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Ferromagnetic Ordering of Energy Levels for $U_q(\\mathfrak{sl}_2)$ Symmetric Spin\n Chains

Bruno Nachtergaele, Ng, Stephen, Shannon L. Starr

Published May 26, 2011• 5 citations
OpenAlex

Abstract

We consider the class of quantum spin chains with arbitrary\n $U_q(\\mathfrak{sl}_2)$-invariant nearest neighbor interactions, sometimes called\n $\\textrm{SU}_q(2)$ for the quantum deformation of $\\textrm{SU}(2)$, for $q>0$. We derive\n sufficient conditions for the Hamiltonian to satisfy the property we call {\\em\n Ferromagnetic Ordering of Energy Levels}. This is the property that the ground state energy\n restricted to a fixed total spin subspace is a decreasing function of the total spin. Using\n the Perron-Frobenius theorem, we show sufficient conditions are positivity of all\n interactions in the dual canonical basis of Lusztig. We characterize the cone of positive\n interactions, showing that it is a simplicial cone consisting of all non-positive linear\n combinations of "cascade operators," a special new basis of $U_q(\\mathfrak{sl}_2)$\n intertwiners we define. We also state applications to interacting particle processes.

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Ferromagnetic Ordering of Energy Levels for Uq(mathfraksl2)U_q(\\mathfrak{sl}_2)Uq​(mathfraksl2​) Symmetric Spin\n Chains
We consider the class of quantum spin chains with arbitrary\n Uq(mathfraksl2)U_q(\\mathfrak{sl}_2)Uq​(mathfraksl2​)-invariant nearest neighbor interactions, sometimes called\n textrmSUq(2)\\textrm{SU}_q(2)textrmSUq​(2) for the quantum deformation of textrmSU(2)\\textrm{SU}(2)textrmSU(2), for q>0q>0q>0. We derive\n sufficient conditions for the Hamiltonian to satisfy the property we call {\em\n Ferromagnetic Ordering of Energy Levels}. This is the property that the ground state energy\n restricted to a fixed total spin subspace is a decreasing function of the total spin. Using\n the Perron-Frobenius theorem, we show sufficient conditions are positivity of all\n interactions in the dual canonical basis of Lusztig. We characterize the cone of positive\n interactions, showing that it is a simplicial cone consisting of all non-positive linear\n combinations of "cascade operators," a special new basis of Uq(mathfraksl2)U_q(\\mathfrak{sl}_2)Uq​(mathfraksl2​)\n intertwiners we define. We also state applications to interacting particle processes.
We prove ferromagnetic ordering of energy levels for XXX Heisenberg chains with spins of arbitrary magnitude, thus extending our previous result for the spin-1/2 chain. Ferromagnetic ordering means that the minimum energies in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin. This result provides a ferromagnetic analogue of the well-known theorem by Lieb and Mattis about ordering of energy levels in antiferromagnetic and ferrimagnetic systems on bipartite graphs.
Dedicated to Elliott Lieb on the occasion of his seventieth birthday Abstract We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the Lieb-Mattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest energies in each subspace of fixed total spin are strictly ordered according to the total spin, with the lowest, i.e., the ground state, belonging to the maximal total spin subspace. Our main result is a proof of this conjecture for the spin-1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof has two main ingredients. The first is an extension of a result of Koma and Nachtergaele which shows that monotonicity as a function of the total spin follows from the monotonicity of the ground state energy in each total spin subspace as a function of the length of the chain. For the second part of the proof we use the Temperley-Lieb algebra to calculate, in a suitable basis, the matrix elements of the Hamiltonian restricted to each subspace of the highest weight vectors with a given total spin. We then show that the positivity properties of these matrix elements imply the necessary monotonicity in the volume. Our method also shows that the first excited state of the XXX ferromagnet on any finite tree has one less than maximal total spin.
The Heisenberg ferromagnet has symmetry group SU(2). The property known as ferromagnetic ordering of energy levels (FOEL) states that the minimum energy eigenvalue among eigenvectors with total spin s is monotone decreasing as a function of s. While this property holds for certain graphs such as open chains, in this note we demonstrate some counterexamples. We consider the spin 1/2 model on rings of length 2n for n = 2, 3, …, 8, and show that the minimum energy among all spin singlets is less than or equal to the minimum energy among all spin triplets, which violates FOEL. This also shows some counterexamples to the “Aldous ordering” for the symmetric exclusion process. We also review some of the literature related to these examples.
We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψψψ be an arbitrary two-qubit state. We consider a chain of nnn qubits with open boundary conditions and Hamiltonian Hn(ψ)H_n(ψ)Hn​(ψ) which is defined as the sum of rank-1 projectors onto ψψψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn(ψ)H_n(ψ)Hn​(ψ) is upper bounded by 1/(n−1)1/(n-1)1/(n−1) if the eigenvalues of a certain two-by-two matrix simply related to ψψψ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of nnn (depending only on ψψψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.
We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψ be an arbitrary two-qubit state. We consider a chain of n qubits with open boundary conditions and Hamiltonian Hn(ψ) which is defined as the sum of rank-1 projectors onto ψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn(ψ) is upper bounded by 1/(n − 1) if the eigenvalues of a certain 2 × 2 matrix simply related to ψ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of n (depending only on ψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.
We introduce a real-space exact renormalization-group method to find exactly solvable quantum spin chains and their ground states. This method allows us to provide a complete list for exact solutions within SU(2) symmetric quantum spin chains with $S\ensuremath{\le}4$ and nearest-neighbor interactions, as well as examples with S=5S=5S=5. We obtain two classes of solutions. One of them converges to the fixed points of renormalization group and the ground states are matrix-product states. Another one does not have renormalization fixed points and the ground states are partially ferromagnetic states.