arXiv:2101.11464 · Communications in Mathematical Physics

Dimerization in Quantum Spin Chains with O(n) Symmetry

Jakob E. Björnberg, Peter Mühlbacher, Bruno Nachtergaele, Daniel Ueltschi

Published Jul 14, 2021• 3 citations

Abstract

Abstract We consider quantum spins with $$S\ge 1$$ S ≥ 1 , and two-body interactions with $$O(2S+1)$$ O ( 2 S + 1 ) symmetry. We discuss the ground state phase diagram of the one-dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for S sufficiently large. We also prove the existence of a gap for excitations.

DOI: 10.1007/s00220-021-04148-1

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arXiv:2101.11464 · Communications in Mathematical Physics

Dimerization in Quantum Spin Chains with O(n) Symmetry

Jakob E. Björnberg, Peter Mühlbacher, Bruno Nachtergaele, Daniel Ueltschi

Published Jul 14, 2021• 3 citations

arXiv HTML PDF

Abstract

Abstract We consider quantum spins with

S\ge 1

S ≥ 1 , and two-body interactions with

O(2S+1)

O ( 2 S + 1 ) symmetry. We discuss the ground state phase diagram of the one-dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for S sufficiently large. We also prove the existence of a gap for excitations.

DOI: 10.1007/s00220-021-04148-1

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Dimerization in $O(n)$ -invariant quantum spin chainsDimerization in $O(n)$-invariant quantum spin chains

Jakob E. Björnberg, Kieran Ryan

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Dimerization and Néel Order in Different Quantum Spin Chains Through a Shared Loop Representation

Michael Aizenman, Hugo Duminil-Copin, Simone Warzel

Abstract The ground-states of the spin-Santiferromagnetic chain $H_{{\text {AF}}}$ HAF with a projection-based interaction and the spin-1/2 XXZ-chain $H_{{\text {XXZ}}}$ HXXZ at anisotropy parameter $\Delta =\cosh (\lambda )$ Δ=cosh(λ) share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planarQ-state Potts model at $\sqrt{Q}= 2S+1 =2\cosh (\lambda )$ Q=2S+1=2cosh(λ) . The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground-states of these two models: dimerization for $H_{{\text {AF}}}$ HAF at all $S> 1/2$ S>1/2 , and Néel order for $H_{{\text {XXZ}}}$ HXXZ at $\lambda >0$ λ>0 . The results presented include: (i) a translation to the above quantum spin systems of the results which were recently proven by Duminil–Copin–Li–Manolescu for a broad class of two-dimensional random-cluster models, and (ii) a short proof of the symmetry breaking in a manner similar to the recent structural proof by Ray–Spinka of the discontinuity of the phase transition for $Q>4$ Q>4 . Altogether, the quantum manifestation of the change between $Q=4$ Q=4 and $Q>4$ Q>4 is a transition from a gapless ground-state to a pair of gapped and extensively distinct ground-states.Abstract The ground-states of the spin-Santiferromagnetic chain $$H_{{\text {AF}}}$$ HAF with a projection-based interaction and the spin-1/2 XXZ-chain $$ H_{{\text {XXZ}}}$$ HXXZ at anisotropy parameter $$\Delta =\cosh (\lambda ) $$ Δ=cosh(λ) share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planarQ-state Potts model at $$ \sqrt{Q}= 2S+1 =2\cosh (\lambda )$$ Q=2S+1=2cosh(λ) . The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground-states of these two models: dimerization for $$H_{{\text {AF}}}$$ HAF at all $$S> 1/2$$ S>1/2 , and Néel order for $$ H_{{\text {XXZ}}} $$ HXXZ at $$\lambda >0$$ λ>0 . The results presented include: (i) a translation to the above quantum spin systems of the results which were recently proven by Duminil–Copin–Li–Manolescu for a broad class of two-dimensional random-cluster models, and (ii) a short proof of the symmetry breaking in a manner similar to the recent structural proof by Ray–Spinka of the discontinuity of the phase transition for $$Q>4$$ Q>4 . Altogether, the quantum manifestation of the change between $$Q=4$$ Q=4 and $$Q>4$$ Q>4 is a transition from a gapless ground-state to a pair of gapped and extensively distinct ground-states.

arXiv:2002.02543Annales Henri PoincaréJun 16, 202010 citations

Abstract We consider quantum spins with

S\ge 1

S ≥ 1 , and two-body interactions with

O(2S+1)

Dimerization in

O(n)

-invariant quantum spin chains

We establish dimerization in

O(n)

-invariant quantum spin chains with large enough

n

, in a much larger region of the phase diagram than previous results. This entails identifying two distinct infinite volume ground states which are gapped, are translations by one unit of each other, and which both have exponentially decaying correlations. We show these states are the limits of finite-volume, finite temperature states as the graph tends to

\mathbb{Z}

and temperature tends to zero in any order. Our method relies on a probabilistic representation of the quantum system in terms of random loops, and an adaptation of a method developed for loop

O(n)

models on the hexagonal lattice by Duminil-Copin, Peled, Samotij and Spinka.

We discuss spin-

S

antiferromagnetic Heisenberg chains with three-spin interactions, next-nearest-neighbor interactions, and bond alternation. First, we prove rigorously that there exist parameter regions of the exact dimerized ground state in this system. This is a generalization of the Majumdar-Ghosh model to arbitrary

S

. Next, we discuss the ground-state phase diagram of the models by introducing several effective field theories and the universality classes of the transitions are described by the level-

2S

\mathrm{SU}(2)

Wess-Zumino-Witten model and the Gaussian model. Finally, we determine the phase diagrams of

S=1

and

S=3/2

systems by using exact diagonalization and level spectroscopy.

Abstract The ground-states of the spin-Santiferromagnetic chain

H_{{\text {AF}}}

HAF with a projection-based interaction and the spin-1/2 XXZ-chain

H_{{\text {XXZ}}}

HXXZ at anisotropy parameter

\Delta =\cosh (\lambda )

Δ=cosh(λ) share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planarQ-state Potts model at

\sqrt{Q}= 2S+1 =2\cosh (\lambda )

Q=2S+1=2cosh(λ) . The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground-states of these two models: dimerization for

H_{{\text {AF}}}

HAF at all

S> 1/2

S>1/2 , and Néel order for

H_{{\text {XXZ}}}

HXXZ at

\lambda >0

λ>0 . The results presented include: (i) a translation to the above quantum spin systems of the results which were recently proven by Duminil–Copin–Li–Manolescu for a broad class of two-dimensional random-cluster models, and (ii) a short proof of the symmetry breaking in a manner similar to the recent structural proof by Ray–Spinka of the discontinuity of the phase transition for

Q>4

Q>4 . Altogether, the quantum manifestation of the change between

Q=4

Q=4 and

Q>4

Q>4 is a transition from a gapless ground-state to a pair of gapped and extensively distinct ground-states.