparrow\}_{k>0}}
\, .  \label{eq:ModelWFProduct}
\end{equation} \end{widetext} This ansatz is schematically depicted on Fig.~\ref{fig:AnstazConstruction}a. Here $\bra{\eta}$ and $\ket{\mu}$ are the two states in the auxiliary space fixing the left and right boundary conditions. We fix the gauge of the MPS by choosing a basis for the CFT auxiliary space, which agrees with the structure discussed earlier and in which we have $\bra{\eta}=\bra{\eta_L}\otimes\bra{\eta_\perp}$ and $\ket{\mu}=\ket{\mu_L}\otimes \ket{\mu_\perp}$. It is worth mentioning some important features of this ansatz. First, the use of Halperin and Laughlin site-independent MPS matrices allows to consider an infinite cylinder and enables the use of efficient infinite-MPS (iMPS) algorithms~\cite{DMRGmicroscopicHamiltonian,DMRGmulticomponent}. Fig.~\ref{fig:AnstazConstruction}b shows the spin-resolved densities of this variational ansatz on an infinite cylinder of perimeter $L=25\ell_B$. As in Fig.~\ref{fig:MicroscopicTestInterfaceED}b, they smoothly interpolate between the polarized Laughlin bulk at filling factor $\nu_L=1/3$ and the Halperin unpolarized bulk at filling factor $\nu_H= \frac{1}{5}+\frac{1}{5}$. We recover the typical bulk densities and the spin SU(2) symmetry of the Halperin (332) state after a few magnetic lengths. We can also observe that the finite size effects on the density quickly disappear with increasing $L$. The correlation lengths for the bulks are respectively $\xi_H = 1.28 \ell_B$ for the Halperin 332 and $\xi_L = 1.38 \ell_B$ for the Laughlin 1/3 states~\cite{MyFirstArticle}. The ripples disappear for $L/\max (\xi_H,\xi_L) \geq 15 $. Another source of finite size effects is the truncation of the infinite CFT Hilbert space in our computations. In practice, we truncate the auxiliary space with respect to the conformal dimension. Our truncation parameter, denoted as $P_{\rm max}$, is a logarithmic measure of the bond dimension (see Refs.~\cite{RegnaultConstructionMPS} and \cite{MyFirstArticle} for a precise definition). The truncation of the auxiliary space is constrained by the entanglement area law~\cite{ReviewEntanglementLaflorencie}, the bond dimension should grow exponentially with the cylinder perimeter $L$ to accurately describes the model WFs (at least in the gapped bulks). Thus, as an empirical rule, $P_{\rm max}$ should grow linearly with the cylinder perimeter. This is what prevents us from reaching the thermodynamic limit $L/\ell_B \to \infty$. Using both charge conservation and rotation symmetry along the cylinder perimeter provides additional refinements to the iMPS algorithm~\cite{ZaletelMongMPS,DMRGmicroscopicHamiltonian,DMRGmulticomponent}. They can be implemented all along the cylinder, and importantly across the interface, by keeping track of the quantum numbers of the CFT states.

Due to the block structure of the MPS matrices~\cite{MyFirstArticle}, the boundary conditions $\bra{\eta}$ and $\ket{\mu}$ naturally separates the different charge and momentum sectors all along the cylinder. In particular, they separate the different topological sectors and ensure that no local measurement can discriminate them. For instance on the Halperin side, we can create five distinct bulk WFs (since $\det \mathbf{K}=5$~\cite{WenZeeKmatrix}) corresponding to the topological degeneracy of the state on genus one surfaces. While for the Halperin state, the U(1)-charge of both $\ket{\mu_L}$ and $\ket{\mu_\perp}$ should be fixed to determine the topological sector, only the one of $\bra{\eta_L}$ is required to fix the topological sector of the Laughlin phase. The remaining bosonic degree of freedom, $\bra{\eta_\perp}$ at the edge of the Laughlin bulk constitutes a knob to dial the low-lying excitations of the one dimensional edge mode at the interface. Note indeed that the Laughlin iMPS matrices~\eqref{eq:iMPSlaughlin} act as the identity on $\bra{\mu_\perp}$ and propagate the state all the way to the interface.

In the following sections, we put our model wavefunctions Eq.~\eqref{eq:ModelWFProduct} to the test and we establish that they indeed capture the universal features of the interface. We confirm that the expected intrinsic topological order is recovered in the bulks away from the interface by extracting the relevant topological entanglement entropies. We characterize the interface gapless mode as a chiral Luttinger liquid. We first extract the interface central charge $c=1$ through the entanglement entropy.  We then extract the corresponding compactification radius $R_{\perp} = \sqrt{15}$ by identifying the spin and charge of the interface  elementary excitations in the many-body spectrum.  Moreover, we compare our model states with finite size studies and characterize the low energy features of the spectrum thanks to the boundary state $\bra{\eta_\perp}$.

\begin{figure*}
	\centering
	\includegraphics[width=0.8\textwidth]{UniversalFeatures.pdf}
	\caption{\textbf{Universal features of the interface ansatz:} a) Levin-Wen subtraction scheme (top) to get rid of the spurious area law coming from the patch boundaries along the cylinder perimeter together with corner contributions to the EE. The position and width $w$ of the rectangular extension are selected to fully include the gapless mode at the interface. (bottom) $S_\mathcal{A}(\ell,w=3.25\ell_B)$ for different cylinder perimeters computed at $P_{\rm max}=12$. They all fall on top of the CFT prediction Eq.~\eqref{eq:CriticalTheoryEEfromCFT} with $c=1$ (black line), pointing toward a critical chiral edge mode at the interface. The agreement improves with increasing $L$, as expected since universal effects are unraveled at $L/\ell_B \to \infty$. The inset shows the derivative $\partial_\ell S_\mathcal{A}(\ell,w)$ and the fit with the central charge as the only free parameter (dashed red). The grayed area corresponds to points for which $\ell$ or $L-\ell$ is smaller than three times the Halperin 332 bulk correlation length~\cite{MyFirstArticle}, points in this areas are discarded to mitigate finite-size effects. We find $c_{\rm Fit}=1.1(1)$, in close agreement with the theoretical prediction (the fitting procedure at $P_{\rm max}=11$ gives the same result). b) Charge and spin excess are localized at the interface when excited states are addressed via the MPS boundary U(1)-charge $n_\perp$. Inset shows how to extract the charge excess (gray shaded area) from the charge densities $\rho_c$ at different $n_\perp$, computed at $P_{\rm max}=11$. Each excess follows a linear relation with respect to $n_\perp$ with extremely good accuracy. The charge (resp. spin) excess has a slope  $0.0666(1)\simeq 1/15$ (resp. $0.3332(2) \simeq 1/3$). This indicates that the elementary excitations of the $c=1$ critical theory at the interface carry a fractional charge $e/15$ and a fractional spin $1/3$ in unit of the electron spin.}
	\label{fig:UniversalFeatures}
\end{figure*}

\paragraph*{Universal Features of the Trial State ---} Effective one-dimensional theories similar to the ones of Refs.~\cite{SchoutensNassMR,KaneFibonacciCoset} predict that the gapless interface is described by the free bosonic CFT $\varphi^\perp$ of central charge $c=1$ and compactification radius $R_\perp = \sqrt{15}$. Remarkably, this is neither an edge mode of the Halperin state nor of the Laughlin state. It is a direct consequence of the edge reconstruction due to interactions which are kept constant across the interface (see Eq.~\eqref{eq:HamiltonianFermionic}). The full characterization of the bulk universal properties and the interface critical theory is the main result of Ref.~\cite{Interface_bosons}. We briefly discuss such a characterization in the context of the fermionic Laughlin 1/3 - Halperin (332) interface.

Local operators such as the density cannot probe the topological content of the bulks. We thus rely on the entanglement entropy (for a review, see Ref.~\cite{ReviewEntanglementStrongly}) to analyze the topological features of our model WF. All the relevant theoretical framework required for the computation of Real-Space Entanglement Spectrum (RSES)~\cite{RSESdubail,RSESsterdyniak,RSESRodriguezSimonSlingerland} has been summed up in the Methods. Consider a bipartition $\mathcal{A}-\mathcal{B}$ of the system defined by a cut perpendicular to the cylinder axis at a position $x$. The RSES and the corresponding Von Neumann EE $S_\mathcal{A}(L,x)$ are computed for various cylinder perimeters $L$. We find that $S_\mathcal{A}(L,x)$ obey an area law~\cite{ReviewEntanglementLaflorencie} for any position of the cut $x$:\begin{equation}
S_\mathcal{A}(L,x)= \alpha(x) L - \gamma(x) \, . \label{eq:AreaLawAndCorrection}
\end{equation} Far away from the transitions, the constant correction to the area law converges to the Topological Entanglement Entropy (TEE)~\cite{TopoCorrectionKitaev,TopoCorrectionLevinWen} of the Laughlin $\left(\gamma(x\to +\infty)=\log \sqrt{3}\right)$ and Halperin $\left(\gamma(x\to -\infty)=\log \sqrt{5}\right)$ states. Near the interface, the EE still follows Eq.~\eqref{eq:AreaLawAndCorrection} as was recently predicted for such a rotationally invariant bipartition~\cite{TaylorHughesTransitionSPT}. The correction $\gamma(x)$ smoothly interpolates between its respective Laughlin and Halperin bulk values (see Fig.~\ref{fig:AnstazConstruction}c). Hence, it contains no universal signature of the critical mode at the interface between the two topologically ordered phases. The same conclusion holds for the area law coefficient $\alpha(x)$.

In order to obtain signature from the interface critical theory, we need to break the translation symmetry along the cylinder perimeter. We thus compute the RSES for a bipartition for which the part $\mathcal{A}$ consists of a rectangular patch of length $\ell$ along the compact dimension and width $w$ along the $x$-axis. To fully harness the power of the iMPS approach, it is convenient to add a half infinite cylinder to the rectangular patch (see Fig.~\ref{fig:UniversalFeatures}a). The contribution of the interface edge mode is isolated with a Levin-Wen addition subtraction scheme~\cite{TopoCorrectionLevinWen} depicted in Fig.~\ref{fig:UniversalFeatures}a and more thoroughly discussed in Ref.~\cite{Interface_bosons}. Noticing that the critical contribution is counted twice, the 1D prediction for a chiral CFT of central charge $c$ with periodic boundary conditions reads~\cite{CFTentanglementEntropyCardy} \begin{equation}
S_\mathcal{A}(\ell,w) - S_\mathcal{A}(L/2,w)  = 2 \,  \dfrac{c}{6} \log \left[ \sin \left( \dfrac{\pi \ell}{L} \right)\right] \, . \label{eq:CriticalTheoryEEfromCFT}
\end{equation} We vary the length $\ell$ along the compact dimension of the cylinder while keeping $w$ constant, we fit the numerical derivative $\partial_\ell S_\mathcal{A}(\ell,w)$ with the theoretical prediction using the central charge $c$ as the only fitting parameter (the derivative removes the area law contribution arising from the cut along $x$). We minimize finite size effects by keeping only the points for which $\ell$ and $L-\ell$ are both greater than three times the Halperin bulk correlation length~\cite{MyFirstArticle} and consider the largest perimeter that reliably converge $L=13\ell_B$. We extract a central charge $c=1.1(1)$ in agreement with the universal expectation. The inset of Fig.~\ref{fig:UniversalFeatures}a shows the numerical data, the result of the fit and the theoretical expectation Eq.~\eqref{eq:CriticalTheoryEEfromCFT} which all nicely agree.

In order to fully characterize the gapless mode circulating at the interface, we now extract the charges of its elementary excitations which are related to the compactification radius $R_\perp$. As previously mentioned, excited states of the critical theory are numerically controlled by the U(1)-charge of $\bra{\eta_\perp}$. For each of these excited states, we compute the spin resolved densities and observe that the excess of charge and spin are localized around the interface. They stem from the gapless interface mode observed in Fig.~\ref{fig:UniversalFeatures}a and we plot the charge and spin excess as a function of $n_\perp$ in Fig.~\ref{fig:UniversalFeatures}b. Elementary excitations at the transition carries fractional charge $e/15$ and a fractional spin $1/3$ in unit of the electron spin. More generally, the transition between Laughlin 1/$m$ state and an Halperin $(m,m,m-1)$ state should host quasiparticles of charge $e/(m(2m-1))$. This exactly fits the elementary spin and charge content of $\varphi^\perp$ (see Eq.~\eqref{eq:BasisChangeBosonsSpinCharge}). The interface gapless mode is described by a chiral Luttinger liquid, \emph{i.e.} a compact bosonic conformal field theory whose elementary excitations agree with the value $R=\sqrt{15}$ of the compactification radius.

\begin{table}
	\centering
	\caption{\textbf{Combining root partitions: } Right and left MPS boundary charges $(n_{\rm L},n_\perp)$ to recover the glued Laughlin 1/3 and Halperin 332 root partitions. $\emptyset$, $\downarrow$ and $\uparrow$ respectively denote an empty orbitals or an occupied orbital with a spin down or up. Using the first line and the first columns, the total root configuration should be understood as ...$\emptyset \; \emptyset \downarrow \emptyset \; \emptyset \downarrow \emptyset \; \emptyset \downarrow - \emptyset \; \emptyset  \uparrow  \emptyset \downarrow \emptyset \; \emptyset \uparrow \emptyset \downarrow  $... for the gluing of $\left[\emptyset \; \emptyset  \downarrow \right]$ and $\left[\emptyset \; \emptyset \; \uparrow \emptyset \downarrow \right]$. Note that defining the root configuration by, \textit{e.g.}, $\left[\emptyset \; \emptyset  \uparrow  \emptyset  \downarrow \right]$ instead of $\left[\emptyset \; \emptyset  \downarrow  \emptyset  \uparrow \right]$ is arbitrary due to the SU(2) singlet nature of the Halperin 332 state. We refer to Ref.~\cite{EstienneRootHalperin} for more details of the Halperin states root configurations.}
	\label{tab:GlueRootTogether}
	\resizebox{\linewidth}{!}{%
	\begin{tabular}{|c||c|c|c|c|c|}
		\hline
		$(n_{\rm L},n_\perp)$   & $ \; \left[\emptyset \; \emptyset \uparrow \emptyset \downarrow \right] \;$ & $ \; \left[\emptyset \downarrow \emptyset \; \emptyset \uparrow \right] \;$ & $ \;\left[\emptyset \uparrow \emptyset \downarrow \emptyset \right] \;$ & $ \;\left[\downarrow \emptyset \; \emptyset \uparrow \emptyset \right] \;$ & $ \;\left[\uparrow \emptyset \downarrow \emptyset  \; \emptyset \right] \;$ \\ \hline \hline
		$\left[\emptyset \; \emptyset \downarrow \right]$ & $(0,0)$  & $(0,3)$ & $(0,6)$ & $(0,9)$ & $(0,12)$ \\ \hline
		$\left[\emptyset \downarrow \emptyset \right]$  & $(1,-5)$ & $(1,-2)$ & $(1,1)$ & $(1,4)$ & $(1,7)$ \\ \hline
		$\left[\downarrow \emptyset \; \emptyset \right]$ &$(2,-10)$ & $(2,-7)$ & $(2,-4)$ & $(2,-1)$ & $(2,2)$ \\ \hline
	\end{tabular}}
\end{table}

\paragraph*{Comparison with Exact Diagonalization in Finite Size ---} To go beyond these universal properties and test the relevance of our model WF at a microscopic level, we now compare it to finite size calculations. We first investigate in more details the model WF to get a better microscopic understanding of the interface and the role of the MPS boundary states' quantum numbers. The electronic operators $\mathcal{W}^\uparrow$ and $\mathcal{W}^\downarrow$ generate the charge lattice from a unit cell composed of 5 inequivalent sites~\cite{MyFirstArticle}. Physically, they correspond to the ground state degeneracy of the Halperin (332) state on the torus (or the infinite cylinder) which is known to be $|\det \mathbf{K}|=5$~\cite{WenZeeKmatrix}. The choice of $n_\perp$ modulo five determines the topological sector of the Halperin bulk far from the transition. An identical analysis involving the spin down electronic operator $\mathcal{W}^\downarrow$ only shows that the Laughlin topological sector is selected by the value of $n_{\rm L}$ modulo 3. Loosely speaking, these degeneracies give 15 different ways of gluing the two bulks together which lead to the observed fractional charge in Fig.~\ref{fig:UniversalFeatures}b and the compactification radius $R_\perp=\sqrt{15}$. This intuition is rigorous in the thin torus limit $L\ll \ell_B$~\cite{RelationRootThinTorus0,RelationRootThinTorus1} where the bulk physics are dominated by their respective root partitions~\cite{RootPartitionBernevig,RootPartitionSpinBernevig}. In the CFT language, we may understand it as a renormalization procedure. Because the Virasoro zero-th mode $L_0$~\cite{YellowBook} only appears with a prefactor $\left(\frac{2\pi\ell_B}{L}\right)^2$, all excitations above the CFT ground state becomes highly energetic and we can trace them out. This is exactly what the truncation at $P_{\rm max}=0$ does. From here, we may look at the U(1)-charges $(n_{\rm L},n_\perp)$ of the boundary state $\ket{\mu}$ which produce a non-zero coefficient for a given Halperin root partition (\textit{i.e.} when $P_{\rm max}=0$). We have performed the study for both the Halperin and the Laughlin bulks, and we summarize our results on the possible ways of gluing together the root partitions of these states in Tab.~\ref{tab:GlueRootTogether}. These insights on the role of boundary states may be used to understand the low energy features of finite size calculations. The choice of the U(1)-charges selects one of the possible root configurations given in Tab.~\ref{tab:GlueRootTogether}. The root configuration fixes the reference for the center of mass angular momentum of the system $K_y$ in finite size. Low energy excitations on top of a given U(1)-charge choice are obtained by dialing the $\varphi^{\rm L}$ and $\varphi^\perp$ descendants.

\begin{table}
	\centering
	\caption{ \textbf{Comparison with ED ground states: } Overlap between the MPS variational ansatz (at $P_{\rm max}=11$) for the $(n_{\rm L},n_\perp)=(0,0)$ and $P_\mu=P_\eta=0$ boundary conditions and the corresponding ED ground state for different system sizes characterized by the particle numbers $(2N_{\rm H},N_{\rm L})$ on a cylinder $L=12\ell_B$. The number of orbitals are fixed to $N_{\rm orb}^{\rm L}=3N_{\rm L}-2$ and $N_{\rm orb}^{\rm H}=5N_{\rm H}$. Due to the dimension of the many-body Hilbert space considered (415 203 170 for the largest systems), the overlaps are computed over a significant fraction of the vectors weights (we keep all the coefficient with a magnitude greater than $10^{-5}$). The norms of the truncated ED $\ket{\psi_{\rm trunc}^{\rm ED}}$ and MPS $\ket{\psi_{\rm trunc}^{\rm MPS}}$ vectors, which can be evaluated rigorously, give an estimate for the possible error.}
	\label{tab:OverlapEDMPS}
	\begin{tabular}{|c|c|c|c|}
		\hline \rule{0pt}{10pt}
		$(2N_{\rm H},N_{\rm L})$ & $|| \; \ket{\psi_{\rm trunc}^{\rm ED}} \; ||$ & $|| \; \ket{\psi_{\rm trunc}^{\rm MPS}} \; ||$ & $ | \braket{\psi_{\rm trunc}^{\rm ED}}{\psi_{\rm trunc}^{\rm MPS}}|$  \\[5pt]
		\hline
		(6,3) &  1.000 & 1.000 & 0.998 \\ \hline
		(6,4) &  1.000 & 1.000 & 0.997 \\ \hline
		(8,4) &  1.000 & 1.000 & 0.997 \\ \hline
	\end{tabular}
\end{table}

Using the root configuration to relate the MPS boundary indices to the finite size parameters, we are now able to provide convincing numerical evidence that our ansatz should capture the low energy physics of the Hamiltonian Eq.~\eqref{eq:HamiltonianFermionic}. For this purpose, we have performed extensive Exact Diagonalization (ED) of the Hamiltonian Eq.~\eqref{eq:HamiltonianFermionic} for $N_{\rm L}+N_{\rm H}$ spin down and $N_{\rm H}$ spin up particles in $N_{\rm orb}^{\rm L}+N_{\rm orb}^{\rm H}$ orbitals, $N_{\rm orb}^{\rm L}$ of which are fully polarized. We would like to show that the low energy features detaching from the continuum in the spectrum of Eq.~\eqref{eq:HamiltonianFermionic}, which is depicted in Fig.~\ref{fig:EdNicolas} for $N_{\rm L}=N_{\rm H}=3$. Let us first fix the level descendant of the MPS boundary conditions $P_\mu=P_\eta=0$ and selects some U(1)-charges appearing in Tab.~\ref{tab:GlueRootTogether}, to describe the states which persist in the thin torus limit $L\ll \ell_B$. We observe that, when $N_{\rm orb}^{\rm L}=3N_{\rm L}-2$ and $N_{\rm orb}^{\rm H}=5N_{\rm H}$, the ED ground state is the unique state detaching from the continuum (see green symbols in Fig.~\ref{fig:EdNicolas}). It has exactly the total momentum expected from the glued root partition selected by the $(0,0)$ boundary charges. Our MPS ansatz with these boundary conditions shows extremely high overlap with the corresponding ED ground states (see Tab.~\ref{tab:OverlapEDMPS}). Fig.~\ref{fig:MicroscopicTestInterfaceED}b shows the spin-resolved densities of the ED ground state of the largest reachable system sizes. Both the bulks and interface physics are displayed in the ED study and its very high overlap with our ansatz shows that this latest correctly captures the interface physics at a microscopic level.

\begin{figure*}
	\centering
	\includegraphics[width=1.\linewidth]{Resultats_ED_Nicolas.pdf}
	\caption{\textbf{Identifying low-energy excitations:} Energy spectrum (using a logarithmic scale) of Eq.~\eqref{eq:HamiltonianFermionic} for a system of 9 particles (6 spin down, 3 spin up) on a cylinder of perimeter $L=12\ell_B$ with $N_{\rm orb}^{\rm H}=15$ orbitals and $N_{\rm orb}^{\rm L}=7$ (green symbols), $N_{\rm orb}^{\rm L}=8$ (orange symbols) or $N_{\rm orb}^{\rm L}=9$ (blue lines). In the first case, only one of the root configuration of Tab.~\ref{tab:GlueRootTogether} can be produced. Increasing the number of polarized orbitals, we allow for excitations of the Laughlin edge gapless mode, that we can track with the MPS boundary quantum number $P_\eta^L$ (a). The low lying branches for $N_{\rm orb}^{\rm L}=8$ or $9$ are all connected to some gluing conditions (see Tab.~\ref{tab:GlueRootTogether}). We have indicated them as $(n_{\rm L},n_\perp)^0$, together with the overlap with the ED states targeted (Main Figure). Halperin edge excitations are also present in the spectrum and, as shown in (b), we can locate them in the spectrum thanks to the MPS ansatz by varying $P_\mu$. Finally, we can discriminate the interface gapless mode excited states by comparing the ED eigenvectors with our MPS model WF when $P_\eta^\perp \neq 0$. The corresponding states and overlaps are labeled $(n_{\rm L},n_\perp)^{P_\eta^\perp}$. We can for instance follow the two first excitations starting from $(2,-4)^{0}$, and observe a very steep dispersion relation for the interface critical mode. In general, all low lying excitations may be reproduced with our ansatz for mixed excitation $(P_\mu \neq 0, \, P_\eta^L \neq 0, P_\eta^\perp \neq 0 )$. All the overlaps presented were computed at truncation parameter $P_{\rm max}=12$ and as a rule of thumb we observe that the closer to the continuum the poorer the MPS ansatz performs.}
	\label{fig:EdNicolas}
\end{figure*}

When the number of polarized orbitals is increased, several low energy branches separate from the continuum (orange and blue markers in Fig.~\ref{fig:EdNicolas}). Changing the boundary U(1)-charges of the MPS model WF as prescribed in Tab.~\ref{tab:GlueRootTogether} while keeping $P_\mu=P_\eta=0$, we could identify the root partition dominating the low energy features in each branch in the thin torus limit. These states are labeled by $(n_{\rm L},n_\perp)$ in Fig.~\ref{fig:EdNicolas} and their overlaps with the corresponding MPS model WF is always above 0.977. The momentum transfer required to go from one gluing condition to another is extensive with the number of particles. Thus for an infinite system, the different branches clearly separate in the spectrum, while they may overlap (and do in many cases) for the system considered as can be seen for the state $(1,1)$ in Fig.~\ref{fig:EdNicolas}. Furthermore, we are able to access and identify the low energy excitations above the different root configurations with the MPS model ansatz. The latter arises at the edge of the gapped FQH droplets and are of three distinct types: Laughlin edge excitations, Halperin edge excitations and excitations of the interface gapless mode. We now exemplify each of these three cases, and show at the same time how to characterize the states in the many-body spectrum.

\begin{description}
	\item[Halperin Edge Excitations] Let us fix the MPS boundary charges to $(n_{\rm L},n_\perp)=(2,2)$, whose corresponding root partition (see Tab.~\ref{tab:GlueRootTogether}) is expected to appear for a center of mass momentum $K_y = 22.5$. In this momentum sector, we find one eigenvector of Eq.~\eqref{eq:HamiltonianFermionic} with high overlap with the model WF for $P_\mu=P_\eta=0$, which slightly detaches from the continuum. Changing the momentum of the Halperin boundary state $P_\mu$, the MPS model WFs acquire a momentum $K_y+P_\mu$. These excitations are used to describe quasihole excitations at the edge of the Halperin bulks~\cite{RootPartitionSpinBernevig,EstienneRootHalperin}. We compute the weight of the ED eigenvectors close-by in the space generated by all MPS model WFs with $P_\eta=0$ and $P_\mu=1$ or $P_\mu=2$. As shown in Fig.~\ref{fig:EdNicolas}a, they clearly belong to this subspace. Thus, our MPS model WFs allowed us to characterize without ambiguity the excitations in the branch starting at $K_y=22.5$ as Halperin gapless edge mode excitations (note the counting 1-2-5-...).

	\item[Laughlin Edge Excitations] The same tools may be used to probe excitations of the Laughlin gapless edge mode. We split $P_\eta=P_\eta^L+P_\eta^\perp$, where $P_\eta^\perp$ (resp. $P_\eta^L$) denotes the descendant level of the state $\bra{\eta}$ with respect to the $\varphi^\perp$ (resp. $\varphi^{\rm L}$) boson and we keep $P_\eta^\perp=0$ for now. Starting from the state $(n_{\rm L},n_\perp)=(0,0)$ at $K_y=40.5$, we could reproduce the excitations at $K_y-P_\eta^L$ as depicted in Fig.~\ref{fig:EdNicolas}b. Finite size effects limit the number of accessible descendants to $P_\eta^L=3$ in the ED spectrum. However, it is clear from the computed overlaps that the considered states are edge excitations of the Laughlin droplet. The same analysis can be repeated all over the spectrum.

	\item[Interface Excitations] Finally, and more interestingly, we were able to localize the excitations due to the interface gapless mode (see Fig.~\ref{fig:EdNicolas}). This time, we keep $P_\mu=P_\eta^L=0$ and vary $P_\eta^\perp$. We really want to highlight the difficulty to find those states from a pure ED approach, especially considering that each interface gapless mode excitation changes the energy by an order of magnitude.  We attribute this large interface mode velocity to the sharpness of the transition described by our ansatz, \textit{i.e.} to the change from $\mu_\uparrow=0$ to $\mu_\uparrow=\infty$ over an inter-orbital distance (see Supplementary Fig.~2 for further investigations).

\end{description}

While we have only considered one kind of excitations at a time, generic low energy states in the spectrum are characterized by non-zero $P_\mu$, $P_\eta^L$ and $P_\eta^\perp$. While the ED spectrum does not distinguish between the Laughlin and Halperin bulk excitations and the excitations of interface modes, the high overlap between the MPS states with the low lying part of the ED spectra help us discriminating these different types of excitations. This makes the proposed model states valuable tools even for finite size studies.

\section*{Discussion}

We have considered the fermionic interface between the Laughlin 1/3 and Halperin (332) states, relevant for condensed matter experiments. Indeed experimental realizations of this transition can be envisioned in graphene. There, the valley degeneracy leads to a spin singlet state at $\nu=2/5$~\cite{JainGrapheneTwoFifthIsHalperin,ExperimentalTwoFifthGraphene} while the system at $\nu=1/3$ is spontaneously valley-polarized~\cite{ExperimentalTwoFifthGraphene,OneThirdPolarizedGraphene,OneThirdPolarizedGrapheneBis}. Thus, changing the density through a top gate provides a direct implementation of our setup.

In Ref.~\cite{Interface_bosons}, we introduced a family of model states to describe the Laughlin-Halperin interface. Their universal properties were established using quantum entanglement measures, and the emerging gapless mode at the interface was characterized. It is described by a chiral Luttinger liquid, \emph{i.e.} a compact bosonic conformal field theory whose elementary excitations agree with the value $R=\sqrt{15}$ of the compactification radius.

The main result of this work is the thorough microscopic validation of our model wavefunctions. We introduced an experimentally relevant microscopic Hamiltonian that captures the physics of this interface, which we then analysed using exact diagonalization simulations on large-size systems. We found that the family of model states we introduced in Ref.~\cite{Interface_bosons} performs exceedingly well, reproducing the low-energy states of the microscopic model with extraordinarily good overlaps. Furthermore, these model states provide a powerful tool to identify the nature of the low-energy states obtained through exact diagonalization. In particular they allow to disentangle the interface modes from the Laughlin and Halperin edge modes,  a notoriously difficult task to carry out from finite size exact diagonalization.

Our interface model state therefore yields a bridge between the microscopic, experimentally relevant model and its low-energy effective description in terms of interfaces between topological quantum field theories.

\section*{Methods}

\paragraph*{Entanglement Entropy and MPS: Derivation ---}\label{SupMatSec:EEandMPSderivation} We turn to the computation of the Real Space Entanglement Spectrum (RSES)~\cite{RSESdubail,RSESsterdyniak,RSESRodriguezSimonSlingerland} for MPS ansatz considered. We recall that the electronic operator modes have the same commutation relations as the creation and annihilation operators. We will use these relations extensively to compute the RSES. For clarity, we focus primarily on the Laughlin case, the generalization to the Halperin case or to the Laughlin-Halperin interface only involves additional indices without involving any new technical step. For clarity, we will remove the spin index from the discussion whenever they are not needed. It is also useful to work with the site-dependent representation of the Laughlin 1/$m$ state~\cite{RegnaultConstructionMPS}:
\begin{equation}  \label{eq:NiceWrittingMPS}
\resizebox{\linewidth}{!}
$\begin{split}
&\ket{\Phi_{\alpha_R}^{\alpha_L}} = \\ & \sum_{\{n_k^\downarrow\}} \bra{\alpha_L} \mathcal{O}_{\rm bkg}^{\rm L} \left[ \prod_{k=0}^{N_\phi} \dfrac{1}{n_k^\downarrow !} \left( \mathcal{W}_{-k}^\downarrow \right)^{n_k^\downarrow} \otimes \left( c_{k}^\dagger \right)^{n_k^\downarrow} \right] \left( \ket{\alpha_R} \otimes \ket{\Omega} \right) \, ,
\end{split}$ } \end{equation}  where $c_{k}^\dagger$ creates a particle on orbital $k$ (see Eq.~\eqref{eq:LLLbasiscylinder}). The afore-mentioned site independent MPS representation Eq.~\eqref{eq:iMPSlaughlin} comes from spreading of the background charge~\cite{ZaletelMongMPS} and requires a shift of the MPS-boundary state U(1) charges~\cite{RegnaultConstructionMPS}, which we will keep implicit here.

\paragraph{Real Space Bipartition ---}

Under a a generic real space bipartition $\mathcal{A}-\mathcal{B}$, the LLL orbitals Eq.~\eqref{eq:LLLbasiscylinder} are decomposed as $c_{k}^\dagger  = d_{k,{\cal A}}^\dagger +d_{k,{\cal B}}^\dagger$ with
\begin{equation}
d_{k,{\cal I}}^\dagger = \int_{\vec{r} \in {\cal I}} {\rm d}^2 \vec{r} \,  \psi_k (\vec{r}) c_{k}^\dagger (\vec{r}) \; , \quad \mathcal{I} \in \{{\cal A},{\cal B}\} \, .
\end{equation} The sets $\{ d_{k,{\cal A}} \}$ and $\{d_{k,{\cal B}}\}$ span two disjoint Hilbert spaces of respective vacua $\ket{\Omega_{\cal A}}$ and $\ket{\Omega_{\cal B}}$ but are in general not orthonormal:
\begin{equation} \left\{ d_{k,{\cal I}} , d_{\ell,{\cal I}'}^\dagger \right\} = \delta_{{\cal I},{\cal I}'} \int_{\vec{r} \in {\cal I}} {\rm d}^2 \vec{r} \, \psi_k^*(\vec{r}) \psi_\ell (\vec{r})  \label{eq:CommutationRelationSplitCreator}
\end{equation} where $\{.,.\}$ denotes the anticommutator. For a cut preserving the rotation symmetry along the cylinder perimeter ${\cal A}=\{(x',y') | x'<x , 0\leq y' \leq L \}$, the overlaps in the right hand side of Eq.~\eqref{eq:CommutationRelationSplitCreator} are diagonal and take the form \begin{equation}\label{eq:RotSymCoefErrorFct} \begin{split}
g_{k,{\cal A}} &= \int_{\vec{r} \in {\cal A}} {\rm d}^2 \vec{r} \, \psi_k^*(\vec{r}) \psi_k (\vec{r}) \\ &= \sqrt{\dfrac{1}{\pi\ell_B} \int_{x'<x} \text{d}x' \, \exp{ \left( -\dfrac{(x'-x_k)^2}{\ell_B^2} \right) }}
\end{split} \end{equation} These overlaps can still be computed analytically for some bipartitions breaking the rotation symmetry. In that case, $\{d_{k,{\cal A}}\}$ can be decomposed over an orthonormal basis $\{\tilde{c}_{\mu,{\cal A}}\}$ as \begin{equation}
d_{k,{\cal A}}^\dagger = \sum_{\mu=0}^{N_\phi} \alpha_{k,\mu} \,  \tilde{c}_{\mu,{\cal A}}^\dagger \label{eq:SplitCreationNonRot}
\end{equation} where the coefficient $\{\alpha_{k,\mu}\}$ are obtained either analytically or numerically from the known overlaps between LLL orbitals over the region ${\cal A}$.

\paragraph{Split and Swap Procedure ---}

Using the decomposition $c_{k,\downarrow}^\dagger  = d_{k,{\cal A}}^\dagger +d_{k,{\cal B}}^\dagger$ together with the commutation relations of Eq.~\eqref{eq:CommutationRelationSplitCreator}, and introducing a closure relation $\sum_{\beta \in \mathcal{H}_{\rm CFT}}  \ket{\beta} \bra{\beta}$, with $\mathcal{H}_{\rm CFT}$ the auxiliary space, \textit{i.e.} the CFT Hilbert space, the Schmidt decomposition of Eq.~\eqref{eq:NiceWrittingMPS} $  \ket{\Phi_{\alpha_R}^{\alpha_L}} = \sum_{\beta \in \mathcal{H}_{\rm CFT}} \ket{\phi_\beta^{\cal B}} \otimes \ket{\phi_\beta^{\cal A}}$ onto the partition ${\cal A}-{\cal B}$ is found to be:
\begin{widetext}
\begin{equation}  \label{eq:SuppMatSchmidt} \begin{aligned}
\ket{\phi_\beta^{\cal B}} &=  \sum_{\{n_k^\downarrow\}} \bra{\alpha_L} \mathcal{O}_{\rm bkg}^{\rm L} \left[ \prod_{k=0}^{N_\phi} \dfrac{1}{n_k^\downarrow !} \left( \mathcal{W}_{-k}^\downarrow \right)^{n_k^\downarrow} \otimes \left( c_{k}^\dagger \right)^{n_k^\downarrow} \right] \left( \ket{\alpha_R} \otimes \ket{\Omega} \right) \, , \\ \ket{\phi_\beta^{\cal B}} &= \sum_{\{n_k^\downarrow\}} \bra{\alpha_L} \mathcal{O}_{\rm bkg}^{\rm L} \left[ \prod_{k=0}^{N_\phi} \dfrac{1}{n_k^\downarrow !} \left( \mathcal{W}_{-k}^\downarrow \right)^{n_k^\downarrow} \otimes \left( c_{k}^\dagger \right)^{n_k^\downarrow} \right] \left( \ket{\alpha_R} \otimes \ket{\Omega} \right) \, ,
\end{aligned} \end{equation} where we have set $\bra{\tilde{\alpha_L}}=\bra{\alpha_L} \mathcal{O}_{\rm bkg}^{\rm L}$. This step is described in great details for the rotationally symmetric case in Ref.~\cite{ZaletelMongMPS,MyFirstArticle}. From now on, we focus on subspace ${\cal A}$, the derivation being exactly the same for the subspace ${\cal B}$. The occupation numbers $\{n_k^{\cal A}\}$ are equivalently described by ordered lists of occupied orbitals $\lambda=(\lambda_1,\cdots ,\lambda_{N_e})$, with $N_e$ the number of electrons in the system: $N_\phi \geq \lambda_1 > \cdots > \lambda_{N_e} \geq 0$. Because of the commutation relation of the vertex operator modes, we may also write \begin{equation}
\ket{\phi_\beta^{\cal A}} = \sum_{\lambda_1 , \cdots , \lambda_{N_e}} \bra{\beta} \left[ \prod_{j=1}^{N_e} \mathcal{W}_{-\lambda_j}^\downarrow  \otimes d_{\lambda_j,{\cal A}}^\dagger \right] \left( \ket{\alpha_R} \otimes \ket{\Omega_{\cal A}} \right) \, ,
\end{equation} where the sum runs over unordered lists of integers $\{\lambda\}$. Plugging the orthonormal basis with Eq.~\eqref{eq:SplitCreationNonRot} and reordering the various terms, we find
\begin{equation}
\ket{\phi_\beta^{\cal A}} = \sum_{\mu_1 , \cdots , \mu_{N_e}} \bra{\beta} \left[ \prod_{j=0}^{N_\phi} \left( \sum_{\lambda=0}^{N_\phi} \alpha_{\lambda,\mu_j} \mathcal{W}_{-\lambda}^\downarrow  \right) \otimes \tilde{c}_{\mu_j,{\cal A}}^\dagger \right] \left( \ket{\alpha_R} \otimes \ket{0_{\cal A}} \right) \, .
\end{equation} A similar reasoning helps us to finally expressing the state $\ket{\phi_\beta^{\cal A}}$ in the occupation basis $\tilde{n}_k^{\cal A}$ relative to the new physical space spanned by the orthonormal basis $\{\tilde{c}_{k,{\cal A}}\}$. We find the MPS expression
\begin{equation}
\ket{\phi_\beta^{\cal A}}  = \sum_{\{\tilde{n}_k^{\cal A}\}} \braOket{\beta}{K_{\cal A}^{\tilde{n}_{N_\phi}^{\cal A}}[N_\phi] \cdots K_{\cal A}^{\tilde{n}_{0}^{\cal A}}[0]}{\alpha_R} \; \ket{\tilde{n}_{N_\phi}^{\cal A} \cdots \tilde{n}_{0}^{\cal A}} \; , \quad K_{\cal A}^{\tilde{n}}[j] = \dfrac{1}{\sqrt{\tilde{n}!}} \left( \sum_{\lambda=0}^{N_\phi} \alpha_{\lambda,j} \mathcal{W}_{-\lambda}^\downarrow \right)^{\tilde{n}} \, . \label{eq:MPSbreakrotationinv} \end{equation} Here, we have used the commutation relations of the vertex operator modes to swap the matrices in order to derive Eq.~\eqref{eq:MPSbreakrotationinv}, a site dependent representation of $\ket{\phi_\beta^{\cal A}}$ onto the orthonormal basis $\{\tilde{c}_{k,{\cal A}}\}$.

\paragraph{Spreading the Background Charge ---}

The last step of the derivation consists in spreading the background charge in order to find back the iMPS matrices Eq.~\eqref{eq:iMPSlaughlin} far away from the cut. The Laughlin background charge $\mathcal{O}_{\rm bkg}^{\rm L}$ can only be spread over $N_{\rm orb}$ orbitals, but the bipartition has introduced twice more matrices ($N_{\rm orb}$ in both parts ${\cal A}$ and ${\cal B}$). Although any allocation of the background charge over these matrices is acceptable, we append $U_{\rm L}$ to the first (resp. last) $N_{\rm orb}/2$ matrices of ${\cal A}$ (resp. ${\cal B}$). The product of matrices appearing in Eq.~\eqref{eq:MPSbreakrotationinv} can be split into two parts \begin{equation}
\left(  F_{{\cal A}}^{(\tilde{n}_{N_\phi}^{\cal A})}[N_\phi,N_{\rm orb}/2] \cdots F_{{\cal A}}^{(\tilde{n}_{N_{\rm orb}/2}^{\cal A})}[N_{\rm orb}/2,1] \right)
\left( F_{{\cal A}}^{(\tilde{n}_{N_{\rm orb}/2-1}^{\cal A})}[N_{\rm orb}/2-1,0] U_{\rm L}  \cdots   F_{{\cal A}}^{(\tilde{n}_{0}^{\cal A})}[0,0] U_{\rm L} \right)
\end{equation}\end{widetext} %
where we have defined \begin{equation}
F_{{\cal A}}^{\tilde{n}}[j,q] =  \dfrac{1}{\sqrt{\tilde{n}!}} \left( \sum_{\lambda=0}^{N_\phi} \alpha_{\lambda,j} \mathcal{W}_{-(\lambda-j)+q}^\downarrow \right)^{\tilde{n}} \, .
\end{equation} For a partition preserving the rotation symmetry around the cylinder axis, we have $\alpha_{k,r} = \delta_{k,r} g_{k,{\cal A}}$ (see Eq.~\eqref{eq:RotSymCoefErrorFct}) and we thus recover the tensor of Refs.~\cite{ZaletelMongMPS,RegnaultConstructionMPS,MyFirstArticle}. For the rectangular patch described in the main text, the off diagonal weights $\alpha_{k,r}$ decay rapidly for orbitals far from the cut (typically like Gaussian factors multiplied by cardinal sine functions) so that we can approximate $\alpha_{k,r}\simeq \delta_{k,r} g_{k,{\cal A}}$. In other words, the rotational symmetry is recovered after a large enough number of orbitals. Moreover, far away from the cut in the iMPS part of the product, $g_{k,{\cal A}}=1$ and we get back the site independent matrices $F_{{\cal A}}^{(\tilde{n})}[k,0] U_{\rm L} = B_{\rm L}^{(\tilde{n})}$. Similarly when $ g_{k,{\cal A}}=0$, \textit{i.e.} far away from the cut in the site-dependent part of the product, the matrices reduces to $ F_{{\cal A}}^{\tilde{n}}[j,q]  \simeq \delta_{\tilde{n},0} {1 \hspace{-1.0mm}  {\bf l}}$, with $1 \hspace{-1.0mm}  {\bf l}$ being the identity operator over the auxiliary space. This shows that the translation invariance along the cylinder axis is recovered far away from the transition. We can thus work on the infinite cylinder and take $N_{\rm orb}\to +\infty$, by switching to the site independent matrices far away from the cut. Numerically, we have considered up to 50 orbitals in the site-dependent region to ensure that when the iMPS is glued to take the limit $N_{\rm orb}\to +\infty$, we always satisfy the condition $|\alpha_{k,r} - \delta_{k,r}| < 10^{-10}$.

\section*{Data Availability}

Raw data and additional results supporting the findings of this study are included in Supplementary Information and are available from the corresponding author on request. The exact diagonalization data for finite size comparisons have been generated using the software \href{http://nick-ux.lpa.ens.fr/diagham/wiki/index.php/Main_Page}{"DiagHam"} (under the GPL license).

\section*{Acknowledgement}

We thank E. Fradkin, J. Dubail, A. Stern and M.O. Goerbig for enlightening discussions. We are also grateful to B.A. Bernevig and P. Lecheminant for useful comments and collaboration on previous works. VC, BE and NR were supported by the grant ANR TNSTRONG No. ANR-16-CE30-0025 and ANR TopO No. ANR-17-CE30-0013-01.

\section*{Author Contributions}

VC, NC and NR developed the numerical code. VC, NC and NR performed the numerical calculations. VC, NC, NR and BE contributed to the analysis of the numerical data and to the writing of the manuscript.

\section*{Competing Interests}

The authors declare no competing interests.

redpajama_set_name: RedPajamaArXiv


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text:
https://www.doc.govt.nz/parks-and-recreation/places-to-go/marlborough/places/kaikoura-area/things-to-do/tracks/kaikoura-peninsula-walkway/nature-and-conservation/
Kaikoura area
Kaikōura Peninsula Walkway
Nature and conservation
Kaikōura is the only place on earth where the endangered Hutton's shearwater breed.
Kaikōura Peninsula is made of limestone and siltstone laid down beneath the sea about 60 million years ago – but it has been exposed to the elements for a mere 180,000 years. Once an island, it is now linked to the mainland by debris eroded from the Kaikōura mountains. These mountains are rising faster than any other mountains in New Zealand (10 mm per year) but erosion keeps their height fairly constant.
Periods of rapid uplift have formed the steep-sided promontories, ideal for pā sites, and have twisted the neatly layered limestone into unusual shapes. The relatively young rocks have been worn into many interesting forms by the pounding sea. In less active periods, the sea has cut large tidal platforms in the softer sandstone.
Offshore is a very deep underwater canyon system called the Hikurangi Trench. It comes unusually close to shore at Kaikōura, where it is known as the Kaikōura Canyon. The canyon floor collects sediments that will form tomorrow's rocks and may appear in millions of years' time as new mountains.
Jutting out around 4 km from the shore and sculpted by numerous rocky headlands and small, semi-sheltered embayments, the peninsula presents a wide variety of aspects depending on the weather.
Rocky outcrops, wavecut mudstone platforms incised with guts and channels and tortured-looking limestone reefs are interspersed with boulder reefs and small, crescent-shaped, stony beaches. The reefs, so prominent at low tide, continue for hundreds of metres offshore, eventually breaking up into flat expanses of pebbles, which eventually terminate in sand and mud further offshore
The peninsula is a biological nodal point, a place where 'north meets south' or, more accurately, where 'warm meets cold'. Here, the distributions of typically northern and southern species overlap – the seaweeds show a strong southern affinity, while the animals show a warmer, more northern influence. Add to this the rich variety of intertidal and sub-tidal habitats, and you have one of the most biologically diverse locations of the entire east coast of the South Island.
Firmly anchored to the rocks, bull kelp forms a distinctive fringe along the low-tide mark, its dense mass of leathery fronds swirling snake-like in the waves. Kelp and numerous other seaweeds thrive in the pulsating light zone of Kaikōura's nutrient-rich coastal waters and, along with phytoplankton (microscopic plants), form the beginning of the marine food chain.
Koura (crayfish) shelter in rocky crevices and under boulders around much of the peninsula; these are some of Kaikōura's best known marine creatures and lend their name to the town. Ranging from barnacles, limpets and crabs to paua, sponges and fish, the multitude of marine animals found at Kaikōura Peninsula is simply remarkable.
Hutton's shearwater colony
This colony, adjacent to the walkway, provides a sanctuary for these unique birds and a special opportunity for visitors to experience the delight of an active sea bird colony, especially at night when the birds return from feeding at sea.
Thee 500 m long, state-of-the-art predator-proof fence keeps out introduced mammals such as rats, cats, possums and stoats as these burrow-nesting birds are particularly vulnerable to predators.
More about Hutton's shearwater/Kaikōura tītī.

redpajama_set_name: RedPajamaCommonCrawl


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redpajama_set_name: RedPajamaGithub


text:
Standard Chartered Bank Expands Branch Network … Opens New Branch in Lagos
Standard Chartered Bank Nigeria has opened another branch located at Plot 1681 Sanusi Fafunwa Street, Victoria Island, Lagos. The new branch, the fifth branch opened by the bank in 2008, is a reflection of the Bank's strong business growth in Nigeria and a result of its strategic expansion drive in the country.
The branch was declared open by Mr Steve Bertamini, Group Executive Director/Chief Executive Officer, Consumer Banking of Standard Chartered Bank Plc. According to Mr Bertamini
"It gives me great pleasure to open this branch in Nigeria because Nigeria is pivotal to the growth and development of Standard Chartered Bank's franchise in Africa. The strength of the Nigerian market has been reflected in our Bank's business growth in Africa which has been exponential and Nigeria is currently the fastest growing and most profitable of our franchises in Africa.
We are proud of the achievements of Standard Chartered Bank in Nigeria over the past few years and I am particularly impressed with the huge potential of the retail banking market in Nigeria.
With an international footprint stretching over 50 countries, Standard Chartered Bank is uniquely positioned to capitalise on cross-border business for fast growing markets like Nigeria and we remain committed to making this happen."
Also speaking at the opening ceremony of the Sanusi Fafunwa branch, Christopher Knight, Managing Director/CEO, Standard Chartered Bank, Nigeria said:
"This is an auspicious period to capitalize on the extraordinary growth dynamics in the country and the establishment of this new branch in Lagos is a key element in executing Standard Chartered Bank's strategy to grow the business and outperform our competitors.
We fully intend to consistently deliver on our brand promise of being the right partner to our customers by providing more branches and channels to access the world class products and services that Standard Chartered Bank is known for globally. Already we have six more branches in various parts of the country earmarked for commissioning in the next 2 - 3 months."


Many students join school programs to obtain academic- or service credits.
But one Morristown High School group prides itself on not giving its members credit.
and its selfless approach was honored last month by the New Jersey State Board of Education.
ALL IN was among 10 student groups from across the state recognized "for academic success as well as leadership qualities and volunteerism both at school and within their communities," the state board said on its website.
Members of ALL IN — who include students with special needs–meet twice a week during lunch and once a month after school to practice peer inclusion, embrace diversity, and dissolve barriers between students of all abilities.
They do it for the sake of doing it, not for credit offered by service-based clubs such as National Honors Society or Key Club, or for resume-building, according to students and school officials.
"It's not appropriate to be earning recognition for hanging out with somebody that may just be different than you," said Jessica Neu, creator of ALL IN and director of pupil services for the Morris School District.
When MHS Key Club students pushed for service hours after the annual ALL IN Holiday Dance, ALL IN Student Board member Erin Joel said no.
"You're not going to get service hours for helping kids feel incorporated with the whole student body," she told them.
ALL IN Student Board member Amy Albin believes that giving out service hours only would sow divisions within the group.
"We're really all one," she said.
Club members are drawn back for reasons less tangible but arguably more valuable than service hours: To "incorporate and involve special education students into school events," said Joel.
"We also want to treat everyone equally, because everyone should be treated equally," added Student Board member Erik Vataker.
"You're giving out a helping hand to show them that they're not different, we're all the same and we can all do the same activities and have fun together," said Junior Rondon, a senior who is on the ALL IN board.
Reaching out to those with special needs has always been a passion for Joel, who has been teaching catechism classes to students with disabilities since 7th grade. Now, her focus is on spending time with students from her school.
"I love helping them get involved and putting a smile on their faces," she said.
ALL IN Student Board receives recognition from the New Jersey State Board of Education for integrating students with different abilities into the general Morristown High School population.
Joel, a junior, was among seven MHS students recognized by the state Board of Education. The others were Albin, Rondon, Andrew Chin and Leo ruiz, all seniors; and sophomores Erik Vataker and Rhys Harris.
ALL IN started at Morristown High in the spring of 2014, and was extended last fall to Frelinghuysen Middle School.
The program has evolved from students sharing lunchtimes and just hanging out, to attending school sporting events and staging an annual holiday dance.
ALL IN has also helped jumpstart creation of teams for track, cross country, swimming and wrestling, run by Unified Sports Special Olympics: Project Unify.
The Special Olympics is the world's largest sports organization for people with intellectual disabilities; its Project Unify initiative empowers schools and educators to provide children of varying abilities a chance to play together as teammates.
Project Unify awarded the Morris School District $1,500 for the 2015-2016 school year, and $15,000 for the next three years, to expand the ALL IN program and inclusive physical education programs.
The group emphasizes the importance of building relationships with people who are different from yourself.
Student Board member Erik Vataker has enjoyed how ALL IN made the transition into high school so easy for him.
"This is one of the places where you can feel like yourself," he said.
Eric's mom, Beth Vataker, said "Teenage years are the time where being part of a group is so important," and that her son loves being a part of ALL IN.
MHS students participate in one of ALL IN's game days.
"We didn't create their leadership skills, we didn't help them create their leadership skills. They naturally had them," said ALL IN Adviser Heidi Heinsohn in reference to the members of the Student Board.
Jessica Neu, the ALL IN creator, said the participants at Morristown High are influencing others there.
Girls Who Code, Melanin Minds, and Student Board member Junior Rondon's Mountain Biking Club are MHS groups that reflect ALL IN's inclusive philosophy.
Most importantly, ALL IN has made lasting impacts on its members.
Joyce Reilly Ruiz, mother of Leo Ruiz, who has cerebral palsy, said that ALL IN has allowed Leo to "have leadership roles in advocacy and inclusion of other students with disabilities," an area of great interest to him.
"Last spring, with the guidance of ALL IN Morris School District staff members, Leo was able to plan and execute a field day with activities designed to be fun for all students," Joyce Ruiz said.
Physical therapist Barbara Miller and occupational therapist Jeff Crane helped Leo to "construct interactive games where students of differing abilities could enjoy time together and appreciate each others strengths," she said.
Another parent of an ALL IN member said her son "talks with pride about his friends from ALL IN… He says 'hi' to them in the hallways, and waves enthusiastically when he sees them outside of school.
"As a child primarily in an enclosed classroom, this network of friends gives him a deeper sense of belonging at MHS and in the broader Morristown community," the mother added.
As the school year comes to a close, senior ALL IN Student Board members are preparing to take their message of inclusion to college.
This fall, Amy Albin will be attending the County College of Morris, Andrew Chin is heading for the University of the Sciences in Philadelphia, Junior Rondon will study at Fairleigh Dickinson University and Leo Ruiz is bound for Marist College.


text:
Autobag invests in clean manufacturing
Will produce 250 million bags and pouches every year
Global bagging specialist Automated Packaging Systems is dedicating 10,000ft2 of its UK factory to a new clean manufacturing facility capable of producing 250 million bags and pouches each year, underlining its commitment to the medical and food sectors.
The new facility will operate under clean manufacturing procedures specially written for the firm.
Colin Rashbrook, QA manager at Automated Packaging Systems, said: "We have long held the BRC/IOP Global Standard for Packaging and Packaging Materials Issue 4, and are proud to have achieved the highest A-rating. But to serve best the medical device and food sectors we're looking to set even higher standards."
Housed within the company's European bag and pouch facility at Malvern in Worcestershire, the clean manufacturing area has been fitted and laid out to ensure that the firm continues to meet and exceed the stringent quality and safety demands for medical devices and food.
Rashbrook added: "By segregating the manufacture of bags and pouches that hold food and products for sterilisation from those for industrial applications, we ensure standards are in line with end-product use. As a result we are a more effective packaging partner, able to demonstrate a complete understanding of the demands and pressures faced by individual customers."
Within 12 months the clean manufacturing area will produce more than a third of all bags and pouches produced at Malvern, and output is projected to rise on the back of significant growth from medical applications.
The company says its new Autobag HD-E medical bagger is a flexible, reliable, automated alternative to bagging by hand for medical devices requiring terminal sterilisation.
Automated Packaging Systems
PackMedical makes its debut
Butterfly pack ensures accurate product dosing


text:
Iyman Faris (a.k.a. Mohammad Rauf; born June 4, 1969) is a Pakistani (formerly American) citizen who served for months as a double agent for the FBI before pleading guilty in May 2003 of providing material support to Al Qaeda. A United States citizen since 1999, he had worked as a truck driver and lived in Columbus, Ohio. As of September 2003, Faris was the "only confessed al Qaeda sleeper caught on U.S. soil." In 2003 he was sentenced to 20 years in prison for providing material support to Al-Qaeda. In February 2020 an American federal court revoked Faris' US citizenship. In August 2020, he was released from a federal prison in Illinois.

Faris was detained in Ohio two weeks after Khalid Sheikh Mohammed was arrested in Pakistan on March 1, 2003. While installed as a double agent for the U.S. government, Faris sent messages to his terrorist commanders by mobile phone and email from an FBI safe house in Virginia. A senior Bush administration official said, "He was sitting in the safe house making calls for us. It was a huge triumph."

Early history

Iyman Faris was born in 1969 in Azad Kashmir, Pakistan. Faris came to the United States as a young man in 1994 on a student visa but never enrolled in school. In 1995, he married Geneva Bowling. Faris became a U.S. citizen in 1999.

He returned to Pakistan the following year, in 2000. His father had just died and his five-year marriage was ending. His ex-wife later said he had suffered from hearing imaginary voices and sudden bouts of believing that somebody was choking him.

Faris admitted to having met Osama bin Laden in Afghanistan. While in the region, Faris was asked to investigate the possibility of using an ultralight aircraft as an "escape plane" for bin Laden. He used an internet cafe in Karachi, Pakistan to research planes, buy cellphones and 2000 sleeping bags for training camps.

While living in Ohio, Faris had been counseled by a local imam for thoughts of suicide. After threatening to jump off a bridge, he was hospitalized for a psychiatric evaluation. Faris' five-year marriage ended in 2000.

Brooklyn Bridge plot
In late 2001, while in Pakistan, Faris went to a travel agency to have some expired airline tickets to Yemen re-issued for several unknown colleagues. He claimed to be a preacher from Tablighi Jamaat, a missionary group.

In early 2002, Faris was introduced to an operative identified only as "C-2". He learned of a plot allegedly involving the simultaneous destruction of the Brooklyn Bridge by cutting through cables with blowtorches, and a second group that would derail a train in Washington D.C. Faris' investigations into obtaining the necessary tools for the dual-operation involved asking a friend where he might purchase welding equipment, and researching the structure of the bridge on the internet. He concluded that the operation was unlikely, and allegedly sent a message back to Pakistan calling off the plot, stating that "The weather is too hot".

Working for FBI
On March 19, 2003, Faris was visited by two FBI agents and an anti-terror officer, who confronted him with testimony from Khalid Sheikh Mohammed about his contacts with al-Qaeda, and the results of an intercepted telephone call. Faris was reportedly accommodating, and agreed to let the agents search his apartment the following day.

Confronted by the FBI in April, Faris confessed. He agreed to cooperate with the FBI and work as a double agent, reporting to them. He was ordered to leave his home in Columbus and was assigned to a safe house in Virginia, from which he would appear to continue discussions with his contacts, reporting the information back to the FBI.

During this period, Faris told the agents that Majid Khan, a Baltimore youth who worked at his father's gas station, had referred to Khalid Sheikh Mohammed as an "uncle" and spoken of a desire to kill Pakistani dictator Pervez Musharraf. He later totally recanted this in a written statement provided in 2007 to Khan's Combatant Status Review Tribunal, saying his accusation was "an absolute lie," and he had been coerced into making the claims.

The United States took Khan into custody in Pakistan and sent him to a CIA black site in Afghanistan, where he was interrogated, allegedly tortured and held for several years. In September 2006, he was transferred to military custody at Guantanamo Bay as a high-value detainee.

Criminal charges
The Department of Justice charged Faris in federal court with terrorism charges. On May 1, 2003, in a plea bargain, Faris pleaded guilty to both charges: providing material support and resources to Al Qaeda and conspiracy for providing the terrorist organization with information about possible U.S. targets for attack. The documents relating to the charges were not unsealed until June 19, 2003. Shortly after, the Department of Justice announced that it had used Faris as a double agent, who had worked under orders from the FBI. Faris is the "only confessed al Qaeda sleeper caught on U.S. soil."

On September 25, 2003, Faris sought to withdraw his guilty plea, claiming that while he admitted to meeting with Khalid Sheikh Mohammed, he had been seeking information about al-Qaeda for a book he wanted to write. He argued that Mohammed had given false information to authorities as revenge, for Faris had refused to be recruited into al-Qaeda by his lieutenant. The appeal was rejected. On October 28, 2003, Faris was sentenced to 20 years' imprisonment by District Court Judge Leonie Brinkema.

According to the New York Times, the Brooklyn Bridge plot was uncovered with assistance from the NSA eavesdropping program. The Times delayed publishing the story for a year ostensibly at the request of the government. The story was one of several that appeared in December 2005 by Eric Lichtblau and James Risen that won the Pulitzer Prize for National Reporting awarded by Columbia University.

Faris' new lawyer, David B. Smith, announced that he was looking into a lawsuit against George W. Bush, who personally authorized the wiretaps.

In 2004, the Department of Justice charged Nuradin Abdi in a bomb plot. It reported that Abdi had said that Faris had picked him up from the Columbus airport.

References

External links
  United States v. Faris (PDF), FindLaw
 "Iyman Faris Sentenced for Providing Material Support to Al Qaeda", press release, Department of Justice, October 28, 2003.
 James Risen and Eric Lichtblau, "Bush Lets U.S. Spy on Callers Without Courts", New York Times, 16 December 2005.

Pakistani emigrants to the United States
Pakistani Islamists
American people of Azad Kashmiri descent
Terrorist incidents in the United States in 2003
Living people
1969 births
American truck drivers
People from Columbus, Ohio
Inmates of ADX Florence

redpajama_set_name: RedPajamaWikipedia


I was that exhausted from expending so much nervous energy on the Saturday under the careful gaze of Franco Ferrero that on the Sunday I was almost too laid back for Mike McClure's liking. Now bear in mind that neither assessor speaks to the other, or the candidate, about the performance they observed until the end of the second day, I really had no idea what the outcome might be. So when Lara Tipper, course director, shook my hand and said "Congratulations Mark, you've passed" I had to ask her to say again, and slowly, what she'd just announced!
It was a tremendous relief to drive away from Moelfre with Lara's words ringing in my ears and the comprehensive written feedback I'd received from Mike and Franco sitting on the passenger seat. I felt a huge sense of satisfaction knowing that the hard work and preparation had paid off. It was just such a damn shame that my mobile phone had no power, so some poor pedestrian had to endure me dancing on the pavement as the realisation of my achievement sunk in.
On reflection I feel that all the circumstances came together at the right time for me in order that this might have a happy ending. Paul and Richard made an immense effort to commit to a long term development programme. In particular, Richard overcame a back injury at the 11th hour to ensure the team was complete for the Sunday. They both showed an unbending faith in my abilities which I think cemented the coach/student relationship so well. Nick Cunliffe did his uppermost to offer guidance, humour, wisdom and limitless support whenever I asked for it. He was there to pick me up when I thought my paddling mojo had been lost and kept me in line when my ideas got too fanciful. Without these folk it may have been a different story.
It goes without saying that there are many others who have contributed to my success. Gordon Brown provided several doses of no-nonsense advice. Jen Kleck made numerous helpful observations about her assessment experiences that I know played a pivotal role in my own planning. I'm grateful to Aled Williams of Tiderace Kayaks and Mike Webb of Rockpool for the use their boats. Also Aled for his unique insights into boat handling which will always make him a 6* paddler in my eyes. Sid Sinfield certainly played a part in his own inimitable way as have many others who I hope don't take umbrage for not being named here. Fair to say, a big thank you goes out to one and all, fellow coaches and students alike for being there along the way.
Ive been reading your blog for a while and the "Long term student development" series are one of the best things I have yet read on a blog.
A great achievement. Well done from both of us.
That's AWSOME! Congrats on getting certified as a 1* coach! Oh wait, sorry 5*.
Congrats, that is some award!!
Surely this must be the end of the beginning!
Hi Mark, congratulations - job well done!
CONGRATULATIONS on becoming a ****** coach .... that's a long, hard road to travel nowadays. You deserve it.
I would like to congratulate Mark on his achievement of gaining the L5 (Sea) Coach award.
As a 'long term student' and friend of Mark I have had a glimpse of the work and commitment needed to achieve L5. Mark has been the catalyst to move my personal paddling skills and confidence forward and whatever one's views of the BCU or coaching scheme its good to know there are such skilled people at the top of the coaching ladder.
Well done, Mark! And I look forward to our next paddle …. And a beer!