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Spontaneous $SU(2)$ symmetry breaking in ground state of quantum spin chain

Anilesh Mohari

Published Sep 21, 2005• 2 citations
OpenAlex

Abstract

In this paper we prove that there exists no translation invariant pure state of $\clb=\otimes_{\mathbb{Z}}M_d(\IC)$ that is real, lattice symmetric, reflection positive with a twist and SU(2) invariant when $d$ is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnetic model with ${1 \over 2}$-odd integer spin degrees of freedom admits spontaneous symmetry breaking at its ground states.

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Spontaneous SU(2)SU(2)SU(2) symmetry breaking in ground state of quantum spin chain
In this paper we prove that there exists no translation invariant pure state of \clb=⊗ZMd(\IC)\clb=\otimes_{\mathbb{Z}}M_d(\IC)\clb=⊗Z​Md​(\IC) that is real, lattice symmetric, reflection positive with a twist and SU(2) invariant when ddd is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnetic model with 12{1 \over 2}21​-odd integer spin degrees of freedom admits spontaneous symmetry breaking at its ground states.
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
We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}M_d(C), \;d \ge 2$, where Md(C)M_d(C)Md​(C) is the set of d×dd \times dd×d dimensional matrices over field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$. This result makes it possible to study such a pure state with additional symmetry. We prove that exponentially decaying two point spacial correlation function of a real lattice symmetric reflection positive translation invariant pure state is a split state. Further there exists no translation invariant pure state on $\clb$ that is real, lattice symmetric, refection positive and su(2)su(2)su(2) invariant when ddd is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnets model for 1/2-odd integer spin degrees of freedom admits spontaneous symmetry breaking at it's ground states
SU2(C)SU_2(\mathbb{C})SU2​(C) symmetry in quantum spin chain ground states and Haldane's conjecture
In this paper, we prove that any translation and $SU_2(\IC)$-invariant pure state of $\IM=\otimes_{k \in \IZ}\!M^{(k)}_d(\IC)$, that is also real, lattice symmetric and reflection positive with a certain twist $r_0 \in U_d(\IC)$, is finitely correlated and its two-point spatial correlation function decays exponentially whenever ddd is an odd integer. In particular, the Heisenberg iso-spin anti-ferromagnetic integer spin model admits unique low temperature limiting ground state and its spatial correlation function decays exponentially. The unique low temperature limiting ground state of the Hamiltonian is determined by the unique solution to Clebsch-Gordon inter-twinning isometry between two representations of $SU_2(\IC)$.
In this exposition we investigate further the general methodology proposed in [Mo2] to study properties of the ground states of a translation invariant Hamiltonian for one lattice dimensional quantum spin chain A = ⊗ ZMd, where Md is the matrix of d × d complex matrices. We introduce a notion of quantum detailed balance [Mo1] for a translation invariant state on A and prove that such a pure state is uniformly mixing [BR,Ma2] if and only if the lattice space correlation functions decay exponentially. Furthermore we also prove that a pure lattice symmetric, translation and SU(2) gauge invariant state give rise to a canonical Popescu systems acting on a finite dimensional Hilbert space and thus the lattice space correlation functions of the pure state decay exponentially. As a consequence of these results we conclude that if the ground states for an integer spin SU(2) invariant (2s + 1 = d) detailed balanced Hamiltonian is unique then the state is split. In particular if the ground state for integer
Some measurements have shown that the second-order exchange interaction is non-negligible in ferromagnetic compounds whose microscopic interactions are described by means of half-odd integer quantum spins. In these spin systems the ground state is either ferromagnetic or antiferromagnetic when the bilinear exchange interaction is dominant. Instead, in ferromagnetic systems characterized by bilinear and biquadratic exchange interactions of comparable magnitude, the energy minimum occurs when spins are in a canting ground-state. To this aim, a one-dimensional (1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins. For these 1D and 2D quantum systems a generalization of the Mermin–Wagner–Hohenberg theorem (also known as Mermin–Wagner–Berezinksii or Coleman theorem) is given. It is demonstrated, by means of quantum statistical arguments, based on Bogoliubov’s inequality, that, at any finite temperature, (1) there is absence of long-range order and that (2) the law governing the vanishing of the order parameter is the same as in the bilinear case for both 1D and 2D quantum ferromagnetic systems. The physical implications of the absence of a spontaneous spin symmetry breaking in 1D spin chains and 2D spin lattices modeled via a generalized quantum Heisenberg Hamiltonian are discussed.