Fully solvable finite simplex lattices with open boundaries in arbitrary dimensions

Finite simplex lattice models are used in different branches of science, e.g., in condensed-matter physics, when studying frustrated magnetic systems and non-Hermitian localization phenomena; or in chemistry, when describing experiments with mixtures. An $n$-simplex represents the simplest possible polytope in $n$ dimensions, e.g., a line segment, a triangle, and a tetrahedron in one, two, and three dimensions, respectively. In this work, we show that various fully solvable, in general non-Hermitian, $n$-simplex lattice models with open boundaries can be constructed from the high-order field-moments space of quadratic bosonic systems. Namely, we demonstrate that such $n$-simplex lattices can be formed by a dimensional reduction of highly degenerate iterated polytope chains in $(k>n)$-dimensions, which naturally emerge in the field-moments space. Our findings indicate that the field-moments space of bosonic systems provides a versatile platform for simulating real-space $n$-simplex lattices exhibiting non-Hermitian phenomena, and it yields valuable insights into the structure of many-body systems exhibiting similar complexity. Among a variety of practical applications, these simplex structures can offer a physical setting for implementing the discrete fractional Fourier transform, an indispensable tool for both quantum and classical signal processing.

Figure 1

Iterative polytope chains in the field-moments (FMs) space of bosonic (a) dimers and (b) trimers, formed by the Cartesian products of 1-polytope (a line) and 2-polytope (a triangle), respectively. (a) 1-polytope (a line) for the first-order FMs of the dimer; 2-polytope (a square) for its second-order FMs; 3-polytope (a cube) for its third-order FMs; and 4-polytope (a hypercube, shown via its 3D Schlegel diagram) for its fourth-order FMs. (b) 2-polytope (a triangle) for the first-order FMs of the trimer, and 4-polytope (a 4D duoprism, shown via its 3D Schlegel diagram) for its second-order FMs. Numbered vertices correspond to the elements of the FMs vector $\langle {⨂}_1^k\widehat{\mathrm{\Psi }}\rangle$, with vertex potentials and the link weights corresponding to the diagonal and off-diagonal elements of the related evolution matrix $M_k$, respectively.

Figure 2

Tensor-products tree of the eigenvectors of the evolution matrices $M_m$ (with $m=0,1,2,3$) for a quadratic dimer, according to Eq. (6). The index $m=0$ corresponds to the identity element. This tree is a solution for the iterative polytope chains in Fig. 1.

Figure 3

Formation of iterative simplex chains (ISCs) by folding iterative polytope chains in the FMs space of bosonic three-mode systems. (a) A 2-simplex, i.e., a triangle, formed in the first-order FMs, which coincides with the $P_{i_1}$ polytope. The links, denoted by $\alpha ,\beta ,\gamma$ of different colors, correspond to the coupling strengths between the three FMs, according to Eq. (10). (b) ISC in the second-order FMs space. By folding the corresponding symmetric 4-polytope in Fig. 1, one arrives at an asymmetric triangular 2-simplex consisting of six vertices, according to Eq. (8). (c) ISC in the $N\mathrm{th}$-order FMs space. The ISCs thus form a finite triangular non-Hermitian lattice with $(N+1)(N+2)/2$ vertices in the high-order FMs space. Formally, the presented 2-simplex chain can be considered in a reversed order, that is, a lattice in panel (c) can be reduced into a lattice in panel (a), which highlights the simple origin of the seemingly complex lattice in (c). Note that such formed asymmetric simplex lattices in the FMS of quadratic bosonic systems can be additionally symmetrized (see Appendix pp5 for details).

Figure 4

Real (a) and imaginary (b) parts of the eigenvalues $\lambda$ of the matrix $M_3$, governing third-order FMs, at the EP $g=\mathrm{\Delta }$, under the perturbation $\epsilon$, which enables resolving a DDEP. The matrix $M_3$ has a single EP of fourth order (presented by the coalescence of four solid curves, characterized by the quartic-root dependence on perturbation, $\sqrt[4]{\epsilon }$). Apart from it, under the perturbation, the initial DDEP splits into two EPs of second order (presented by the merge of two eigenvalues, shown by the two dashed and two dotted curves, respectively, which are characterized by the square-root dependence of the perturbation, $\sqrt{\epsilon }$).

Figure 5

An element of the triangular non-Hermitian lattice model in Fig. 3, described by the non-Hermitian Hamiltonian in Eq. (D2), illustrating the nearest-neighbor interactions for a site with indices $(j,k,l)$ (shown in the center). The site potentials and couplings $f_q^{\xi }$ are given in Eqs. (D3) and (D4), respectively.

Figure 6

Formation of iterated simplex chains by folding iterated polytope chains in the field-moments space of bosonic four-mode systems. (a) A 3-simplex, i.e., a tetrahedron, formed in the first-order FMs, which coincides with the $P_{i_1}$ polytope. The links, denoted by Greek letters $\alpha ,\beta ,\gamma ,\mu ,\xi ,\nu$, and highlighted by different colors, correspond to the coupling strengths between the three FMs. (b) An ISC in the second-order FMs space. By folding the corresponding symmetric 6-polytope, one obtains an asymmetric 3-simplex, namely a tetrahedral-octahedral honeycomb in the high-order FMs space of the four-mode bosonic system.

Figure 7

Symmetrization of the asymmetric 2-simplex lattice (left panel), also shown in Fig. 3, to the symmetric 2-simplex lattice (right panel). In other words, the asymmetric intersite couplings (depicted as double arrows on the left panel) become symmetric (shown by single arrows on the right panel). This mapping can be realized via an appropriate similarity transformation, or, alternatively, by choosing an appropriate basis when projecting polytopes onto simplexes, formed in the field-moments space of quadratic systems. The edges of the symmetric 2-simplex lattice can describe a quantum angular momentum operator with different quantum numbers. See more details in Appendix pp5.