Fractional Quantum Hall States: Infinite Matrix Product Representation and its Implications

TL;DR

The paper develops a rigorous infinite matrix product state (iMPS) framework for Laughlin and related fractional quantum Hall wavefunctions via chiral CFT correlators, enabling exact control of expansion coefficients in Slater-determinant bases and revealing a renewal structure. It introduces root partitions, monomial-symmetric boundary charges, and a geometry-dependent weight alongside a geometry-independent factorization, yielding explicit bounds on correlators and an entanglement-gap result on thin cylinders, plus exponential clustering of local observables. The core technical achievements include a detailed operator algebra for W_m, a combinatorial theory of partitions and dominance, and a factorization mechanism that ties cylinder geometry to planar roots through irreducible segments. These results provide a rigorous foundation for understanding the structure and correlations of fractional quantum Hall states in cylinder and planar geometries, with implications for gappedness and potential adiabatic connections to truncated pseudopotentials. Overall, the work offers quantitative tools for analyzing incompressible quantum liquids via iMPS, with clear links between CFT correlators, partition combinatorics, and geometric entanglement properties.

Abstract

We present a novel matrix product representation of the Laughlin and related fractional quantum Hall wavefunctions based on a rigorous version of the correlators of a chiral quantum field theory. This representation enables the quantitative control of the coefficients of the Laughlin wavefunction times an arbitrary monomial symmetric polynomial when expanded in a Slater determinant or permanent basis. It renders the properties, such as factorization and the renewal structure, inherent in such fractional quantum Hall wavefunctions transparent. We prove bounds on the correlators of the chiral quantum field theory and utilize this representation to demonstrate the exponential decay of connected correlations and a gap in the entanglement spectrum on a thin cylinder.

Figures (7)

• Figure 1: An example of the occupation configuration $\mathbf{m}$ of a root partition $\lambda_N^{(q)}(b)$ with $q = 3$ and $N = 6$ and $b = (1, 1, 2 , 3, 3 , 5)$. The bottom line illustrates the orbitals and their occupation with red balls symbolizing particles. The top line is the equivalent tiling picture with monomer and void tiles on which the occupations are imprinted. The dashed lines mark separations of elementary blocks, which carry exactly one monomer and potentially frontal voids.
• Figure 2: An example of twice two elementary squeezing moves on a root with $q = 2$ in $a.)$ the occupation picture with red balls symbolizing particles and $b.)$ in the tiling picture with occupation numbers printed on the tiles. The orange vertical lines mark segmentations, which are consistent with the squeezings. In each picture, the top line represents the root, and the bottom line represents the result after the two squeezing moves.
• Figure 3: The root tiling from Figure \ref{['fig:tiling']} with a particular segmentation of length three with $N_1=1, N_2=3, N_3=2$. The orange lines indicate the renewal points, and the orange blocks in the last line correspond to the term with $r = 3$ and $N_1=1, N_2=3, N_3=2$ in the representation \ref{['eq:fact']} with irreducible factors indicated on the blocks.
• Figure 5: The concatenation of the root partitions $\lambda^{(3)}_2( b^{(1)}) = ( 2,6)$ corresponding to $b^{(1)} = (2,3)$ and $\lambda^{(3)}_3( b^{(2)}) (0,6,9)$ and $b^{(2)} = (0,3,3)$ in $a.)$ the tiling picture, $b.)$ the occupation picture. The concatenated root is $\lambda^{(3)}_5( b) = (2,6,9,15,18)$ with $b= ( b^{(1)}, b^{(2)}) = (2,3,3,6,6)$.
• Figure 6: The three cases in Lemma \ref{['lm:partitions PsiAB']}: $a.)$ when $L_1<L_2$. The orange blocks correspond to segmentations of two partitions $\lambda, \mu\in \mathcal{M}_{b,N}^{(AB)}$ with $\langle \Phi_\lambda, AB \Phi_\mu \rangle\neq 0$. The renewal points of blocks that do not intersect the supports of $A$ or $B$ coincide. The dashed line indicates the transition when the segments intersect the supports. Illustrated in $b.)$ and $c.)$ are cases with an elementary block which is longer than or equal to $d_{AB}/3$. The endpoints of those blocks, indicated again by dashed lines, correspond to renewal points common to all partitions in $\mathcal{M}_{b,N}^{(AB)}$.

Theorems & Definitions (79)

• Definition 1.1
• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Corollary 1.4
• Definition 1.2
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8