The Ground State of
the S=1 Antiferromagnetic
Heisenberg Chain is
Topologically Nontrivial
if Gapped
Hal Tasaki
webinar @ YouTube / July 2024

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a unique gapped ground state that is
topologically trivial

t h is is e x p e c te da unique gapped ground state that is
topologically nontrivial

gapless or degenerate ground states(s)the S = 1 antiferromagnetic Heisenberg chain<latexit sha1_base64="6gWOIxcPrC3srIUKcri7oQtkry0=">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</latexit>

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we r ig o ro us ly r u le o u t t h is p o s s ibil it

a unique gapped ground state that is
topologically trivial

t h is is e x p e c te da unique gapped ground state that is
topologically nontrivial

gapless or degenerate ground states(s)a rigorous and nontrivial result about the
ground state of the S = 1 antiferromagnetic
Heisenberg chainy!

introduction/motivation
main results
idea of the proof

the ground state of the spin S
antiferromagnetic
Heisenberg
chain
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1
3
S = 2 , 1, 2 , 2, . . .
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Haldane 1981, 1983, 1983

half-odd integer S
unique gapless ground state
integer S
unique gapped ground state

O(1/L
E = O(1)the picture of symmetry-protected topological (SPT)phases Gu,Wen 2009, Pollmann, Turner, Berg, Oshikawa 2010, 2012even S belongs to a trivial SPT phase
odd S
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AF Heisenberg chain Ĥ
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unique gapped g.s.

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j2Z {s Ŝ j · Ŝ j+1 + (1energy gap E 1st

E GS

trivialgap

gapless point?

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unique gapped g.s.

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sha1_base64="usEhJX/6iZrJoL9ND0L3BPTG3Sc=">AAAB6HicbZDJSgNBEIZr4hbjFpebl8YgeAozXvRmQESPCZgFkiH0dGqSNj0L3T1CHPIEXjwo4tUH8EE8evOYN7GzHDTxh4aP/6+iq8qLBVfatr+tzNLyyupadj23sbm1vZPf3aupKJEMqywSkWx4VKHgIVY11wIbsUQaeALrXv9ynNfvUSoehbd6EKMb0G7Ifc6oNlZFtfMFu2hPRBbBmUHh4nN0/XHwMCq381+tTsSSAEPNBFWq6dixdlMqNWcCh7lWojCmrE+72DQY0gCVm04GHZJj43SIH0nzQk0m7u+OlAZKDQLPVAZU99R8Njb/y5qJ9s/dlIdxojFk04/8RBAdkfHWpMMlMi0GBiiT3MxKWI9KyrS5Tc4cwZlfeRFqp0XHLjoVp1C6gqmycAhHcAIOnEEJbqAMVWCA8AjP8GLdWU/Wq/U2Lc1Ys559+CPr/QepfpFF</latexit>

there seems to be a phase transition

nontrivial SPT phase of S = 1 chain<latexit sha1_base64="rpr0IgQtRvfmrDyBpill60/2oiw=">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</latexit>

AF Heisenberg chain Ĥ
<latexit sha1_base64="7mJJ7Xeh4xXbswFP/cSYVmj515E=">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</latexit>

(0)
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trivial modelinterpolating model<latexit sha1_base64="WbvzvJANbmwheuoasj3WVtRj7YM=">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</latexit>Ĥ (s) =

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unique gapped g.s.

unique gapped g.s.

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j |0ijs)(Ŝjz )2 } s 2 [0, 1]<latexit sha1_base64="8LEYqnieAe7nE14ViN/kTqbE6c4=">AAAB+XicbVDLSsNAFL2pr1pfVZduhhZBUErioroMunFZwT4gDWUynbRDJ5MwMymE0L9w6caFIm79E3f9G6ePhbYeuHA4517uvSdIOFPatqdWYWNza3unuFva2z84PCofn7RUnEpCmyTmsewEWFHOBG1qpjntJJLiKOC0HYzuZ357TKVisXjSWUL9CA8ECxnB2kgMFLwBAwEe2HAFDvi9ctWu2XOgdeIsSdWtdC+fp27W6JW/u/2YpBEVmnCslOfYifZzLDUjnE5K3VTRBJMRHlDPUIEjqvx8fvkEnRulj8JYmhIazdXfEzmOlMqiwHRGWA/VqjcT//O8VIe3fs5EkmoqyGJRmHKkYzSLAfWZpETzzBBMJDO3IjLEEhNtwiqZEJzVl9dJ67rm1Gv1R6fq3sECRTiDClyYIG/AhQdoQBMIjOHFxPxu5dar9WF9LloL1nLmFP7A+voBx+OTgA==</latexit>

j2Z {s Ŝ j · Ŝ j+1 + (1energy gap E 1st<latexit sha1_base64="A2FML3uBgz0gpTVT+aRWmzXDzxA=">AAACFXicbVC7SgNBFL3rM4mvVcHGZjAINoZdi2gZFNEyonlgsobZySQZMvtgZlYMy/6EjfgB1rY2ForYCnZ+jU4ehSYeuHA4517uvccNOZPKsr6MqemZ2bn5VDqzsLi0vGKurpVlEAlCSyTggai6WFLOfFpSTHFaDQXFnstpxe0e9f3KNRWSBf6F6oXU8XDbZy1GsNJSYG7AMVxBDA8gwAMENkhQkMDumH4C55A0zKyVswZAk8QekWwhHd5fPt58FxvmZ70ZkMijviIcS1mzrVA5MRaKEU6TTD2SNMSki9u0pqmPPSqdePBVgra10kStQOjyFRqovydi7EnZ81zd6WHVkeNeX/zPq0WqdeDEzA8jRX0yXNSKOFIB6keEmkxQonhPE0wE07ci0sECE6WDzOgQ7PGXJ0l5L2fnc/kzO1s4hCFSsAlbsKMj3ocCnEIRSkDgFp7gBV6NO+PZeDPeh61TxmhmHf7A+PgBXdaalA==</latexit>

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s plot by Emil Aagaard<latexit sha1_base64="OxOLrntTln7cSloH7d+WEH80VQ0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m86LEggscWTFtoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8MBVcG9f9dkobm1vbO+Xdyt7+weFR9fikrZNMMfRZIhLVDalGwSX6hhuB3VQhjUOBnXByO/c7T6g0T+SDmaYYxHQkecQZNVZq6UG15tbdBcg68QpSgwLNQfWrP0xYFqM0TFCte56bmiCnynAmcFbpZxpTyiZ0hD1LJY1RB/ni0Bm5sMqQRImyJQ1ZqL8nchprPY1D2xlTM9ar3lz8z+tlJroJci7TzKBky0VRJohJyPxrMuQKmRFTSyhT3N5K2JgqyozNpmJD8FZfXiftq7rn1r2WV2vcFXGU4QzO4RI8uIYG3EMTfGCA8Ayv8OY8Oi/Ou/OxbC05xcwp/IHz+QPenIz3</latexit><latexit
sha1_base64="tAoCvPhjunlb5GZUJXJaRhoYV9Q=">AAAB6HicbZC7SgNBFIbPeo3xFrW0GQyCVdi10c6AiJYJmAskS5idnE3GzM4uM7NCWPIENhaK2Pok1pZ2lr6Jk0uhiT8MfPz/Ocw5J0gE18Z1v5yl5ZXVtfXcRn5za3tnt7C3X9dxqhjWWCxi1QyoRsEl1gw3ApuJQhoFAhvB4HKcN+5RaR7LWzNM0I9oT/KQM2qsVdWdQtEtuRORRfBmULz4+L5+B4BKp/DZ7sYsjVAaJqjWLc9NjJ9RZTgTOMq3U40JZQPaw5ZFSSPUfjYZdESOrdMlYazsk4ZM3N8dGY20HkaBrYyo6ev5bGz+l7VSE577GZdJalCy6UdhKoiJyXhr0uUKmRFDC5QpbmclrE8VZcbeJm+P4M2vvAj105LnlryqVyxfwVQ5OIQjOAEPzqAMN1CBGjBAeIAneHbunEfnxXmdli45s54D+CPn7QfYE4/o</latexit>
sha1_base64="usEhJX/6iZrJoL9ND0L3BPTG3Sc=">AAAB6HicbZDJSgNBEIZr4hbjFpebl8YgeAozXvRmQESPCZgFkiH0dGqSNj0L3T1CHPIEXjwo4tUH8EE8evOYN7GzHDTxh4aP/6+iq8qLBVfatr+tzNLyyupadj23sbm1vZPf3aupKJEMqywSkWx4VKHgIVY11wIbsUQaeALrXv9ynNfvUSoehbd6EKMb0G7Ifc6oNlZFtfMFu2hPRBbBmUHh4nN0/XHwMCq381+tTsSSAEPNBFWq6dixdlMqNWcCh7lWojCmrE+72DQY0gCVm04GHZJj43SIH0nzQk0m7u+OlAZKDQLPVAZU99R8Njb/y5qJ9s/dlIdxojFk04/8RBAdkfHWpMMlMi0GBiiT3MxKWI9KyrS5Tc4cwZlfeRFqp0XHLjoVp1C6gqmycAhHcAIOnEEJbqAMVWCA8AjP8GLdWU/Wq/U2Lc1Ys559+CPr/QepfpFF</latexit>

there seems to be a phase transition

nontrivial SPT phase of S = 1 chain<latexit sha1_base64="WbvzvJANbmwheuoasj3WVtRj7YM=">AAAC3XicjVLLbtNAFL12eZTwCnTJZkSE1KqQ2l0UNpWismFZVNJWTYI1Hk+aofOwZsZIqWWpG4RAiC1bNv4RlizZ8TfcOFmElgXX8vj4nHPv3HmkuRTOR9HvIFy5dv3GzdVbrdt37t67337w8NCZwjLeZ0Yae5xSx6XQvO+Fl/w4t5yqVPKj9OzlTD96z60TRr/x05yPFD3VYiwY9UiZ9jnUMAEKHkp4BRW8xe86ONhAvIuagwIUJMi+gyEI0Diqxj+BFJ8STtBZobNEbw1Pl+rVqBuQkKEyxaz5XwkHTUaCFWtgqBp0/3/WrJNNiBFvYqcxPGu6XV+qcLBYRw0W8wmcL2bbQHYb2Sppd6Ju1AS5CuIF6PS2fnz/+DPb3U/av4aZYYXi2jNJnRvEUe5HJbVeMMmr1rBwPKfsjJ7yAUJNFXejsjmdijxBJiNjY/HVnjTsckZJlXNTlaJTUT9xl7UZ+S9tUPjxi1EpdF54rtl8onEhiTdkdtQkE5YzL6cIKLMCeyVsQi1lHi9ECzchvrzkq+BwuxvvdHdex53eHsxjFR7B42brn0MPL80+9IEFJ8FF8Cn4HCbhh/BL+HVuDYNFzhr8FeG3PyEWw3U=</latexit>Ĥ (s) =Ps)(Ŝjz )2 } s 2 [0, 1]<latexit sha1_base64="8LEYqnieAe7nE14ViN/kTqbE6c4=">AAAB+XicbVDLSsNAFL2pr1pfVZduhhZBUErioroMunFZwT4gDWUynbRDJ5MwMymE0L9w6caFIm79E3f9G6ePhbYeuHA4517uvSdIOFPatqdWYWNza3unuFva2z84PCofn7RUnEpCmyTmsewEWFHOBG1qpjntJJLiKOC0HYzuZ357TKVisXjSWUL9CA8ECxnB2kgMFLwBAwEe2HAFDvi9ctWu2XOgdeIsSdWtdC+fp27W6JW/u/2YpBEVmnCslOfYifZzLDUjnE5K3VTRBJMRHlDPUIEjqvx8fvkEnRulj8JYmhIazdXfEzmOlMqiwHRGWA/VqjcT//O8VIe3fs5EkmoqyGJRmHKkYzSLAfWZpETzzBBMJDO3IjLEEhNtwiqZEJzVl9dJ67rm1Gv1R6fq3sECRTiDClyYIG/AhQdoQBMIjOHFxPxu5dar9WF9LloL1nLmFP7A+voBx+OTgA==</latexit>

j2Z {s Ŝ j · Ŝ j+1 + (1gapless(critical)
point
n o n t r iv ia lt r iv ia l
unique gapped g.s. sunique gapped g.s.
<latexit sha1_base64="flJqh3zIAXf1CWWaHy/6TEggDnE=">AAAB6HicbZC7SgNBFIbPxltcb1FLm8UgWIVdi2gjBm0sEzAXSJYwOzmbjJmdXWZmhbDkCWwsFLHVh7G3Ed/GyaXQ6A8DH/9/DnPOCRLOlHbdLyu3tLyyupZftzc2t7Z3Crt7DRWnkmKdxjyWrYAo5ExgXTPNsZVIJFHAsRkMryZ58w6lYrG40aME/Yj0BQsZJdpYNdUtFN2SO5XzF7w5FC/e7fPk7dOudgsfnV5M0wiFppwo1fbcRPsZkZpRjmO7kypMCB2SPrYNChKh8rPpoGPnyDg9J4yleUI7U/dnR0YipUZRYCojogdqMZuY/2XtVIdnfsZEkmoUdPZRmHJHx85ka6fHJFLNRwYIlczM6tABkYRqcxvbHMFbXPkvNE5KXrlUrnnFyiXMlIcDOIRj8OAUKnANVagDBYR7eIQn69Z6sJ6tl1lpzpr37MMvWa/fQh2QQA==</latexit>

0<latexit sha1_base64="h+aNUI7Gw33Te3qpCkvF9GJn7eY=">AAAB6HicbZC7SgNBFIbPxltcb1FLm8UgWIVdi2gjBm0sEzAXSJYwOzmbjJmdXWZmhbDkCWwsFLHVh7G3Ed/GyaXQ6A8DH/9/DnPOCRLOlHbdLyu3tLyyupZftzc2t7Z3Crt7DRWnkmKdxjyWrYAo5ExgXTPNsZVIJFHAsRkMryZ58w6lYrG40aME/Yj0BQsZJdpYNbdbKLoldyrnL3hzKF682+fJ26dd7RY+Or2YphEKTTlRqu25ifYzIjWjHMd2J1WYEDokfWwbFCRC5WfTQcfOkXF6ThhL84R2pu7PjoxESo2iwFRGRA/UYjYx/8vaqQ7P/IyJJNUo6OyjMOWOjp3J1k6PSaSajwwQKpmZ1aEDIgnV5ja2OYK3uPJfaJyUvHKpXPOKlUuYKQ8HcAjH4MEpVOAaqlAHCgj38AhP1q31YD1bL7PSnDXv2Ydfsl6/AdyCj/0=</latexit>

1<latexit sha1_base64="MdH0ceqimViboaLLvFg0HZM2auk=">AAAB5HicbZC7SgNBFIbPxltcb9HWZjAIVmHXItqIQRvLCOYCyRJmJ2eTMbOzy8ysEEKewMZCsRV8GHsb8W2cXApN/GHg4//PYc45YSq4Np737eRWVtfWN/Kb7tb2zu5ewd2v6yRTDGssEYlqhlSj4BJrhhuBzVQhjUOBjXBwPckbD6g0T+SdGaYYxLQnecQZNda69TuFolfypiLL4M+hePnhXqTvX261U/hsdxOWxSgNE1Trlu+lJhhRZTgTOHbbmcaUsgHtYcuipDHqYDQddEyOrdMlUaLsk4ZM3d8dIxprPYxDWxlT09eL2cT8L2tlJjoPRlymmUHJZh9FmSAmIZOtSZcrZEYMLVCmuJ2VsD5VlBl7G9cewV9ceRnqpyW/XCoXK1cwUx4O4QhOwIczqMANVKEGDBAe4RlenHvnyXmdFeaceccB/JHz9gNnfI7X</latexit>

cannot be distinguished by an order parameter
distinct symmetry-protected topological (SPT) phases
there is always a phase transition if the interpolating
Hamiltonians have one of the following symmetries

time-reversal symmetry• ℤ2 × ℤ2 symmetry (π rotations about x, y, z axes)• bond-centered inversion symmetry• U(1) × ℤ2 symmetry(any rotation about z axis + site-centered inversion)•

Pollmann, Turner, Berg, Oshikawa 2010, 2012, Fuji, Pollmann, Oshikawa 2015

topological indicesgapless(critical)
point
n o n t r iv ia lt r iv ia l
unique gapped g.s. sunique gapped g.s.
<latexit sha1_base64="flJqh3zIAXf1CWWaHy/6TEggDnE=">AAAB6HicbZC7SgNBFIbPxltcb1FLm8UgWIVdi2gjBm0sEzAXSJYwOzmbjJmdXWZmhbDkCWwsFLHVh7G3Ed/GyaXQ6A8DH/9/DnPOCRLOlHbdLyu3tLyyupZftzc2t7Z3Crt7DRWnkmKdxjyWrYAo5ExgXTPNsZVIJFHAsRkMryZ58w6lYrG40aME/Yj0BQsZJdpYNdUtFN2SO5XzF7w5FC/e7fPk7dOudgsfnV5M0wiFppwo1fbcRPsZkZpRjmO7kypMCB2SPrYNChKh8rPpoGPnyDg9J4yleUI7U/dnR0YipUZRYCojogdqMZuY/2XtVIdnfsZEkmoUdPZRmHJHx85ka6fHJFLNRwYIlczM6tABkYRqcxvbHMFbXPkvNE5KXrlUrnnFyiXMlIcDOIRj8OAUKnANVagDBYR7eIQn69Z6sJ6tl1lpzpr37MMvWa/fQh2QQA==</latexit>

0<latexit sha1_base64="h+aNUI7Gw33Te3qpCkvF9GJn7eY=">AAAB6HicbZC7SgNBFIbPxltcb1FLm8UgWIVdi2gjBm0sEzAXSJYwOzmbjJmdXWZmhbDkCWwsFLHVh7G3Ed/GyaXQ6A8DH/9/DnPOCRLOlHbdLyu3tLyyupZftzc2t7Z3Crt7DRWnkmKdxjyWrYAo5ExgXTPNsZVIJFHAsRkMryZ58w6lYrG40aME/Yj0BQsZJdpYNbdbKLoldyrnL3hzKF682+fJ26dd7RY+Or2YphEKTTlRqu25ifYzIjWjHMd2J1WYEDokfWwbFCRC5WfTQcfOkXF6ThhL84R2pu7PjoxESo2iwFRGRA/UYjYx/8vaqQ7P/IyJJNUo6OyjMOWOjp3J1k6PSaSajwwQKpmZ1aEDIgnV5ja2OYK3uPJfaJyUvHKpXPOKlUuYKQ8HcAjH4MEpVOAaqlAHCgj38AhP1q31YD1bL7PSnDXv2Ydfsl6/AdyCj/0=</latexit>

1<latexit sha1_base64="MdH0ceqimViboaLLvFg0HZM2auk=">AAAB5HicbZC7SgNBFIbPxltcb9HWZjAIVmHXItqIQRvLCOYCyRJmJ2eTMbOzy8ysEEKewMZCsRV8GHsb8W2cXApN/GHg4//PYc45YSq4Np737eRWVtfWN/Kb7tb2zu5ewd2v6yRTDGssEYlqhlSj4BJrhhuBzVQhjUOBjXBwPckbD6g0T+SdGaYYxLQnecQZNda69TuFolfypiLL4M+hePnhXqTvX261U/hsdxOWxSgNE1Trlu+lJhhRZTgTOHbbmcaUsgHtYcuipDHqYDQddEyOrdMlUaLsk4ZM3d8dIxprPYxDWxlT09eL2cT8L2tlJjoPRlymmUHJZh9FmSAmIZOtSZcrZEYMLVCmuJ2VsD5VlBl7G9cewV9ceRnqpyW/XCoXK1cwUx4O4QhOwIczqMANVKEGDBAe4RlenHvnyXmdFeaceccB/JHz9gNnfI7X</latexit>

Inds = 1<latexit sha1_base64="/Q+FFBaJwfe5LH6J7aRxmMhQD7c=">AAACAXicbVA9SwNBEJ2LXzF+RS1tFoNgFe5Eoo0QtNEugvmA5Ah7e5tkye7esbsnhCM2/hUtLBSx9V/Y+W/cJFdo4oOBx3szzMwLYs60cd1vJ7e0vLK6ll8vbGxube8Ud/caOkoUoXUS8Ui1AqwpZ5LWDTOctmJFsQg4bQbDq4nfvKdKs0jemVFMfYH7kvUYwcZKAlJ4BgUCENyAhBDG0AUNF+B1iyW37E6BFomXkRJkqHWLX50wIomg0hCOtW57bmz8FCvDCKfjQifRNMZkiPu0banEgmo/nX4wRkdWCVEvUrakQVP190SKhdYjEdhOgc1Az3sT8T+vnZjeuZ8yGSeGSjJb1Es4MhGaxIFCpigxfGQJJorZWxEZYIWJsaEVbAje/MuLpHFS9irlyu1pqXqZxZGHAziEY/DgDKpwDTWoA4EHeIJXeHMenRfn3fmYteacbGYf/sD5/AGeIJKD</latexit>

Inds =<latexit sha1_base64="Oqa371TJcZr0whghRZftRjNjGFU=">AAACA3icbVA9SwNBEJ3zM8avqKXNYhBsDHci0UYI2mgXwXxAcoS9vU2yZHfv2N0TwhGw8adoY6GIrX/Czn/jJrlCEx8MPN6bYWZeEHOmjet+OwuLS8srq7m1/PrG5tZ2YWe3rqNEEVojEY9UM8CaciZpzTDDaTNWFIuA00YwuBr7jXuqNIvknRnG1Be4J1mXEWysJCGFJ1AgAMENSAhhBB3QcAHH4HUKRbfkToDmiZeRImSodgpf7TAiiaDSEI61bnlubPwUK8MIp6N8O9E0xmSAe7RlqcSCaj+d/DBCh1YJUTdStqRBE/X3RIqF1kMR2E6BTV/PemPxP6+VmO65nzIZJ4ZKMl3UTTgyERoHgkKmKDF8aAkmitlbEeljhYmxseVtCN7sy/OkflLyyqXy7WmxcpnFkYN9OIAj8OAMKnANVagBgQd4hld4cx6dF+fd+Zi2LjjZzB78gfP5AzTdksM=</latexit>1

the t wo phases are distinguished by a “topological index”
2
Ind ∈ H (G, U(1)) (G is the symmetry group)“topological” index is defined for a locally-unique gappedground state on the infinite chain, and is invariant undera smooth modification of the model with symmetry G
matrix product states Pollmann, Turner, Berg, Oshikawa 2010, 2012
general unique gapped ground states Ogata 2018
Ind = 1 and Ind = − 1 in the ground states of the trivial
model and the Affleck-Kennedy-Lieb-Tasaki (AKLT) model
prototypical solvable model in the “Haldane phase”
Affleck, Kennedy, Lieb, Tasaki 1987

Haldane gap and SPT phasesge ne ra l t h e o r ygeneral picture of SPT phases by Gu and Wen and
Pollmann, Turner, Berg, and Oshikawa
well-defined indices and index theorem by Ogatae xam p le s

AKLT model provides a rigorous example of the
Haldane gap and topologically nontrivial ground state<latexit sha1_base64="vlIZ2a9vM06GdY9NgnAvgi6n8Y8=">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</latexit>

ĤAKLT =

PL

1
2
{
Ŝ
·
Ŝ
+
(
Ŝ
·
Ŝ
)
j
j+1
j
j+1 }
j=1
3

nothing rigorous has been proved about theantiferromagnetic Heisenberg chain with odd S!!

the goal of the present work

two conjectures about the S = 1 antiferromagnetic
Heisenberg chain<latexit sha1_base64="rpr0IgQtRvfmrDyBpill60/2oiw=">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</latexit>

Ĥ

(1)

=

P

j Ŝ j · Ŝ j+1“Haldane conjecture” (probably very difficult)(1) it has a unique gapped ground state(2) it belongs to a nontrivial SPT phase
we assume (1) and rigorously justify (2)proof is elementary!
standard Lieb-Mattis-type techniqueelementary index theorem by Tasaki
the model with general odd S can be treated similarlyintroduction/motivation
main results
idea of the proofS = 1 model on finite chains<latexit sha1_base64="CTZoyvLuTSNwT5OXTSX6tZzCBOU=">AAAB6XicbVA9SwNBEJ2LXzF+RS1tFoNgY7gTiRYWARsLiyjmA5IQ9jZ7yZK9vWN3TghH/oGNhSK2/iM7/42b5ApNfDDweG+GmXl+LIVB1/12ciura+sb+c3C1vbO7l5x/6BhokQzXmeRjHTLp4ZLoXgdBUreijWnoS950x/dTP3mE9dGROoRxzHvhnSgRCAYRSs9nN31iiW37M5AlomXkRJkqPWKX51+xJKQK2SSGtP23Bi7KdUomOSTQicxPKZsRAe8bamiITfddHbphJxYpU+CSNtSSGbq74mUhsaMQ992hhSHZtGbiv957QSDq24qVJwgV2y+KEgkwYhM3yZ9oTlDObaEMi3srYQNqaYMbTgFG4K3+PIyaZyXvUq5cn9Rql5nceThCI7hFDy4hCrcQg3qwCCAZ3iFN2fkvDjvzse8NedkM4fwB87nDw6hjQw=</latexit>L<latexit sha1_base64="Ob0/Csmucrxvg/IAWHRTTcWVksk=">AAAB6XicbVA9SwNBEJ2LXzF+RS1tFoNgY7gLEi0sAjaWUcwHJEfY28wlS/b2jt09IYT8AxsLRWz9R3b+GzfJFZr4YODx3gwz84JEcG1c99vJra1vbG7ltws7u3v7B8XDo6aOU8WwwWIRq3ZANQousWG4EdhOFNIoENgKRrczv/WESvNYPppxgn5EB5KHnFFjpYeLSq9YcsvuHGSVeBkpQYZ6r/jV7ccsjVAaJqjWHc9NjD+hynAmcFrophoTykZ0gB1LJY1Q+5P5pVNyZpU+CWNlSxoyV39PTGik9TgKbGdEzVAvezPxP6+TmvDan3CZpAYlWywKU0FMTGZvkz5XyIwYW0KZ4vZWwoZUUWZsOAUbgrf88ippVspetVy9vyzVbrI48nACp3AOHlxBDe6gDg1gEMIzvMKbM3JenHfnY9Gac7KZY/gD5/MH5yqM8g==</latexit>

2<latexit sha1_base64="NqwpgvdDH2W2kkNDJYc3O6Uzpp8=">AAAB6XicbVA9SwNBEJ3zM8avqKXNYhBsDHci0cIiYGMZxXxAcoS9zVyyZG/v2N0TwpF/YGOhiK3/yM5/4ya5QhMfDDzem2FmXpAIro3rfjsrq2vrG5uFreL2zu7efungsKnjVDFssFjEqh1QjYJLbBhuBLYThTQKBLaC0e3Ubz2h0jyWj2acoB/RgeQhZ9RY6eHc65XKbsWdgSwTLydlyFHvlb66/ZilEUrDBNW647mJ8TOqDGcCJ8VuqjGhbEQH2LFU0gi1n80unZBTq/RJGCtb0pCZ+nsio5HW4yiwnRE1Q73oTcX/vE5qwms/4zJJDUo2XxSmgpiYTN8mfa6QGTG2hDLF7a2EDamizNhwijYEb/HlZdK8qHjVSvX+sly7yeMowDGcwBl4cAU1uIM6NIBBCM/wCm/OyHlx3p2PeeuKk88cwR84nz/lpozx</latexit>

10<latexit sha1_base64="pDtm4Zf7rOJ2bU5iVNjBd50GbxQ=">AAAB6HicbVA9SwNBEJ3zM8avqKXNYhCswp1ItLAI2FgmYD4gOcLeZi5Zs7d37O4J4cgvsLFQxNafZOe/cZNcoYkPBh7vzTAzL0gE18Z1v5219Y3Nre3CTnF3b//gsHR03NJxqhg2WSxi1QmoRsElNg03AjuJQhoFAtvB+G7mt59QaR7LBzNJ0I/oUPKQM2qs1HD7pbJbcecgq8TLSRly1Pulr94gZmmE0jBBte56bmL8jCrDmcBpsZdqTCgb0yF2LZU0Qu1n80On5NwqAxLGypY0ZK7+nshopPUkCmxnRM1IL3sz8T+vm5rwxs+4TFKDki0WhakgJiazr8mAK2RGTCyhTHF7K2EjqigzNpuiDcFbfnmVtC4rXrVSbVyVa7d5HAU4hTO4AA+uoQb3UIcmMEB4hld4cx6dF+fd+Vi0rjn5zAn8gfP5A3rdjLk=</latexit>

1<latexit sha1_base64="l7VcU+nygq5viS2FSlVDLBpVGo8=">AAAB6HicbVA9SwNBEJ3zM8avqKXNYhCswp1ItLAI2FgmYD4gOcLeZi5Zs7d37O4J4cgvsLFQxNafZOe/cZNcoYkPBh7vzTAzL0gE18Z1v5219Y3Nre3CTnF3b//gsHR03NJxqhg2WSxi1QmoRsElNg03AjuJQhoFAtvB+G7mt59QaR7LBzNJ0I/oUPKQM2qs1PD6pbJbcecgq8TLSRly1Pulr94gZmmE0jBBte56bmL8jCrDmcBpsZdqTCgb0yF2LZU0Qu1n80On5NwqAxLGypY0ZK7+nshopPUkCmxnRM1IL3sz8T+vm5rwxs+4TFKDki0WhakgJiazr8mAK2RGTCyhTHF7K2EjqigzNpuiDcFbfnmVtC4rXrVSbVyVa7d5HAU4hTO4AA+uoQb3UIcmMEB4hld4cx6dF+fd+Vi0rjn5zAn8gfP5A3xhjLo=</latexit>

2<latexit sha1_base64="zqbTmRDptXYJen4+cJHcx7FQOC0=">AAAB6HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6CRAuLgI1lAuYDkiPsbeaSNXt7x+6eEEJ+gY2FIrb+JDv/jZvkCk18MPB4b4aZeUEiuDau++3kNja3tnfyu4W9/YPDo+LxSUvHqWLYZLGIVSegGgWX2DTcCOwkCmkUCGwH47u5335CpXksH8wkQT+iQ8lDzqixUqPSL5bcsrsAWSdeRkqQod4vfvUGMUsjlIYJqnXXcxPjT6kynAmcFXqpxoSyMR1i11JJI9T+dHHojFxYZUDCWNmShizU3xNTGmk9iQLbGVEz0qveXPzP66YmvPGnXCapQcmWi8JUEBOT+ddkwBUyIyaWUKa4vZWwEVWUGZtNwYbgrb68TlqVslctVxtXpdptFkcezuAcLsGDa6jBPdShCQwQnuEV3pxH58V5dz6WrTknmzmFP3A+fwB95Yy7</latexit>L<latexit sha1_base64="Nesb43eLSY/AIzKyL85Jm2/YsuU=">AAAB6HicbVA9SwNBEJ2LXzF+RS1tFoNgFe5EooVFwMbCIgHzAckR9jZzyZq9vWN3Twghv8DGQhFbf5Kd/8ZNcoUmPhh4vDfDzLwgEVwb1/12cmvrG5tb+e3Czu7e/kHx8Kip41QxbLBYxKodUI2CS2wYbgS2E4U0CgS2gtHtzG89odI8lg9mnKAf0YHkIWfUWKl+3yuW3LI7B1klXkZKkKHWK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSSSPU/mR+6JScWaVPwljZkobM1d8TExppPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nXpM8VMiPGllCmuL2VsCFVlBmbTcGG4C2/vEqaF2WvUq7UL0vVmyyOPJzAKZyDB1dQhTuoQQMYIDzDK7w5j86L8+58LFpzTjZzDH/gfP4ApU2M1Q==</latexit>

Hamiltonian on the finite open chain {−L, ⋯, L}
PL 1(1)
ĤL = j= L Ŝ j · Ŝ j+1<latexit sha1_base64="tN7Liy/rviTeqb6oMpQ+l8MVI5M=">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</latexit>

symmetry••••h(Ŝ z L + ŜLz )magnetic field at the edges

time-reversal symmetryℤ2 × ℤ2 symmetry (π rotations about x, y, z axes)
bond-centered inversion symmetry

U(1) × ℤ2 symmetry Fuji, Pollmann, Oshikawa 2015
(any rotation about z axis + inversion about 0)

(1)
GS
̂
lemma: H L has a unique ground state | ΦL ⟩
L
ẑ
GS
GS
it satisfies ( ∑Sj ) | ΦL ⟩ = | ΦL ⟩j=−L

assumption

Hamiltonian on the finite open chain {−L, ⋯, L}
PL 1(1)
ĤL = j= L Ŝ j · Ŝ j+1<latexit sha1_base64="tN7Liy/rviTeqb6oMpQ+l8MVI5M=">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</latexit>h(Ŝ z L + ŜLz )GS
1stEL , EL the ground state and the 1st excited energies

highly nontrivial and unprovenassumption: there are constants h > 0, γ > 0, and L0
1stGS
such that EL − EL ≥ γ for any L ≥ L0
expected hold with γ ≃ 0.41 and h = 1 (Haldane conjecture)
the boundary field h ≳ 1 is necessary to suppress gaplessexcitations at the boundaries

main result

̂
A
for
any
local
operator
L"1P(1)
ω a locally-unique gapped g.s. of Ĥ (1) = j2Z Ŝ j · Ŝ j+1<latexit sha1_base64="9dXGgkNys5g2Diy+wYiyyPLBNI4=">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</latexit>! (1) (Â) = lim hGS
L |Â|

(1)

GS
L i

<latexit sha1_base64="6gWOIxcPrC3srIUKcri7oQtkry0=">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</latexit>

Index defined in Tasaki 2018

theorem: Ind[ω ] = − 1
interpolating model<latexit sha1_base64="WbvzvJANbmwheuoasj3WVtRj7YM=">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</latexit>

Ĥ

(s)

=

P

j2Z {s Ŝ j · Ŝ j+1 + (1z 2s)(Ŝj ) }s 2 [0, 1]<latexit sha1_base64="8LEYqnieAe7nE14ViN/kTqbE6c4=">AAAB+XicbVDLSsNAFL2pr1pfVZduhhZBUErioroMunFZwT4gDWUynbRDJ5MwMymE0L9w6caFIm79E3f9G6ePhbYeuHA4517uvSdIOFPatqdWYWNza3unuFva2z84PCofn7RUnEpCmyTmsewEWFHOBG1qpjntJJLiKOC0HYzuZ357TKVisXjSWUL9CA8ECxnB2kgMFLwBAwEe2HAFDvi9ctWu2XOgdeIsSdWtdC+fp27W6JW/u/2YpBEVmnCslOfYifZzLDUjnE5K3VTRBJMRHlDPUIEjqvx8fvkEnRulj8JYmhIazdXfEzmOlMqiwHRGWA/VqjcT//O8VIe3fs5EkmoqyGJRmHKkYzSLAfWZpETzzBBMJDO3IjLEEhNtwiqZEJzVl9dJ67rm1Gv1R6fq3sECRTiDClyYIG/AhQdoQBMIjOHFxPxu5dar9WF9LloL1nLmFP7A+voBx+OTgA==</latexit>

(0)
(s)
̂
Ind[ω ] = 1
ω the ground state of H
(s0)corollary: there is s0 ∈ (0,1) such that ω is not a(s)unique gapped ground state or ω is discontinuous at s0
(s)

(1)

remark: ω is critical if the assumption is not valid
the existence of a phase transition in the ground states(s)̂of H has been proved without any assumptions!!

introduction/motivation
main results
idea of the proof

twist operator

Bloch 1940’s, Lieb, Schultz, Mattis 1961, Affleck Lieb 1986

t wist angle with gradient α ∈ [0,4 − π]
8
>
<0,
(↵)
✓j = ⇡ + ↵j,>:2⇡,<latexit sha1_base64="rwlikVRJV+4LhOjt/KprtM58Jcw=">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</latexit>

⇡
↵;

j

⇡
⇡

j

;
↵
↵
⇡
j
↵,

t wist operator on finite chain⇥(↵)
ÛL = exp
<latexit sha1_base64="P8v7b1Y0sXHCM2Ge/fGMAQXzJZA=">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</latexit>i(↵) z ⇤
j= L ✓j ŜjPL

when α > 0, we take the L ↑ ∞ limit
⇥
P
(↵)
Û
= exp i j2Z\[<latexit sha1_base64="/KiCEONhdJixbqzqp+3AExaZKBE=">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</latexit>(↵) z ⇤Ŝj⇡ ⇡ ✓j,]↵ ↵<latexit sha1_base64="cjDC00hvKMIzNUUQwhpehwuCy60=">AAACL3icbVDLTgIxFL3jE/GFunTTODFxI5khBt2YEN24xCiPBEbSKR2odB5pOyY44XPYufFX2BijMW79CwvMAsGbtD0959y097gRZ1JZ1ruxtLyyurae2chubm3v7Ob29qsyjAWhFRLyUNRdLClnAa0opjitR4Ji3+W05vaux3rtiQrJwuBe9SPq+LgTMI8RrDQV5kyg8AAJnAKDAgwh0ucQuoBBafYOBhN1CAJ8QPCs7y141PvljMuGQStnWnlrUmgR2CkwIa1yKzdqtkMS+zRQhGMpG7YVKSfBQjHC6SDbjCWNMOnhDm1oGGCfSieZzDtAx5ppIy8UegUKTdjZjgT7UvZ9Vzt9rLpyXhuT/2mNWHkXTsKCKFY0INOHvJgjFaJxeKjNBCWK9zXARDD9V0S6WGCidMRZHYI9P/IiqBbydjFfvD0zS1dpHBk4hCM40UGeQwluoAwVIPACI/iAT+PVeDO+jO+pdclIew7gTxk/v86/nhI=</latexit>ei2⇡ Ŝjz= 1̂

local operator on the infinite chain
(α)
(α)
̂
̂
UL and U are continuous in α(α)̂
topological
index
in
terms
of
U
P
<latexit sha1_base64="WbvzvJANbmwheuoasj3WVtRj7YM=">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</latexit>

Ĥ

(s)

=

ω

(s)

(s)
̂
the ground state of H

j2Z {s Ŝ j · Ŝ j+1 + (1z 2s)(Ŝj ) }s 2 [0, 1]<latexit sha1_base64="8LEYqnieAe7nE14ViN/kTqbE6c4=">AAAB+XicbVDLSsNAFL2pr1pfVZduhhZBUErioroMunFZwT4gDWUynbRDJ5MwMymE0L9w6caFIm79E3f9G6ePhbYeuHA4517uvSdIOFPatqdWYWNza3unuFva2z84PCofn7RUnEpCmyTmsewEWFHOBG1qpjntJJLiKOC0HYzuZ357TKVisXjSWUL9CA8ECxnB2kgMFLwBAwEe2HAFDvi9ctWu2XOgdeIsSdWtdC+fp27W6JW/u/2YpBEVmnCslOfYifZzLDUjnE5K3VTRBJMRHlDPUIEjqvx8fvkEnRulj8JYmhIazdXfEzmOlMqiwHRGWA/VqjcT//O8VIe3fs5EkmoqyGJRmHKkYzSLAfWZpETzzBBMJDO3IjLEEhNtwiqZEJzVl9dJ67rm1Gv1R6fq3sECRTiDClyYIG/AhQdoQBMIjOHFxPxu5dar9WF9LloL1nLmFP7A+voBx+OTgA==</latexit>(s)

Tasaki 2018

theorem: if ω is a locally-unique gapped ground state andis invariant under inversion j → − j, then the following
index is well-defined. the index is invariant when s is(s)varied, provided that ω satisfies the above conditions.
<latexit sha1_base64="GnDVDM6xdu4D0Zak1i2ZcRaFKzQ=">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</latexit>

Ind[! (s) ] = lim ! (s) (Û (↵) ) 2 {1, 1}
↵#0

(α)
̂
the (sign of) the ground state expectation value of U

trivial modelP

Nakamura, Todo 2002

<latexit sha1_base64="7mJJ7Xeh4xXbswFP/cSYVmj515E=">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</latexit>Ĥ (0) =<latexit sha1_base64="lfHvMTarorE04J44f4vNSuBjBZs=">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</latexit>

z 2
(
Ŝ
j)
j

<latexit sha1_base64="UqJTAE7PnrOUia3BzAq0KPlOaEU=">AAAB/3icbVC7SgNBFL0bX3F9RS1tBoOgTdi1iDZi0MYygnlAsobZyWwyZHZ2mZkVwprCX1HEQhFbv8DeRvwbJ49CEw9cOJxzL/fe48ecKe0431Zmbn5hcSm7bK+srq1v5Da3qipKJKEVEvFI1n2sKGeCVjTTnNZjSXHoc1rze+dDv3ZDpWKRuNL9mHoh7ggWMIK1kTg8QgQhUOgAhmtIYR8cOIBBK5d3Cs4IaJa4E5I//bBP4ocvu9zKfTbbEUlCKjThWKmG68TaS7HUjHA6sJuJojEmPdyhDUMFDqny0tH9A7RnlDYKImlKaDRSf0+kOFSqH/qmM8S6q6a9ofif10h0cOylTMSJpoKMFwUJRzpCwzBQm0lKNO8bgolk5lZEulhiok1ktgnBnX55llQPC26xULx08qUzGCMLO7BronThCEpwAWWoAIFbuIdneLHurCfr1Xobt2asycw2/IH1/gP/CpU7</latexit>

(0)

N

!
j |0ij
⇥
⇤
P
(↵)
(0)
(↵)
(0)
z
! (Û ) = ! (exp i j ✓j Ŝj ) = 1<latexit sha1_base64="sUs67MKnlNhMIVkKHds/qj6Hsjo=">AAACF3icbZC7SgNBFIbPxluMt1ULC5vFIFjFXYtoIwRtLCOYCyRLmJ1MkjGzM8vMrBDW1L6AjZ2NL2EjqIjYaefbOLkUmnhgmJ/vP4eZ8wcRo0q77reVmpmdm19IL2aWlldW1+z1jbISscSkhAUTshogRRjlpKSpZqQaSYLCgJFK0D0d+JUrIhUV/EL3IuKHqM1pi2KkDRL2FjxAABTaIECbOwQCChpwCdfgGk8CAm5cZrihDTvr5txhOdPCG4tsYf/5/uajeVxs2F/1psBxSLjGDClV89xI+wmSmmJG+pl6rEiEcBe1Sc1IjkKi/GS4V9/ZNaTptIQ0h2tnSH9PJChUqhcGpjNEuqMmvQH8z6vFunXkJ5RHsSYcjx5qxczRwhmE5DSpJFiznhEIS2r+6uAOkghrE2XGhOBNrjwtygc5L5/Ln3vZwgmMKg3bsAN74MEhFOAMilACDLfwCC/wat1ZT9ab9T5qTVnjmU34U9bnD+Dom/8=</latexit><latexit sha1_base64="uey2xeZLo4JqBrnIRANNCBDQnFI=">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</latexit>

Ind[!

(0)

]=1

(1)

proof that Ind[ω ] = − 1

8
>
<0,
(↵)
✓j = ⇡ + ↵j,>:2⇡,<latexit sha1_base64="rwlikVRJV+4LhOjt/KprtM58Jcw=">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</latexit>(0)
when α = 0, we have θj = π⇥(0)ÛL = exp
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h

PL

i

z
Ŝ
j= L j )|(0)GS
L |ÛL |⇥(0) z ⇤
j= L ✓j Ŝj = exp

PL

GS
L i=|

GS
L i=

GS
L i

i⇡

from lemma

j

⇡
↵;

⇡
⇡

j

↵
↵;
⇡
j
↵,

⇤
z

PL

j= L Ŝj1<latexit sha1_base64="PaZXLAVxOWxxcC7uvzVhCAKfu0M=">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</latexit>

we have proved lim limh
L"1 ↵#0(↵)
GS
L |ÛL |

GS
L i=

1

<latexit sha1_base64="eG0a6NVNzA0jEEWWEsoWpJRiai4=">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</latexit>

definition of the index Ind[! ] = lim lim h(1)↵#0 L"1(↵)
GS
L |ÛL |the assumption on the gap allows
us to exchange the limits!
lemma: for α ∈ [0,γ/18] and sufficiently large L<latexit sha1_base64="xPcGyBK7qRA0k7haTBUYI6FJA4c=">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</latexit>

h

(↵)
GS
L |ÛL |illustration by Chisato Naruse

GS
L i2[

1,

1
1
]
[
[
3
3 , 1]

GS
L i

summary
two conjectures about the S = 1 antiferromagnetic
Heisenberg chain<latexit sha1_base64="rpr0IgQtRvfmrDyBpill60/2oiw=">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</latexit>

Ĥ

(1)

=

P

j Ŝ j · Ŝ j+1(1) it has a unique gapped ground state(2) it belongs to a nontrivial SPT phase

we proved (2), assuming (1)the proof makes use of the elementary index
theory based on the t wist operator
it is desirable to prove also (1)it seems we need a new idea