Theoretical Physics Seminar

We write down a class of two-dimensional quantum spin-1/2 Hamiltonians with bond-dependent couplings whose eigenspectra are exactly solvable via the Jordan-Wigner transformation. The general structure corresponds to a suitable grid composed of XY or XX-Ising spin chains and ZZ-Ising spin chains and is generalizable to higher dimensions. They can host stacks of one-dimensional spinless fermion liquids with gapless excitations and power-law correlations coexisting with ordered spin moments (localized spinless fermions). Bond-dependent couplings thus allow for an embedding of one-dimensional spinless fermion (Tomonoga-Luttinger) liquids and solids and also Majorana excitations in higher dimensions. One of the grid structures hosts an unusual magnetic ground state where Ising order/disorder coexists with a classical paramagnet at zero temperature. It has extensive entropy reminiscent of classical spin ices and the SYK model and can be considered a simpler variant of the Kitaev spin liquid including a two-step entropy release through a different mechanism. The proposed grid structure may provide an architecture for quantum engineering with controllable qubits. These results are based on Ref. [1].

We then discuss aspects of a two-dimensional lattice $S=\frac{1}{2}$ quantum Hamiltonian with bond-dependent couplings and related variants with quantum spin liquidity introduced recently by the author that grew out of the previous work [1]. These models have a mutually <<anticommuting>> algebra of extensively many local $Z_2$ conserved charges. This mutual algebra is like the algebra of quantum spin-$\frac{1}{2}$ local degrees of freedom however arising in the structure of the \emph{local conserved charges}. As provable consequences, these models have finite residual entropy density in the ground state with a simple but non-trivial degeneracy counting and  concomitant quantum spin liquidity, as elaborated in Ref.~[2]. The spin liquidity relies also on a geometrically site-interlinked character that is natural for <<anticommuting>> local conserved charges composed of spin-$\frac{1}{2}$ operators or Paulis, in contrast to e.g. the bond-interlinked character of the local $Z_2$ conserved plaquette charges of the Kitaev honeycomb spin-$\frac{1}{2}$ model which have a mutually commuting algebra. We will discuss the connections and differences of this kind of quantum spin liquidity in relation to many-body topological order found in some gapped quantum spin liquids -- the canonical example being the Kitaev toric code which belongs to the more general class of Levin-Wen or string net constructions with mutually commuting algebras of conserved charges. If time permits, we will make several new exact statements on the many-body order present in this class of «anticommuting» quantum spin liquids and some conjectures pertaining to them.

[1] arXiv:2406.17034,  A solvable embedding mechanism for one-dimensional spinless and Majorana fermions in higher- dimensional spin-1/2 magnets


[2] A theorem on extensive ground state entropy, spin liquidity and some related models, S. Pujari, 
https://scipost.org/submissions/scipost_202502_00016v1/