Off-DiagonalLong Range Order
Implies
Vanishing Charge Gap
Hal Tasaki and Haruki Watanabe
brief introduction @ YouTube / May 2021

previously known for lattice bosons withhard core interaction (Tian 1992)

general inequality that relates the longrange order (LRO) or the off-diagonal longrange order (ODLRO) with the charge gap in
a quantum many-body systems with U(1)symmetry
applies to a very general class of systems
including interacting particles (bosons or
fermions) on lattice or in continuum, andvarious quantum spin systems
different from the Goldstone theorem

two typical applications
interacting bosons
magnetization plateau

interacting bosons

bosons in a box (continuum or lattice) with volume V<latexit sha1_base64="bh3Mm3phLGr2V6p/YY2ebeRbMr8=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKewGUY9BL54kgnlAsoTZySQZMju7zPQqYcl/ePGgqFf/xZt/4yTZgyYWNBRV3XR3BbEUBl3328mtrK6tb+Q3C1vbO7t7xf2DhokSzXidRTLSrYAaLoXidRQoeSvWnIaB5M1gdD31mw9cGxGpexzH3A/pQIm+YBStNIB3GAIFhBRuYdItltyyOwNZJl5GSpCh1i1+dXoRS0KukElqTNtzY/RTqlEwySeFTmJ4TNmIDnjbUkVDbvx0dvWEnFilR/qRtqWQzNTfEykNjRmHge0MKQ7NojcV//PaCfYv/VSoOEGu2HxRP5EEIzKNgPSE5gzl2BLKtLC3EjakmjK0QRVsCN7iy8ukUSl75+XK3VmpepXFkYcjOIZT8OACqnADNagDAw1P8AKvzqPz7Lw5H/PWnJPNHMIfOJ8//gOQPg==</latexit>N̂ total particle number operatorĤ non-pathological Hamiltonian such that [Ĥ, N̂ ] = 0<latexit sha1_base64="RzkRy2acJRsv0XyqfQCjnOwMYqk=">AAAB9XicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9Vj04rGC/YA2lM120y7dbMLuRCmh/8OLB0W9+l+8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXBtRKzucZxwP6IDJULBKFppAO8wBAoIGdzCpFcquxV3BrJMvJyUIUe9V/rq9mOWRlwhk9SYjucm6GdUo2CST4rd1PCEshEd8I6likbc+Nns6gk5tUqfhLG2pZDM1N8TGY2MGUeB7YwoDs2iNxX/8zophld+JlSSIldsvihMJcGYTCMgfaE5Qzm2hDIt7K2EDammDG1QRRuCt/jyMmlWK95FpXp3Xq5d53EU4BhO4Aw8uISajbMODWCg4Qle4NV5dJ6dN+dj3rri5DNH8AfO5w/035A4</latexit><latexit sha1_base64="z4OeJF8PcaknPeZ+uSWbfhZjDMg=">AAACD3icbVC7SgNBFL3rM8bXakqbwSBYSNgNojZC0MZKIpgHJEuYncwmQ2Znl5lZISz5Axs/wh+wsVDE1tbOv3GSbBETDwyce8693LnHjzlT2nF+rKXlldW19dxGfnNre2fX3tuvqyiRhNZIxCPZ9LGinAla00xz2owlxaHPacMfXI/9xgOVikXiXg9j6oW4J1jACNZGiuwCtOAZ+oBBQwo3MIKTmfrW1B5cgtOxi07JmQAtEjcjRchQ7djf7W5EkpAKTThWquU6sfZSLDUjnI7y7UTRGJMB7tGWoQKHVHnp5J4ROjJKFwWRNE9oNFFnJ1IcKjUMfdMZYt1X895Y/M9rJTq48FIm4kRTQaaLgoQjHaFxOKjLJCWaDw3BRDLzV0T6WGKiTYR5E4I7f/IiqZdL7lmpfHdarFxlceTgAA7hGFw4h4oJuQo1IPAIL/AG79aT9Wp9WJ/T1iUrmynAH1hfvxyHlcY=</latexit>

ground state |GSN i g.s. energy EN with N particles<latexit sha1_base64="TOMMwD1H8S/RbAR1smNcBgZYCFA=">AAACDXicbVA9SwNBEJ3zM8avM5Y2i0GwCndB1DJooZVENB+QHGFvs0mW7O0du3tCOPMHbPwV9jYWitja2/lv3Fyu0MQHA4/3ZpiZ50ecKe0439bC4tLyympuLb++sbm1be8U6iqMJaE1EvJQNn2sKGeC1jTTnDYjSXHgc9rwh+cTv3FHpWKhuNWjiHoB7gvWYwRrI4V2Ae4hgSeQEACCC7iBMXTgKlUwCOgDB9qxi07JSYHmiZuRImSoduyvdjckcUCFJhwr1XKdSHsJlpoRTsf5dqxohMkQ92nLUIEDqrwk/WaMDozSRb1QmhIaperviQQHSo0C33QGWA/UrDcR//Nase6degkTUaypINNFvZgjHaJJNKjLJCWajwzBRDJzKyIDLDHRJsC8CcGdfXme1Msl97hUvj4qVs6yOHKwB/twCC6cQAUuoQo1IPAAz/AKb9aj9WK9Wx/T1gUrm9mFP7A+fwCVhJWV</latexit><latexit sha1_base64="ACi1f7BhQyZMrUNaVAlo4x21768=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKewGUY9BETxJRPOAZAmzk0kyZHZ2mekVwpJP8OJBEa9+kTf/xkmyB00saCiquunuCmIpDLrut5NbWV1b38hvFra2d3b3ivsHDRMlmvE6i2SkWwE1XArF6yhQ8lasOQ0DyZvB6HrqN5+4NiJSjziOuR/SgRJ9wSha6eGme9ctltyyOwNZJl5GSpCh1i1+dXoRS0KukElqTNtzY/RTqlEwySeFTmJ4TNmIDnjbUkVDbvx0duqEnFilR/qRtqWQzNTfEykNjRmHge0MKQ7NojcV//PaCfYv/VSoOEGu2HxRP5EEIzL9m/SE5gzl2BLKtLC3EjakmjK06RRsCN7iy8ukUSl75+XK/VmpepXFkYcjOIZT8OACqnALNagDgwE8wyu8OdJ5cd6dj3lrzslmDuEPnM8f7C2NkQ==</latexit>

charge gap

nonzero charge gap means the system is insulating

1 @µ
2EN '
V @⇢

<latexit sha1_base64="ei0qMwAMIU+wwQj5udyLvZLj4CU=">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</latexit>

N = EN +1 + EN

1

const > 0

|GSN i has no ODLRO<latexit sha1_base64="TOMMwD1H8S/RbAR1smNcBgZYCFA=">AAACDXicbVA9SwNBEJ3zM8avM5Y2i0GwCndB1DJooZVENB+QHGFvs0mW7O0du3tCOPMHbPwV9jYWitja2/lv3Fyu0MQHA4/3ZpiZ50ecKe0439bC4tLyympuLb++sbm1be8U6iqMJaE1EvJQNn2sKGeC1jTTnDYjSXHgc9rwh+cTv3FHpWKhuNWjiHoB7gvWYwRrI4V2Ae4hgSeQEACCC7iBMXTgKlUwCOgDB9qxi07JSYHmiZuRImSoduyvdjckcUCFJhwr1XKdSHsJlpoRTsf5dqxohMkQ92nLUIEDqrwk/WaMDozSRb1QmhIaperviQQHSo0C33QGWA/UrDcR//Nase6degkTUaypINNFvZgjHaJJNKjLJCWajwzBRDJzKyIDLDHRJsC8CcGdfXme1Msl97hUvj4qVs6yOHKwB/twCC6cQAUuoQo1IPAAz/AKb9aj9WK9Wx/T1gUrm9mFP7A+fwCVhJWV</latexit><latexit sha1_base64="3wRnya+wdL7k5LWcLonyUkFzKyM=">AAACK3icbVDLSgMxFL1T3/VVdekmWgRXZUZEXUlRF65EwWqhLSWTpm0wkwxJRijDfI7gxl9xoQsfuPU/vFO7UOuFhJNzzuXmnjCWwjrff/MKE5NT0zOzc8X5hcWl5dLK6pXViWG8xrTUph5Sy6VQvOaEk7weG06jUPLr8OY4169vubFCq0s3iHkroj0luoJRh5QubcAdREDBQR8E3im+T4CDREwhgzacIdNDJlcMegkw0KDAoiODQ/DbpbJf8YdFxkEwAmUY1Xm79NTsaJZEXDkmqbWNwI9dK6XGCSZ5VmwmlseU3dAebyBUNOK2lQ53zcgWMh3S1QaPcmTI/uxIaWTtIArRGVHXt3+1nPxPaySue9BKhYoTxxX7HtRNJHGa5MGRjjCcOTlAQJkR+FfC+tRQ5jDeIoYQ/F15HFztVIK9ys7Fbrl6NIpjFtZhE7YhgH2owimcQw0DvodHeIFX78F79t69j29rwRv1rMGv8j6/ABJCnTc=</latexit>N<latexit sha1_base64="sQHguySIaKVql+vlvzjnZqgwzcQ=">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</latexit><latexit sha1_base64="NF2zVX9w2jGvcF9/r5YeIhRivEo=">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</latexit>|GSN i has ODLRO<latexit sha1_base64="TOMMwD1H8S/RbAR1smNcBgZYCFA=">AAACDXicbVA9SwNBEJ3zM8avM5Y2i0GwCndB1DJooZVENB+QHGFvs0mW7O0du3tCOPMHbPwV9jYWitja2/lv3Fyu0MQHA4/3ZpiZ50ecKe0439bC4tLyympuLb++sbm1be8U6iqMJaE1EvJQNn2sKGeC1jTTnDYjSXHgc9rwh+cTv3FHpWKhuNWjiHoB7gvWYwRrI4V2Ae4hgSeQEACCC7iBMXTgKlUwCOgDB9qxi07JSYHmiZuRImSoduyvdjckcUCFJhwr1XKdSHsJlpoRTsf5dqxohMkQ92nLUIEDqrwk/WaMDozSRb1QmhIaperviQQHSo0C33QGWA/UrDcR//Nase6degkTUaypINNFvZgjHaJJNKjLJCWajwzBRDJzKyIDLDHRJsC8CcGdfXme1Msl97hUvj4qVs6yOHKwB/twCC6cQAUuoQo1IPAAz/AKb9aj9WK9Wx/T1gUrm9mFP7A+fwCVhJWV</latexit>

const
N 
V

@⇢
@µ

const > 0

Bose-Einstein condensate is always “compressible”

interacting bosons<latexit sha1_base64="sQHguySIaKVql+vlvzjnZqgwzcQ=">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</latexit><latexit sha1_base64="NF2zVX9w2jGvcF9/r5YeIhRivEo=">AAACN3icbZDLSgMxFIbPeLfeqi7dBIvoqsyIqEtRF65EwV6gLSWTZtrQTDIkGaEM8zi+gRtfw51uXCji1jfw9LLwdiDhz3f+Q5I/TKSwzvefvKnpmdm5+YXFwtLyyupacX2janVqGK8wLbWph9RyKRSvOOEkryeG0ziUvBb2z4b92i03Vmh14wYJb8W0q0QkGHWIdHEX7iAGCg56IHDP8HwOHCRqCjm04RKJRHIHERhkbOQxOEVQa1Bg0ZsjrULeLpb8sj8q8lcEE1GCSV21i4/NjmZpzJVjklrbCPzEtTJqnGCS54VmanlCWZ92eQOlojG3rWz075zsIOmQSBtcypER/T6R0djaQRyiM6auZ3/3hvC/XiN10XErEypJHVdsfFGUSuI0GYZIOsJw5uQABWVG4FsJ61FDmcOoCxhC8PvLf0V1vxwclvevD0onp5M4FmALtmEPAjiCE7iAK6hgxPfwDK/w5j14L9679zG2TnmTmU34Ud7nF06zoM8=</latexit>|GSN i has ODLRO<latexit sha1_base64="TOMMwD1H8S/RbAR1smNcBgZYCFA=">AAACDXicbVA9SwNBEJ3zM8avM5Y2i0GwCndB1DJooZVENB+QHGFvs0mW7O0du3tCOPMHbPwV9jYWitja2/lv3Fyu0MQHA4/3ZpiZ50ecKe0439bC4tLyympuLb++sbm1be8U6iqMJaE1EvJQNn2sKGeC1jTTnDYjSXHgc9rwh+cTv3FHpWKhuNWjiHoB7gvWYwRrI4V2Ae4hgSeQEACCC7iBMXTgKlUwCOgDB9qxi07JSYHmiZuRImSoduyvdjckcUCFJhwr1XKdSHsJlpoRTsf5dqxohMkQ92nLUIEDqrwk/WaMDozSRb1QmhIaperviQQHSo0C33QGWA/UrDcR//Nase6degkTUaypINNFvZgjHaJJNKjLJCWajwzBRDJzKyIDLDHRJsC8CcGdfXme1Msl97hUvj4qVs6yOHKwB/twCC6cQAUuoQo1IPAAz/AKb9aj9WK9Wx/T1gUrm9mFP7A+fwCVhJWV</latexit>

const
N 
V

@⇢
@µ

const > 0

Bose-Einstein condensate is always “compressible”
interesting applicationcommensurate supersolid

example: soft-core Bose-Hubbard model with nearest-neighbor
repulsion on the cubic lattice(number of sites)<latexit sha1_base64="Murqkm5MwHrzvTH8hlrUkfHO3Dk=">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</latexit>N=<latexit sha1_base64="leHCf87FfAqOCibwQKdqvMIaB8w=">AAACLXicbVHLSgMxFL3js9bXqEs3oUVwVWaKqBuh6MZlBfuAtpRMmmlDk8mQZIQy9HPcuPFXRHBREbf+hmk7C217IOTknHtJ7kkQc6aN502ctfWNza3t3E5+d2//4NA9Oq5rmShCa0RyqZoB1pSziNYMM5w2Y0WxCDhtBMO7qd94okozGT2aUUw7AvcjFjKCjZWkW4BnCEEBBgKp5bFlCgwwu3N7VjAACeOVnoDEOjfgdd2iV/JmQMvEz0gRMlS77nu7J0kiaGQIx1q3fC82nRQrwwin43w70TTGZIj7tGVphAXVnXQ27RidWaWHQqnsigyaqX87Uiy0HonAVgpsBnrRm4qrvFZiwutOyqI4MTQi84vChCMj0TQ61GOKEsNHlmCimH0rIgOsMDE24LwNwV8ceZnUyyX/slR+uChWbrM4cnAKBTgHH66gAvdQhZr9jBd4gwl8Oq/Oh/PlfM9L15ys5wT+wfn5BWFgnn8=</latexit>

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re
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Ohgoe, Suzuki, Kawashima 20122@⇢
Mott insulator phase=0
broken translation symmetry@µno BEC<latexit sha1_base64="WicHk9OMHzmATLj+XAmGbwWOyAE=">AAACLXicbVHLSgMxFL3js9bXqEs3oUVwVWaKqCspunFZwT6gLSWTZtrQZDIkGaEM/Rw3bvwVEVxUxK2/YdrOQtseCDk5516SexLEnGnjeRNnbX1jc2s7t5Pf3ds/OHSPjutaJorQGpFcqmaANeUsojXDDKfNWFEsAk4bwfBu6jeeqNJMRo9mFNOOwP2IhYxgYyXpFuAZQlCAgUBqeWyZAgPM7tyeFQxAwnilJyCxzg14XbfolbwZ0DLxM1KEDNWu+97uSZIIGhnCsdYt34tNJ8XKMMLpON9ONI0xGeI+bVkaYUF1J51NO0ZnVumhUCq7IoNm6t+OFAutRyKwlQKbgV70puIqr5WY8LqTsihODI3I/KIw4chINI0O9ZiixPCRJZgoZt+KyAArTIwNOG9D8BdHXib1csm/LJUfLoqV2yyOHJxCAc7BhyuowD1UoWY/4wXeYAKfzqvz4Xw53/PSNSfrOYF/cH5+AWLmnoA=</latexit>@⇢> 0 supersolid phase@µbroken translation symmetry
BEC

magnetization plateau
quantum spin system on a lattice with N sites<latexit sha1_base64="JH7P9nCEXH4ZNdiBDNdXGphxihU=">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</latexit><latexit sha1_base64="vqoJg5rD6420ZIvmomNW49/ZaIo=">AAAB+XicbVDLSgNBEOz1GeMrKnjxMhgET2E3B/UY4sVjAuYByRJmJ5NkyOzsMtMbCEv+xIsHRQRP/oJf4M2L3+LkcdDEgoaiqpvuriCWwqDrfjlr6xubW9uZnezu3v7BYe7ouG6iRDNeY5GMdDOghkuheA0FSt6MNadhIHkjGN5O/caIayMidY/jmPsh7SvRE4yilQQ8wQAoIKRwBxPogNvJ5d2COwNZJd6C5Eun1W/xVv6odHKf7W7EkpArZJIa0/LcGP2UahRM8km2nRgeUzakfd6yVNGQGz+dXT4hF1bpkl6kbSkkM/X3REpDY8ZhYDtDigOz7E3F/7xWgr0bPxUqTpArNl/USyTBiExjIF2hOUM5toQyLeythA2opgxtWFkbgrf88iqpFwveVaFY9fKlMsyRgTM4h0vw4BpKNtIK1IDBCB5s0M9O6jw6L87rvHXNWcycwB847z9K85SL</latexit>z
Ĥ0 non-pathological Hamiltonian such that [Ĥ0 , Ŝtot ] = 0
P
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j j
|GSh izĤh = Ĥ0 h Ŝtot
M (h)<latexit sha1_base64="BJA+OgsxpuQcBr3s1355XJ3ZjDE=">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</latexit>

ground state of<latexit sha1_base64="xqDqCB0ubX43COYjMDchSuU05gg=">AAACDXicbVA9SwNBEJ3zM8avM5Y2S4IgCOEuhVoGLbSMaD4gOcLeZpMs2ds7dveEcOYP2Nj4F+xtLBSxtbfLv3FzSaGJDwYe780wM8+POFPaccbW0vLK6tp6ZiO7ubW9s2vv5WoqjCWhVRLyUDZ8rChnglY105w2Iklx4HNa9wcXE79+R6ViobjVw4h6Ae4J1mUEayOFdg7uIYFnkBAAgku4gRG0oZ8qGAT0gANt2wWn6KRAi8SdkUI53zp+GpeHlbb93eqEJA6o0IRjpZquE2kvwVIzwuko24oVjTAZ4B5tGipwQJWXpN+M0KFROqgbSlNCo1T9PZHgQKlh4JvOAOu+mvcm4n9eM9bdMy9hIoo1FWS6qBtzpEM0iQZ1mKRE86EhmEhmbkWkjyUm2gSYNSG48y8vklqp6J4US9duoXwOU2TgAPJwBC6cQhmuoAJVIPAAL/AG79aj9Wp9WJ/T1iVrNrMPf2B9/QDZUZi7</latexit><latexit sha1_base64="n6vvQJ9JI/xinDNfrfeKb8z46Wk=">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</latexit><latexit sha1_base64="kRYIKtwlKTN92f4vxEA12LeRd38=">AAAB63icbVDLSgNBEOyNryS+ol4EL4tBiJewm4PmGPTiRYhgHpAsYXYymwyZmV1mZoW4BPwCLx4U8eoP+CneBD/G2SQHTSxoKKq66e7yI0aVdpwvK7Oyura+kc3lN7e2d3YLe/tNFcYSkwYOWSjbPlKEUUEammpG2pEkiPuMtPzRZeq37ohUNBS3ehwRj6OBoAHFSKfSdWl42isUnbIzhb1M3Dkp1g7vv3MPHxf1XuGz2w9xzInQmCGlOq4TaS9BUlPMyCTfjRWJEB6hAekYKhAnykumt07sE6P07SCUpoS2p+rviQRxpcbcN50c6aFa9FLxP68T66DqJVREsSYCzxYFMbN1aKeP230qCdZsbAjCkppbbTxEEmFt4smbENzFl5dJs1J2z8qVG5NGFWbIwhEcQwlcOIcaXEEdGoBhCI/wDC8Wt56sV+tt1pqx5jMH8AfW+w9uWpDM</latexit><latexit sha1_base64="ZNrvE7J3dCU4RAjKlHBF3uEg6f0=">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</latexit>z
Ŝtot |GSh i = M (h)|GSh ih<latexit sha1_base64="F4wbJfTWIbdtD1Xh0Cr71cA0ims=">AAAB6HicbZC7SgNBFIbPeo3xFrUUZDEIVmE3haYzYGOZgLlAsoTZydlkzOzsMjMrhCWllY2FIrY+RSofws5n8CWcXApN/GHg4//PYc45fsyZ0o7zZa2srq1vbGa2sts7u3v7uYPDuooSSbFGIx7Jpk8Uciawppnm2IwlktDn2PAH15O8cY9SsUjc6mGMXkh6ggWMEm2sar+TyzsFZyp7Gdw55K8+xtXvh5NxpZP7bHcjmoQoNOVEqZbrxNpLidSMchxl24nCmNAB6WHLoCAhKi+dDjqyz4zTtYNImie0PXV/d6QkVGoY+qYyJLqvFrOJ+V/WSnRQ8lIm4kSjoLOPgoTbOrInW9tdJpFqPjRAqGRmVpv2iSRUm9tkzRHcxZWXoV4suBeFYtXJl0swUwaO4RTOwYVLKMMNVKAGFBAe4RlerDvryXq13malK9a85wj+yHr/AZSJkTE=</latexit>

when h is within a magnetization plateau
X<latexit sha1_base64="9OG1cmi6wsRd8/jJamzZoqsN64M=">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</latexit>

j,k

y y
x x
⇣j ⇣k hGSh |(Ŝj Ŝk + Ŝj Ŝk )|GSh i  const N

for any ⇣j 2 R with |⇣j |  1<latexit sha1_base64="WXAwcFr1CcqkB5xE5tdM6OJx7Bw=">AAACAXicbVC7TgJBFL3rE8EHamkzEU2syC6FWhJtLDGRRwKEzA6zMDI7u5mZJcEVG39FCwuNsbCx9A/s/BCtHRYKBU8yybnn3Js797ghZ0rb9qc1N7+wuLScWklnVtfWN7KbWxUVRJLQMgl4IGsuVpQzQcuaaU5roaTYdzmtur3TkV/tU6lYIC70IKRNH3cE8xjB2kg+XMM9XAEFDRhacJnU3NROK5uz83YCNEucCckV975e3/uZ71Ir+9FoByTyqdCEY6Xqjh3qZoylZoTTYboRKRpi0sMdWjdUYJ+qZpxcMET7RmkjL5DmCY0S9fdEjH2lBr5rOn2su2raG4n/efVIe8fNmIkw0lSQ8SIv4kgHaBQHajNJieYDQzCRzPwVkS6WmGgTWtqE4EyfPEsqhbxzmC+cmzROYIwU7MAuHIADR1CEMyhBGQjcwB08wpN1az1Yz9bLuHXOmsxswx9Ybz+Y75eO</latexit><latexit sha1_base64="iWYLyFt8yFgL7RzDUGyhIMJsHo8=">AAACEXicbVA9T8MwEL2Ur1K+ysfGElGQmKqkAzBWsDAWRAGpRJXjOq2p40T2pVKJ+hdY+A38A4TEAEIwsrHxQ2DGbRmg5Syf3r13J/ueHwuu0XE+rMzE5NT0THY2Nze/sLiUX1451VGiKKvSSETq3CeaCS5ZFTkKdh4rRkJfsDO/fdDXzzpMaR7JE+zGzAtJU/KAU4KGivJrcAtXwACBQB0uTcVBmhyaGqEFvjkpHEOvni84RWcQ9jhwf0ChvPl5/9qZ+6rU8+8XjYgmIZNIBdG65joxeilRyKlgvdxFollMaJs0Wc1ASUKmvXSwUc/eMkzDDiJlrkR7wP6eSEmodTf0TWdIsKVHtT75n1ZLMNjzUi7jBJmkw4eCRNgY2X177AZXjKLoGkCo4uavNm0RRSgaE3PGBHd05XFwWiq6O8XSkXFjH4aRhXXYgG1wYRfKcAgVqAKFa7iDR3iybqwH69l6GbZmrJ+ZVfgT1ts3Ww2bfg==</latexit>

no long-range order in the xy directions

detailsbosons in a boxsummary
inequality that relates the long range order (LRO) or
the off-diagonal long range order (ODLRO) with the
charge gap in a large class of quantum many-bodysystems with U(1) symmetry
previously known for lattice bosons withhard core interaction (Tian 1992)

bosons or fermions on lattice or in continuum
if the charge gap is nonzero there is no ODLRO
if there is ODLRO the charge gap is vanishing andthe ground state is “compressible”
quantum spin systems
a ground state within a magnetization plateau
does not have transverse LRO