Uniqueness of the translationally invariant ground state in quantum spin systems

Walter Felipe Wreszinski

Goldstone’s theorem is proved for quantum spin systems of finite range with rotationally invariant Hamiltonian under the assumptions that the ground state of the infinite system is unique, is invariant under a subset of the translation group (which depends on the system), and has long range order.

Ola Bratteli, Akitaka Kishimoto, Derek W. Robinson

Anilesh Mohari

Though our primary motivation is to study properties of the ground states of a Hamiltonian for quantum spin chain, in this exposition we propose a general methodology valid for a translation invariant state. To that end we study the associated Popescu systems [Po,BJKW] representing the translation invariant state and find an useful criteria for the state to be pure. As a simple consequence of this criteria we prove a surprising result which says that there exists no translation invariant lattice symmetric and SU(2)-gauge invariant pure state for half-odd integer spin chain. As a consequence of this result we prove that ground state for half-odd integer spin chain is not unique for any SU(2)-gauge invariant translation and lattice symmetric Hamiltonian. In other words this exhibits that there is a phase transition in the ground state for anti-ferromagnetic isotropic Heisenberg half-odd integer spin chain. 2 It has been known in physics literature that the spin 1

A Galindo, R. Tarrach

Libor Caha, Daniel Nagaj

Investigating translationally invariant qudit spin chains with a low local dimension, we ask what is the best possible tradeoff between the scaling of the entanglement entropy of a large block and the inverse-polynomial scaling of the spectral gap. Restricting ourselves to Hamiltonians with a "rewriting" interaction, we find the pair-flip model, a family of spin chains with nearest neighbor, translationally invariant, frustration-free interactions, with a very entangled ground state and an inverse-polynomial spectral gap. For a ground state in a particular invariant subspace, the entanglement entropy across a middle cut scales as $\log n$ for qubits (it is equivalent to the XXX model), while for qutrits and higher, it scales as $\sqrt{n}$. Moreover, we conjecture that this particular ground state can be made unique by adding a small translationally-invariant perturbation that favors neighboring letter pairs, adding a small amount of frustration, while retaining the entropy scaling.

Hal Tasaki

Using the analytic machinery and numerical inputs of OpenAI's proofs of the Haldane gap, we prove uniqueness of the translation-invariant ground state of the infinite $S=1$ Heisenberg antiferromagnetic chain and of the $J$-$D$ model for sufficiently small $|D|$. Uniqueness is among all translation-invariant ground states in the operator-algebraic sense, without assuming a gap or spin-rotation symmetry for a competing state. The proof combines concentration of the spatial transfer spectrum, an exponentially small low-temperature free-energy correction, and a Golden--Thompson comparison for separated local observables. The unique translation-invariant state is pure, locally unique and gapped, and continuous in $D$. The ground states of even periodic chains and odd open chains with equal positive endpoint fields converge to this state along their full sequences, uniformly in $D$, with a stretched-exponential local rate. Every choice of ground states of open chains without endpoint magnetic fields has the same local limit. These conclusions do not establish uniqueness among arbitrary nontranslation-invariant ground states of the infinite chain. The author formulated the problems and proposed the core strategies. ChatGPT Astra made substantial contributions to developing the proofs and preparing the manuscript.