Algebraic and field-theoretic structure of the corner transfer matrix of the exact continuous matrix product state for the Lieb-Liniger model

Maruyama and Katsura (J. Phys. Soc. Jpn. 79, 073002 (2010)) constructed an exact continuous matrix product state (cMPS) for the $n$-particle Bethe eigenstates of the Lieb-Liniger model, the one-dimensional Bose gas with delta-function interaction, by taking the continuum limit of an algebraic-Bethe-ansatz matrix product state on an integrable lattice discretization; factorizing $F$-matrices are indispensable to the limit and reduce the world-line representation to a five-vertex structure. The paper observes that a corner transfer matrix (CTM) of this cMPS arises as a mapping from the momentum-space auxiliary description into real space and, since Baxter's lattice CTM is known to act as a Lorentz boost, explicitly poses the problem of studying the field theory and algebraic structure behind the CTM of the cMPS. The stated problem is to construct the CTM of the exact cMPS explicitly, derive its algebraic relations with the auxiliary operators $Q_n$, $K_n$, $R_n$ and the $F$-matrices, and decide whether the Lorentz-boost interpretation of the lattice CTM survives in this nonrelativistic continuum setting.

Background

The Lieb-Liniger model describes a one-dimensional gas of bosons with point interaction $V(x,y)=\kappa\,\delta(x-y)$ and is the paradigmatic exactly solvable continuum many-body system: its eigenfunctions are Bethe-ansatz superpositions of plane waves whose exchange amplitudes are the two-body scattering matrix elements. The matrix product state (MPS) is the variational ansatz underlying the density-matrix renormalization group for one-dimensional lattice systems; Verstraete and Cirac extended it to continuum one-dimensional systems as the continuous matrix product state (cMPS), in which a state is written as $\mathrm{Tr}\,\mathcal{P}\exp\int_0^L[M_0(x)+M_1(x)\psi^\dagger(x)]\,dx\,|0\rangle$, with $\mathcal{P}$ a path-ordering symbol, $\psi^\dagger$ the boson creation field, and $M_0$, $M_1$ finite-dimensional matrices acting on an auxiliary space.

Maruyama and Katsura asked whether an exact Bethe eigenstate of the Lieb-Liniger model admits such a form. They started from an integrable lattice discretization, one whose local $L$-operator satisfies the Yang-Baxter relation — the compatibility condition among scattering amplitudes that makes the model exactly solvable and yields commuting transfer matrices — built the Bethe state there by the algebraic Bethe ansatz (the monodromy matrix $T(\lambda)$ is a 2-by-2 matrix of operators $A,B,C,D$, and products of its $B$-operators create the Bethe state from the vacuum), reorganized it as a lattice MPS, and took the continuum limit of lattice spacing $a\to 0$ at fixed particle number $n$. Two ingredients proved indispensable. First, the factorizing $F$-matrices, a change of basis in the auxiliary space (known as a Drinfel'd twist) that renders the relevant local matrices upper triangular, are needed to diagonalize the matrix $D_n$ that must be raised to a diverging power in the limit; the same simplification reduces the graphical (vertex) representation of the Bethe state from infinitely many vertex types to a five-vertex model, in analogy with the six-to-five-vertex reduction for the Heisenberg chain. Second, a boundary operator $Q_n=|\Rightarrow\rangle\langle\Leftarrow|$ acting on the $2^n$-dimensional auxiliary space $\bar{H}$ projects the apparently number-fluctuating exponential onto a fixed particle number. The outcome is an exact cMPS with $M_0=iK_n$, where $K_n=-\sum_{j=1}^{n}(\lambda_j/2)\sigma_j^z$ is built from the Bethe momenta $\lambda_j$ and generates free propagation, and $M_1=R_n$, an operator assembled from the $F$-matrices that plays the role of a reflection operator. In the world-line representation, whose two directions are real space and the discrete momentum labels $k_j=\lambda_j$, the $F$-matrices eliminate one of the three local connections, and for $n=2$ the cMPS collapses to the familiar coordinate-Bethe wavefunction $\Psi(x_1,x_2)\propto e^{ik_1x_1}e^{ik_2x_2}+S(k_1,k_2)\,e^{ik_1x_2}e^{ik_2x_1}$.

For two-dimensional vertex models, Baxter's corner transfer matrix (CTM) is the transfer matrix of one quadrant of the lattice; its product structure underlies exact solvability and the corner-transfer-matrix renormalization group. In the six- and eight-vertex setting the CTM carries a remarkable algebraic meaning: it acts as a Lorentz boost on the scattering data (Thacker), is tied to the boost operator of the algebraic Bethe ansatz (Sogo and Wadati) and to a lattice Virasoro algebra (Itoyama and Thacker), and its eigenstates are themselves Bethe states of an associated spin chain (Frahm and Thacker). Maruyama and Katsura note that, for the statistical-mechanical world-line model underlying their cMPS, a CTM 'is obtained as a mapping from momentum space into real space', that is, by the ninety-degree rotation of the world-line picture exchanging the real-space transfer direction for the momentum-space auxiliary direction. They then pose, as an explicitly interesting problem, the study of the field theory and algebraic structure behind the CTM of the cMPS. Nothing is currently known about this continuum CTM: it has never been constructed, its algebraic relations with $Q_n$, $K_n$, $R_n$ and the $F$-matrices have not been derived, and it is not known whether the Lorentz-boost interpretation of the lattice CTM has any counterpart in this nonrelativistic continuum setting.

Problem Statement

Construct the corner transfer matrix of the exact continuous matrix product state of the $n$-particle Lieb-Liniger Bethe state, the transfer matrix of the rotated world-line quadrant that realizes the mapping of the momentum-space auxiliary description into the real-space transfer direction, and determine the algebraic and field-theoretic structure behind it. A complete answer should: (i) give the explicit form of this CTM in terms of the cMPS data, namely the Bethe momenta $\lambda_j$, the momentum operator $K_n=-\sum_{j}(\lambda_j/2)\sigma_j^z$, the reflection-type operator $R_n$, the boundary operator $Q_n$, and the factorizing $F$-matrices or their continuum limits; (ii) derive the algebraic relations and spectral structure of the CTM, in particular the relations it satisfies with $Q_n$, $K_n$, $R_n$ and the two-body scattering phases; and (iii) settle whether the Lorentz-boost interpretation of Baxter's lattice CTM carries over to this continuum integrable system, by determining the transformation of the cMPS and scattering data that the continuum CTM effects. Any construction must reproduce the known exact cMPS at fixed $n$, $|\Psi\rangle=\mathrm{Tr}_{\bar{H}}\,Q_n\,\mathcal{P}\exp\int_0^L[iK_n+R_n\,\psi^\dagger(x)]\,dx\,|0\rangle$, and must reduce for $n=2$ to the coordinate-Bethe wavefunction $\Psi(x_1,x_2)\propto e^{ik_1x_1}e^{ik_2x_2}+S(k_1,k_2)\,e^{ik_1x_2}e^{ik_2x_1}$.

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