Griffiths-type theorems
for short-range
spin glass models
Chigak Itoi, Hisamitsu Mukaida, and Hal Tasaki
webinar @ YouTube / 2023

introduction
definitions and main results
discussion

Griffiths’ theorem for the
ferromagnetic Ising model

Ising model on the L × ⋯ × L hypercubic latticeP
Hamiltonian HL ( ) =hx,yi x y<latexit sha1_base64="yt0DUbEvxJN9rEdfyh7MjlCWi1g=">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</latexit>

two ways of characterizing theqferromagnetic order<latexit sha1_base64="/cnA1WlrhPiocmf+jA4mEljP598=">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</latexit>

long-range orderµLRO = limL"1

L

P
2d

x,y h x y iL,>0correlation ⟨σxσy⟩β does not vanish as | x − y | ↑ ∞<latexit sha1_base64="fxJFIBV20t6NScjiNxsuPs2RGQE=">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</latexit>

spontaneous magnetization µSM =@f ( ,h)limh#0 @h > 0

the free energy is non-differentiable at h = 0
in nitesimal magnetic eld causes nonzero magnetization
spontaneous symmetry breaking

THEOREM: (Grif ths 1966) µSM
fi

fi

fi

<latexit sha1_base64="wFESj8/n61CNLzSYjRFu3sSVy+M=">AAACIXicbZDLSgMxFIbPeK31VnVpkWARXJUZwcuy6MaFYqv2Au1QMmmmDc1cSDJCGbr0NdyI79AHcONCke7ER/AlTC8LbftD4Mv/n0NyjhNyJpVpfhlz8wuLS8uJleTq2vrGZmpruySDSBBaJAEPRMXBknLm06JiitNKKCj2HE7LTvtikJcfqJAs8O9VJ6S2h5s+cxnBSltBKg0v4EEEdYg1Cc0I7uAauvrWBDojvYJbuIFuPZUxs+ZQaBqsMWRy6V7h53Gvl6+n+rVGQCKP+opwLGXVMkNlx1goRjjtJmuRpCEmbdykVY0+9qi04+GEXXSgnQZyA6GPr9DQ/dsRY0/KjufoSg+rlpzMBuasrBop98yOmR9Givpk9JAbcaQCNFgXajBBieIdDZgIpv+KSAsLTJRealIvwZoceRpKR1nrJHtcsDK5cxgpAbuwD4dgwSnk4BLyUAQCT/AK7/BhPBtvxqfRH5XOGeOeHfgn4/sXCLyegw==</latexit>µLRO

different characterizations of spin glass orderbroadening of the overlap distributionq22
qbr = hR i
hRi > 0<latexit sha1_base64="gzkNSZITZwb/IDUGVVtubiHv7Cc=">AAAB+3icbZDJSgNBEIZr4hbjFvXopTEIegkz4nYMevEYwSyQDKGn05O06VnsrhHDkFfxIqKIV1/Em29jJ5mDJv7Q8PNXFVX9ebEUGm3728otLC4tr+RXC2vrG5tbxe2duo4SxXiNRTJSTY9qLkXIayhQ8masOA08yRve4GpcbzxwpUUU3uIw5m5Ae6HwBaNoojuoQgdewAMOCBQO4R6OOsWSXbYnIvPGyUwJMlU7xa92N2JJwENkkmrdcuwY3ZQqFEzyUaGdaB5TNqA93jI2pAHXbjq5fUQOTNIlfqTMC5FM0t8TKQ20Hgae6Qwo9vVsbRz+V2sl6F+4qQjjBHnIpov8RBKMyBgE6QrFGcqhMZQpYW4lrE8VZWhwFQwEZ/bL86Z+XHbOyqc3J6XKZYYjD3uwb0A6cA4VuDaAa8DgEZ7gFd6skfVsvVsf09aclc3swh9Znz+6M5EE</latexit>

P (q)<latexit sha1_base64="0v61arR599GWo7pQERBS52MFk5c=">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</latexit>q<latexit sha1_base64="iVGWJjUY9T0HmMOpn/tMDZu36y8=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPRi0dI5JHAhswODYzMzq4zsyZkwxd48aAxXv0kb/6NA+xBwUo6qVR1p7sriAXXxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRDOssEpFqBVSj4BLrhhuBrVghDQOBzWB0O/WbT6g0j+S9Gcfoh3QgeZ8zaqxUe+wWS27ZnYEsEy8jJchQ7Ra/Or2IJSFKwwTVuu25sfFTqgxnAieFTqIxpmxEB9i2VNIQtZ/ODp2QE6v0SD9StqQhM/X3REpDrcdhYDtDaoZ60ZuK/3ntxPSv/ZTLODEo2XxRPxHERGT6NelxhcyIsSWUKW5vJWxIFWXGZlOwIXiLLy+TxlnZuyxf1M5LlZssjjwcwTGcggdXUIE7qEIdGCA8wyu8OQ/Oi/PufMxbc042cwh/4Hz+AN7djP8=</latexit>

non-differentiability of the 2-replica free energy<latexit sha1_base64="86EfpwcyGoZY7ZRV7udcTyM5rq4=">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</latexit>

1
2

qjump =
<latexit sha1_base64="OTG0Vn5cdgx6/i671SyH938vDjc=">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</latexit>@f (2) ( , )
lim
@
#0

= 12 hRi+

hRi@f (2) ( , )
lim
@
"0

<latexit sha1_base64="xnPa8O65Z2wR+DmUzGV9+K2BcEw=">AAACFXicbVDLSsNAFL2pr1pfUcGNm2ARLEhJio8uC25cVrAPaGOZTCbt0EkmzEyEEvoTbvwC/8GNC0XcCu78G6dtFtp6YTiHc+7lzj1ezKhUtv1t5JaWV1bX8uuFjc2t7R1zd68peSIwaWDOuGh7SBJGI9JQVDHSjgVBocdIyxteTfzWPRGS8uhWjWLihqgf0YBipLTEzQMI4A5SOIEKlGCs8Qk8IKAAwanmTGOoFV9jqWcW7bI9LWuROBkpQlb1nvnV9TlOQhIpzJCUHceOlZsioShmZFzoJpLECA9Rn3Q0jVBIpJtOrxpbx1rxrYAL/SJlTdXfEykKpRyFnu4MkRrIeW8i/ud1EhVU3ZRGcaJIhGeLgoRZiluTiCyfCoIVG2mCsKD6rxYeIIGw0kEWdAjO/MmLpFkpOxfl85uzYq2axZGHQzjSATtwCTW4hjo0AMMDPMMrvBmPxovxbnzMWnNGNrMPf8r4/AGmEpZg</latexit>f (2) ( , )<latexit sha1_base64="MF2y1bujfBMH8UBCe9Rk+1r/w1g=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hV3xkWPAi8cI5gHJEnpnJ8mQ2dllZlYJS/7DiwdFvfov3vwbJ8keNLGgoajqprsrSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqKGvQWMSqHaBmgkvWMNwI1k4UwygQrBWMbqZ+64EpzWN5b8YJ8yMcSN7nFI2VBvAOAhAiCCAE7JXKbsWdgSwTLydlyFHvlb66YUzTiElDBWrd8dzE+Bkqw6lgk2I31SxBOsIB61gqMWLaz2ZXT8ipVULSj5UtachM/T2RYaT1OApsZ4RmqBe9qfif10lNv+pnXCapYZLOF/VTQUxMphGQkCtGjRhbglRxeyuhQ1RIjQ2qaEPwFl9eJs3zindVuby7KNeqeRwFOIYTOAMPrqEGt1CHBlBQ8AQv8Oo8Os/Om/Mxb11x8pkj+APn8wfINZAV</latexit>>0<latexit sha1_base64="DvNDOZ0Af0ST7xXlMbPqjLxv6ik=">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</latexit>

qjump = qEA = hRi+ > 0 models without magnetic eldliteral breakdown of replica symmetry in the 3-replica system
2<latexit sha1_base64="F8FkEfYKFbWhX6CeAoaLBDiUGi8=">AAAB6HicbZC7SgNBFIbPxluMt6ilIItBsAq7ATWdARvLBMwFkhBmJ2eTMbOzy8ysEJaUVjYWitj6FKl8CDufwZdwcik08YeBj/8/hznneBFnSjvOl5VaWV1b30hvZra2d3b3svsHNRXGkmKVhjyUDY8o5ExgVTPNsRFJJIHHse4Nrid5/R6lYqG41cMI2wHpCeYzSrSxKoVONufknansZXDnkLv6GFe+H47H5U72s9UNaRyg0JQTpZquE+l2QqRmlOMo04oVRoQOSA+bBgUJULWT6aAj+9Q4XdsPpXlC21P3d0dCAqWGgWcqA6L7ajGbmP9lzVj7xXbCRBRrFHT2kR9zW4f2ZGu7yyRSzYcGCJXMzGrTPpGEanObjDmCu7jyMtQKefcif15xcqUizJSGIziBM3DhEkpwA2WoAgWER3iGF+vOerJerbdZacqa9xzCH1nvP0OnkP4=</latexit><latexit sha1_base64="GKpMOa48wyXzTMv8KvWDhAFv2/E=">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</latexit>

qrsb =

lim

@f

#0

(3)

( , ,
@

)

+0
<latexit sha1_base64="SvofpUwiUNbp8sHlQWStUDsAqE8=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoMgCGFXUHMMeBG8RDEPSJYwO+mNQ2Znl5lZIYT8gRcPinj1D/wUycUP8e7kcdDEgoaiqpvuriARXBvX/XIyS8srq2vZ9dzG5tb2Tn53r6bjVDGssljEqhFQjYJLrBpuBDYShTQKBNaD3uXYrz+g0jyWd6afoB/RruQhZ9RY6fbEbecLbtGdgCwSb0YKZfj+GI0y15V2/rPViVkaoTRMUK2bnpsYf0CV4UzgMNdKNSaU9WgXm5ZKGqH2B5NLh+TIKh0SxsqWNGSi/p4Y0EjrfhTYzoiaez3vjcX/vGZqwpI/4DJJDUo2XRSmgpiYjN8mHa6QGdG3hDLF7a2E3VNFmbHh5GwI3vzLi6R2WvTOi2c3No0STJGFAziEY/DgAspwBRWoAoMQHuEZXpye8+S8Om/T1owzm9mHP3DefwAQL5DB</latexit>

1<latexit sha1_base64="XH3XOuLVabmTmD3e9N+w6jcfCp0=">AAAB6HicbZC7SgNBFIbPxluMt6ilIItBsAq7gprOgI1lAuYCyRJmJ2eTMbOzy8ysEJaUVjYWitj6FKl8CDufwZdwcik0+sPAx/+fw5xz/JgzpR3n08osLa+srmXXcxubW9s7+d29uooSSbFGIx7Jpk8Uciawppnm2IwlktDn2PAHV5O8cYdSsUjc6GGMXkh6ggWMEm2sqtvJF5yiM5X9F9w5FC7fx9Wv+8NxpZP/aHcjmoQoNOVEqZbrxNpLidSMchzl2onCmNAB6WHLoCAhKi+dDjqyj43TtYNImie0PXV/dqQkVGoY+qYyJLqvFrOJ+V/WSnRQ8lIm4kSjoLOPgoTbOrInW9tdJpFqPjRAqGRmVpv2iSRUm9vkzBHcxZX/Qv206J4Xz6pOoVyCmbJwAEdwAi5cQBmuoQI1oIDwAE/wbN1aj9aL9TorzVjznn34JevtG0IjkP0=</latexit>

3<latexit sha1_base64="KDHue//9yManPttmBaUs3fpk6Q8=">AAAB6HicbZC7SgNBFIbPxluMt6ilIItBsAq7iprOgI1lAuYCyRJmJyfJmNnZZWZWCEtKKxsLRWx9ilQ+hJ3P4Es4uRSa+MPAx/+fw5xz/IgzpR3ny0otLa+srqXXMxubW9s72d29qgpjSbFCQx7Kuk8UciawopnmWI8kksDnWPP71+O8do9SsVDc6kGEXkC6gnUYJdpY5bNWNufknYnsRXBnkLv6GJW/Hw5HpVb2s9kOaRyg0JQTpRquE2kvIVIzynGYacYKI0L7pIsNg4IEqLxkMujQPjZO2+6E0jyh7Yn7uyMhgVKDwDeVAdE9NZ+Nzf+yRqw7BS9hIoo1Cjr9qBNzW4f2eGu7zSRSzQcGCJXMzGrTHpGEanObjDmCO7/yIlRP8+5F/rzs5IoFmCoNB3AEJ+DCJRThBkpQAQoIj/AML9ad9WS9Wm/T0pQ169mHP7LefwBFK5D/</latexit><latexit sha1_base64="2Kw3m4WDnX0FjmpFlEoNMIcrIdw=">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</latexit>

hR13 i > 0fi= hR12 i<latexit sha1_base64="xeNRz3KNObhmstAvgI8O1KjSZY8=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoPgxbArqDkGvAheopgHJEuYnfTGIbOzy8ysEEL+wIsHRbz6B36K5OKHeHfyOGhiQUNR1U13V5AIro3rfjmZpeWV1bXsem5jc2t7J7+7V9NxqhhWWSxi1QioRsElVg03AhuJQhoFAutB73Ls1x9QaR7LO9NP0I9oV/KQM2qsdHvitvMFt+hOQBaJNyOFMnx/jEaZ60o7/9nqxCyNUBomqNZNz02MP6DKcCZwmGulGhPKerSLTUsljVD7g8mlQ3JklQ4JY2VLGjJRf08MaKR1PwpsZ0TNvZ73xuJ/XjM1YckfcJmkBiWbLgpTQUxMxm+TDlfIjOhbQpni9lbC7qmizNhwcjYEb/7lRVI7LXrnxbMbm0YJpsjCARzCMXhwAWW4ggpUgUEIj/AML07PeXJenbdpa8aZzezDHzjvPxM5kMM=</latexit>

0introduction
definitions and main results
discussion

Edwards-Anderson (EA) model with aperiodic b.c. (in this video)
magnetic fieldΛL : d-dimensional L × ⋯ × L hypercubic lattice
σx = ± 1 : spin on x ∈ ΛL σ = (σx)x∈ΛL : spin con guration
P
P
Hamiltonian HL ( ) =hx,yi Jxy x yx2⇤L hx x<latexit sha1_base64="AHcFaN6+lhYmWpwYPa0a4Uf3Q3w=">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</latexit>

nearest neighbor pairs in ΛL

Jxy = Jyx 2 R random interaction (i.i.d.)hx 2 R random (or non-random) magnetic eld (i.i.d.)<latexit sha1_base64="5WZdyyk5GZ9CARmtiIFAK1BCsXA=">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</latexit><latexit sha1_base64="9VhSljllCg7VyOJaO8Niki41Ncc=">AAACCXicbVC7TsMwFL3hWcorgMTCYlEhMVUJEo+xKgtji+hDaqPKcZ3WquNEtoOooq4sfAcbDAwgxApfwMbCt+A+Bmg5lq+OzrlX9j1+zJnSjvNlzc0vLC4tZ1ayq2vrG5v21nZVRYkktEIiHsm6jxXlTNCKZprTeiwpDn1Oa37vfOjXrqlULBJXuh9TL8QdwQJGsDZSZNvQhRbcwD0wEKaGgEEbzTcnhUsYtOyck3dGQLPEnZBcYbf8zR6LH6WW/dlsRyQJqdCEY6UarhNrL8VSM8LpINtMFI0x6eEObRgqcEiVl442GaADo7RREElzhUYj9fdEikOl+qFvOkOsu2raG4r/eY1EB2deykScaCrI+KEg4UhHaBgLajNJieZ9QzCRzPwVkS6WmGgTXtaE4E6vPEuqR3n3JH9cdnOFIoyRgT3Yh0Nw4RQKcAElqACBW3iAZ3ix7qwn69V6G7fOWZOZHfgD6/0HNgGYqw==</latexit>

fi

fi

fi

the equilibrium state at β in the limit L ↑ ∞standard EA model ( hx = 0 for all x )may exhibit a spin glass phase if d ≥ 3
EA model with a magnetic eld
not known if it exhibits a spin glass phase in d = 3
it probably has a spin glass phase if d > 6

Edwards-Anderson (EA) model with a
we
do
not
discuss
theexistenceortheperiodicb.c.
(in this video)
magnetic field

nature
of
the
spin
glass
phase,butstudy
ΛL : d-dimensional L × ⋯ × L hypercubic latticetherelations
betweendifferent±
σx = 1 : spin on x ∈ ΛL σ = (σx)x∈ΛL : spin con guration
Pof “spin glassPorder”
characterizations
JhHamiltonian H ( ) =<latexit sha1_base64="AHcFaN6+lhYmWpwYPa0a4Uf3Q3w=">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</latexit>

L

hx,yi

xy x yx2⇤Lx xnearest neighbor pairs in ΛL

Jxy = Jyx 2 R random interaction (i.i.d.)hx 2 R random (or non-random)
but magnetic
the nature ofeld
the(i.i.d.)phasethe equilibrium state at β in the limit
L ↑is∞still
controversial!
<latexit sha1_base64="5WZdyyk5GZ9CARmtiIFAK1BCsXA=">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</latexit><latexit sha1_base64="9VhSljllCg7VyOJaO8Niki41Ncc=">AAACCXicbVC7TsMwFL3hWcorgMTCYlEhMVUJEo+xKgtji+hDaqPKcZ3WquNEtoOooq4sfAcbDAwgxApfwMbCt+A+Bmg5lq+OzrlX9j1+zJnSjvNlzc0vLC4tZ1ayq2vrG5v21nZVRYkktEIiHsm6jxXlTNCKZprTeiwpDn1Oa37vfOjXrqlULBJXuh9TL8QdwQJGsDZSZNvQhRbcwD0wEKaGgEEbzTcnhUsYtOyck3dGQLPEnZBcYbf8zR6LH6WW/dlsRyQJqdCEY6UarhNrL8VSM8LpINtMFI0x6eEObRgqcEiVl442GaADo7RREElzhUYj9fdEikOl+qFvOkOsu2raG4r/eY1EB2deykScaCrI+KEg4UhHaBgLajNJieZ9QzCRzPwVkS6WmGgTXtaE4E6vPEuqR3n3JH9cdnOFIoyRgT3Yh0Nw4RQKcAElqACBW3iAZ3ix7qwn69V6G7fOWZOZHfgD6/0HNgGYqw==</latexit>

fi

fi

fi

standard AE model ( hx = 0 for all x )
this
is
also
may exhibit a spin glass phase if d ≥ 3controversial!
AE model with a magnetic eld
not known if it exhibits a spin glass phase in d = 3
it probably has a spin glass phase if d > 6

replica overlap and the broadening
order parameter qbr
expectation value for t wo independent replicas<latexit sha1_base64="U9BBB2Cf0OFT3rzXlvj8j1ZUoYE=">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</latexit>(2)h· · · iL,

=

X
1,

(· · · )2ereplica overlap R = L<latexit sha1_base64="4Y3tDbubMspdttO4bhftIxdrjVY=">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</latexit>

1

HL (

)

e

2

HL (

<latexit sha1_base64="EfKohJ8dKtAqm9tlcDZWnbhmu+s=">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</latexit>)<latexit sha1_base64="gqeqBw5ES/RsQ9/tSCrBXKhQyTc=">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</latexit>

ZL ( )ZL ( )
P
d
1 2
x x

x

1

=(

2

=(

1
x )x2⇤L2x )x2⇤L

random average<latexit sha1_base64="8Os6CYIYmq2UwNNndUx8/Uhnh9c=">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</latexit>

replica overlap distribution P (q) = limL"1 h (R
no
order
some
order
P (q)
P
(q)
P (q)

(2)
q)iL,

<latexit sha1_base64="gzkNSZITZwb/IDUGVVtubiHv7Cc=">AAAB+3icbZDJSgNBEIZr4hbjFvXopTEIegkz4nYMevEYwSyQDKGn05O06VnsrhHDkFfxIqKIV1/Em29jJ5mDJv7Q8PNXFVX9ebEUGm3728otLC4tr+RXC2vrG5tbxe2duo4SxXiNRTJSTY9qLkXIayhQ8masOA08yRve4GpcbzxwpUUU3uIw5m5Ae6HwBaNoojuoQgdewAMOCBQO4R6OOsWSXbYnIvPGyUwJMlU7xa92N2JJwENkkmrdcuwY3ZQqFEzyUaGdaB5TNqA93jI2pAHXbjq5fUQOTNIlfqTMC5FM0t8TKQ20Hgae6Qwo9vVsbRz+V2sl6F+4qQjjBHnIpov8RBKMyBgE6QrFGcqhMZQpYW4lrE8VZWhwFQwEZ/bL86Z+XHbOyqc3J6XKZYYjD3uwb0A6cA4VuDaAa8DgEZ7gFd6skfVsvVsf09aclc3swh9Znz+6M5EE</latexit><latexit sha1_base64="gzkNSZITZwb/IDUGVVtubiHv7Cc=">AAAB+3icbZDJSgNBEIZr4hbjFvXopTEIegkz4nYMevEYwSyQDKGn05O06VnsrhHDkFfxIqKIV1/Em29jJ5mDJv7Q8PNXFVX9ebEUGm3728otLC4tr+RXC2vrG5tbxe2duo4SxXiNRTJSTY9qLkXIayhQ8masOA08yRve4GpcbzxwpUUU3uIw5m5Ae6HwBaNoojuoQgdewAMOCBQO4R6OOsWSXbYnIvPGyUwJMlU7xa92N2JJwENkkmrdcuwY3ZQqFEzyUaGdaB5TNqA93jI2pAHXbjq5fUQOTNIlfqTMC5FM0t8TKQ20Hgae6Qwo9vVsbRz+V2sl6F+4qQjjBHnIpov8RBKMyBgE6QrFGcqhMZQpYW4lrE8VZWhwFQwEZ/bL86Z+XHbOyqc3J6XKZYYjD3uwb0A6cA4VuDaAa8DgEZ7gFd6skfVsvVsf09aclc3swh9Znz+6M5EE</latexit><latexit sha1_base64="gzkNSZITZwb/IDUGVVtubiHv7Cc=">AAAB+3icbZDJSgNBEIZr4hbjFvXopTEIegkz4nYMevEYwSyQDKGn05O06VnsrhHDkFfxIqKIV1/Em29jJ5mDJv7Q8PNXFVX9ebEUGm3728otLC4tr+RXC2vrG5tbxe2duo4SxXiNRTJSTY9qLkXIayhQ8masOA08yRve4GpcbzxwpUUU3uIw5m5Ae6HwBaNoojuoQgdewAMOCBQO4R6OOsWSXbYnIvPGyUwJMlU7xa92N2JJwENkkmrdcuwY3ZQqFEzyUaGdaB5TNqA93jI2pAHXbjq5fUQOTNIlfqTMC5FM0t8TKQ20Hgae6Qwo9vVsbRz+V2sl6F+4qQjjBHnIpov8RBKMyBgE6QrFGcqhMZQpYW4lrE8VZWhwFQwEZ/bL86Z+XHbOyqc3J6XKZYYjD3uwb0A6cA4VuDaAa8DgEZ7gFd6skfVsvVsf09aclc3swh9Znz+6M5EE</latexit>

qbr = 00>qbr

q<latexit sha1_base64="iVGWJjUY9T0HmMOpn/tMDZu36y8=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPRi0dI5JHAhswODYzMzq4zsyZkwxd48aAxXv0kb/6NA+xBwUo6qVR1p7sriAXXxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRDOssEpFqBVSj4BLrhhuBrVghDQOBzWB0O/WbT6g0j+S9Gcfoh3QgeZ8zaqxUe+wWS27ZnYEsEy8jJchQ7Ra/Or2IJSFKwwTVuu25sfFTqgxnAieFTqIxpmxEB9i2VNIQtZ/ODp2QE6v0SD9StqQhM/X3REpDrcdhYDtDaoZ60ZuK/3ntxPSv/ZTLODEo2XxRPxHERGT6NelxhcyIsSWUKW5vJWxIFWXGZlOwIXiLLy+TxlnZuyxf1M5LlZssjjwcwTGcggdXUIE7qEIdGCA8wyu8OQ/Oi/PufMxbc042cwh/4Hz+AN7djP8=</latexit>qorder = broadening of Pβ(q) can be quanti
ed
by
r
qbr =

qR

dq q 2 P(q)

R{ dq q P(2)2
hR iL,(q)}2 = lim
L"1fi<latexit sha1_base64="/FUOvfUrvn3lzo8R1SFsA+EPkrs=">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</latexit>q<latexit sha1_base64="iVGWJjUY9T0HmMOpn/tMDZu36y8=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPRi0dI5JHAhswODYzMzq4zsyZkwxd48aAxXv0kb/6NA+xBwUo6qVR1p7sriAXXxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRDOssEpFqBVSj4BLrhhuBrVghDQOBzWB0O/WbT6g0j+S9Gcfoh3QgeZ8zaqxUe+wWS27ZnYEsEy8jJchQ7Ra/Or2IJSFKwwTVuu25sfFTqgxnAieFTqIxpmxEB9i2VNIQtZ/ODp2QE6v0SD9StqQhM/X3REpDrcdhYDtDaoZ60ZuK/3ntxPSv/ZTLODEo2XxRPxHERGT6NelxhcyIsSWUKW5vJWxIFWXGZlOwIXiLLy+TxlnZuyxf1M5LlZssjjwcwTGcggdXUIE7qEIdGCA8wyu8OQ/Oi/PufMxbc042cwh/4Hz+AN7djP8=</latexit><latexit sha1_base64="iVGWJjUY9T0HmMOpn/tMDZu36y8=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPRi0dI5JHAhswODYzMzq4zsyZkwxd48aAxXv0kb/6NA+xBwUo6qVR1p7sriAXXxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRDOssEpFqBVSj4BLrhhuBrVghDQOBzWB0O/WbT6g0j+S9Gcfoh3QgeZ8zaqxUe+wWS27ZnYEsEy8jJchQ7Ra/Or2IJSFKwwTVuu25sfFTqgxnAieFTqIxpmxEB9i2VNIQtZ/ODp2QE6v0SD9StqQhM/X3REpDrcdhYDtDaoZ60ZuK/3ntxPSv/ZTLODEo2XxRPxHERGT6NelxhcyIsSWUKW5vJWxIFWXGZlOwIXiLLy+TxlnZuyxf1M5LlZssjjwcwTGcggdXUIE7qEIdGCA8wyu8OQ/Oi/PufMxbc042cwh/4Hz+AN7djP8=</latexit>

qbr > 0

⇣

(2)
hRiL,

⌘2

non-differentiability of the 2-replica freeenergy and the jump order parameter qjump

t wo replicas with explicit coupling λ ∈ ℝ
Hamiltonian HL ( 1 , 2 ; ) = HL ( 1 ) + HL ( 2 )1<latexit sha1_base64="NMHENTZNCpuh7HdPqm8SEorJuJ4=">AAADO3icnVI9SzNBEJ479VWjvkYtbQ7lBUUNdwE/QISgjYWFglEhiWFvbxMX926P3T3hOPIv/AsW/gMb/4SdjY2FIrb2Tj4K32ihDiz7zDPPMLMz68eCa+O695Y9MDj0Z3hkNDc2PvF3Mj81faRloigrUymkOvGJZoJHrGy4EewkVoyEvmDH/vlOO358wZTmMjo0acxqIWlGvMEpMUjJ/DXsQh32YAGuwAcJAgLQkELY8zLkNXBoIkOgBafgwfK3tUXYRE6g19YEeC9CDrZ+UXMRln6cVexUW+nr4OoH/XvIUFRJMN+vWs/PuwW3Y85n4PXAfGmuunR5X0r36/m7aiBpErLIUEG0rnhubGoZUYZTwVq5aqJZTOg5abIKwoiETNeyzu5bzj9kAqchFZ7IOB32Y0ZGQq3T0EdlSMyZ7o+1ya9ilcQ0NmoZj+LEsIh2CzUS4RjptD+SE3DFqBEpAkIVx14dekYUoQa/Ww6H4PU/+TM4Kha8tcLqAU5jG7o2ArMwh0v2YB1KuPJ9KAO1bqwH68l6tm/tR/vFfu1KbauXMwP/mf32DtjG2AU=</latexit>

P1log
1, 2 eLd<latexit sha1_base64="cthuatfqhwp/C+zaLbnKCnqmmlg=">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</latexit>

free energy f(2)( , )=

lim

L"1

·

1,

HL (

2
2;

)

if the system has any order, f (2) should be singular at λ = 0<latexit sha1_base64="dbHNZms/OW1CSxl+qNoUu9oYDqU=">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</latexit>

qjump =

1
2

n

lim

@f

(2)

@

( , )

n #0
(2)
1
= 2 hRi , =+0<latexit sha1_base64="mR1x6VgSjd99qIC437NKrjfYY8M=">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</latexit>

qjump > 0 ⟺ f

(2)

lim

@f

(2)

@
"0
o
(2)
hRi , = 0

( , )

o

<latexit sha1_base64="xnPa8O65Z2wR+DmUzGV9+K2BcEw=">AAACFXicbVDLSsNAFL2pr1pfUcGNm2ARLEhJio8uC25cVrAPaGOZTCbt0EkmzEyEEvoTbvwC/8GNC0XcCu78G6dtFtp6YTiHc+7lzj1ezKhUtv1t5JaWV1bX8uuFjc2t7R1zd68peSIwaWDOuGh7SBJGI9JQVDHSjgVBocdIyxteTfzWPRGS8uhWjWLihqgf0YBipLTEzQMI4A5SOIEKlGCs8Qk8IKAAwanmTGOoFV9jqWcW7bI9LWuROBkpQlb1nvnV9TlOQhIpzJCUHceOlZsioShmZFzoJpLECA9Rn3Q0jVBIpJtOrxpbx1rxrYAL/SJlTdXfEykKpRyFnu4MkRrIeW8i/ud1EhVU3ZRGcaJIhGeLgoRZiluTiCyfCoIVG2mCsKD6rxYeIIGw0kEWdAjO/MmLpFkpOxfl85uzYq2axZGHQzjSATtwCTW4hjo0AMMDPMMrvBmPxovxbnzMWnNGNrMPf8r4/AGmEpZg</latexit>f (2) ( , )

0
>
q mp
ju

<latexit sha1_base64="MF2y1bujfBMH8UBCe9Rk+1r/w1g=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hV3xkWPAi8cI5gHJEnpnJ8mQ2dllZlYJS/7DiwdFvfov3vwbJ8keNLGgoajqprsrSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqKGvQWMSqHaBmgkvWMNwI1k4UwygQrBWMbqZ+64EpzWN5b8YJ8yMcSN7nFI2VBvAOAhAiCCAE7JXKbsWdgSwTLydlyFHvlb66YUzTiElDBWrd8dzE+Bkqw6lgk2I31SxBOsIB61gqMWLaz2ZXT8ipVULSj5UtachM/T2RYaT1OApsZ4RmqBe9qfif10lNv+pnXCapYZLOF/VTQUxMphGQkCtGjRhbglRxeyuhQ1RIjQ2qaEPwFl9eJs3zindVuby7KNeqeRwFOIYTOAMPrqEGt1CHBlBQ8AQv8Oo8Os/Om/Mxb11x8pkj+APn8wfINZAV</latexit>

van Enter, Grif ths 1983

is non-differentiable at λ = 0

for the standard EA model without a magnetic eld<latexit sha1_base64="/HU31a4R+IHx8UP9fOOjOakr0ek=">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</latexit>

EA order parameter
fi

fi

(2)
qjump = qEA = hRi , =+0the first and the second theorems
THEOREM 1: in the standard EA model withouta magnetic eld, one has qbr ≤ qEA = qjump
straight for ward extension of Grif ths’ theorem

THEOREM 2: in the general EA model,2(qbr)
one has≤ qjump4
broadening of the overlap distribution implies nondifferentiability of the t wo-replica free energy<latexit sha1_base64="xnPa8O65Z2wR+DmUzGV9+K2BcEw=">AAACFXicbVDLSsNAFL2pr1pfUcGNm2ARLEhJio8uC25cVrAPaGOZTCbt0EkmzEyEEvoTbvwC/8GNC0XcCu78G6dtFtp6YTiHc+7lzj1ezKhUtv1t5JaWV1bX8uuFjc2t7R1zd68peSIwaWDOuGh7SBJGI9JQVDHSjgVBocdIyxteTfzWPRGS8uhWjWLihqgf0YBipLTEzQMI4A5SOIEKlGCs8Qk8IKAAwanmTGOoFV9jqWcW7bI9LWuROBkpQlb1nvnV9TlOQhIpzJCUHceOlZsioShmZFzoJpLECA9Rn3Q0jVBIpJtOrxpbx1rxrYAL/SJlTdXfEykKpRyFnu4MkRrIeW8i/ud1EhVU3ZRGcaJIhGeLgoRZiluTiCyfCoIVG2mCsKD6rxYeIIGw0kEWdAjO/MmLpFkpOxfl85uzYq2axZGHQzjSATtwCTW4hjo0AMMDPMMrvBmPxovxbnzMWnNGNrMPf8r4/AGmEpZg</latexit>

qbr > 0<latexit sha1_base64="gzkNSZITZwb/IDUGVVtubiHv7Cc=">AAAB+3icbZDJSgNBEIZr4hbjFvXopTEIegkz4nYMevEYwSyQDKGn05O06VnsrhHDkFfxIqKIV1/Em29jJ5mDJv7Q8PNXFVX9ebEUGm3728otLC4tr+RXC2vrG5tbxe2duo4SxXiNRTJSTY9qLkXIayhQ8masOA08yRve4GpcbzxwpUUU3uIw5m5Ae6HwBaNoojuoQgdewAMOCBQO4R6OOsWSXbYnIvPGyUwJMlU7xa92N2JJwENkkmrdcuwY3ZQqFEzyUaGdaB5TNqA93jI2pAHXbjq5fUQOTNIlfqTMC5FM0t8TKQ20Hgae6Qwo9vVsbRz+V2sl6F+4qQjjBHnIpov8RBKMyBgE6QrFGcqhMZQpYW4lrE8VZWhwFQwEZ/bL86Z+XHbOyqc3J6XKZYYjD3uwb0A6cA4VuDaAa8DgEZ7gFd6skfVsvVsf09aclc3swh9Znz+6M5EE</latexit>

P (q)f (2) ( , )<latexit sha1_base64="gzkNSZITZwb/IDUGVVtubiHv7Cc=">AAAB+3icbZDJSgNBEIZr4hbjFvXopTEIegkz4nYMevEYwSyQDKGn05O06VnsrhHDkFfxIqKIV1/Em29jJ5mDJv7Q8PNXFVX9ebEUGm3728otLC4tr+RXC2vrG5tbxe2duo4SxXiNRTJSTY9qLkXIayhQ8masOA08yRve4GpcbzxwpUUU3uIw5m5Ae6HwBaNoojuoQgdewAMOCBQO4R6OOsWSXbYnIvPGyUwJMlU7xa92N2JJwENkkmrdcuwY3ZQqFEzyUaGdaB5TNqA93jI2pAHXbjq5fUQOTNIlfqTMC5FM0t8TKQ20Hgae6Qwo9vVsbRz+V2sl6F+4qQjjBHnIpov8RBKMyBgE6QrFGcqhMZQpYW4lrE8VZWhwFQwEZ/bL86Z+XHbOyqc3J6XKZYYjD3uwb0A6cA4VuDaAa8DgEZ7gFd6skfVsvVsf09aclc3swh9Znz+6M5EE</latexit>

P (q)
q

q

0
>
q mp
ju

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the theorems can be proved for any classical spin glassmodel with short-range interaction and bounded spins

the idearof the proof (fortheEA
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qbr = lim

L"1

(2)
2
hR iL,= lim L

d

L"1

S1 x
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Px,y2⇤L2h x y iL,

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X

x2S1
y2S2

bc
y

y

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 max
bc

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X

x2S1

h x i`, ,bc

2

max
bc

X

y2S2

h y i`, ,bc

X
1
2
qEA := lim d max
h x i`, ,bc =`"1 `bc<latexit sha1_base64="o2xcJFm9rcyuqApOabhduCwpvII=">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</latexit>

x2⇤`

lim
#0

2

Tasaki 1989

Edwards-Anderson order parameterfi<latexit sha1_base64="BOvVSULLfKAblBLDstNTUvPs3n4=">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</latexit>`

S2<latexit sha1_base64="nBZDDAWV9hcqLhJv6hiR++vI/kU=">AAAB6nicbVC7SgNBFL3jM8ZXVLCxGQyCVdgN+ChDbCwTYh6QLGF2MpsMmZ1dZmaFsOQTbCwUsbX1L/wCOxu/xcmj0MQDFw7n3Mu99/ix4No4zhdaWV1b39jMbGW3d3b39nMHhw0dJYqyOo1EpFo+0UxwyeqGG8FasWIk9AVr+sObid+8Z0rzSN6ZUcy8kPQlDzglxkq1WrfYzeWdgjMFXibunORLx9Vv/l7+qHRzn51eRJOQSUMF0brtOrHxUqIMp4KNs51Es5jQIemztqWShEx76fTUMT6zSg8HkbIlDZ6qvydSEmo9Cn3bGRIz0IveRPzPaycmuPZSLuPEMElni4JEYBPhyd+4xxWjRowsIVRxeyumA6IINTadrA3BXXx5mTSKBfeycFG1aZRhhgycwCmcgwtXUIJbqEAdKPThAZ7gGQn0iF7Q66x1Bc1njuAP0NsPtq+RIw==</latexit><latexit sha1_base64="AsbnVqJMUQ3+szZGBl8QEeVgz4g=">AAAB83icbVDJSgNBEK2JWxy3qEcvjUHwFGYEl4sY9OIxglkgGUJPpydp0j3TdPcIYchvePGguBz9Du9exL+xsxw08UHB470qquqFkjNtPO/byS0sLi2v5FfdtfWNza3C9k5NJ6kitEoSnqhGiDXlLKZVwwynDakoFiGn9bB/NfLrd1RplsS3ZiBpIHA3ZhEj2FgpgjdQIABBCKRdKHolbww0T/wpKV58uOfy5cuttAufrU5CUkFjQzjWuul70gQZVoYRToduK9VUYtLHXdq0NMaC6iAb3zxEB1bpoChRtmKDxurviQwLrQcitJ0Cm56e9Ubif14zNdFZkLFYpobGZLIoSjkyCRoFgDpMUWL4wBJMFLO3ItLDChNjY3JtCP7sy/OkdlTyT0rHN16xfAkT5GEP9uEQfDiFMlxDBapAQMI9PMKTkzoPzrPzOmnNOdOZXfgD5/0H7j2SrA==</latexit>@f (2) ( , )van Enter, Grif ths 1983

the first and the second theorems
THEOREM 1: in the standard EA model withouta magnetic eld, one has qbr ≤ qEA = qjump
straight for ward extension of Grif ths’ theorem

THEOREM 2: in the general EA model,2(qbr)
one has≤ qjump4
broadening of the overlap distribution implies nondifferentiability of the t wo-replica free energy<latexit sha1_base64="xnPa8O65Z2wR+DmUzGV9+K2BcEw=">AAACFXicbVDLSsNAFL2pr1pfUcGNm2ARLEhJio8uC25cVrAPaGOZTCbt0EkmzEyEEvoTbvwC/8GNC0XcCu78G6dtFtp6YTiHc+7lzj1ezKhUtv1t5JaWV1bX8uuFjc2t7R1zd68peSIwaWDOuGh7SBJGI9JQVDHSjgVBocdIyxteTfzWPRGS8uhWjWLihqgf0YBipLTEzQMI4A5SOIEKlGCs8Qk8IKAAwanmTGOoFV9jqWcW7bI9LWuROBkpQlb1nvnV9TlOQhIpzJCUHceOlZsioShmZFzoJpLECA9Rn3Q0jVBIpJtOrxpbx1rxrYAL/SJlTdXfEykKpRyFnu4MkRrIeW8i/ud1EhVU3ZRGcaJIhGeLgoRZiluTiCyfCoIVG2mCsKD6rxYeIIGw0kEWdAjO/MmLpFkpOxfl85uzYq2axZGHQzjSATtwCTW4hjo0AMMDPMMrvBmPxovxbnzMWnNGNrMPf8r4/AGmEpZg</latexit>

qbr > 0<latexit sha1_base64="gzkNSZITZwb/IDUGVVtubiHv7Cc=">AAAB+3icbZDJSgNBEIZr4hbjFvXopTEIegkz4nYMevEYwSyQDKGn05O06VnsrhHDkFfxIqKIV1/Em29jJ5mDJv7Q8PNXFVX9ebEUGm3728otLC4tr+RXC2vrG5tbxe2duo4SxXiNRTJSTY9qLkXIayhQ8masOA08yRve4GpcbzxwpUUU3uIw5m5Ae6HwBaNoojuoQgdewAMOCBQO4R6OOsWSXbYnIvPGyUwJMlU7xa92N2JJwENkkmrdcuwY3ZQqFEzyUaGdaB5TNqA93jI2pAHXbjq5fUQOTNIlfqTMC5FM0t8TKQ20Hgae6Qwo9vVsbRz+V2sl6F+4qQjjBHnIpov8RBKMyBgE6QrFGcqhMZQpYW4lrE8VZWhwFQwEZ/bL86Z+XHbOyqc3J6XKZYYjD3uwb0A6cA4VuDaAa8DgEZ7gFd6skfVsvVsf09aclc3swh9Znz+6M5EE</latexit>

P (q)f (2) ( , )<latexit sha1_base64="gzkNSZITZwb/IDUGVVtubiHv7Cc=">AAAB+3icbZDJSgNBEIZr4hbjFvXopTEIegkz4nYMevEYwSyQDKGn05O06VnsrhHDkFfxIqKIV1/Em29jJ5mDJv7Q8PNXFVX9ebEUGm3728otLC4tr+RXC2vrG5tbxe2duo4SxXiNRTJSTY9qLkXIayhQ8masOA08yRve4GpcbzxwpUUU3uIw5m5Ae6HwBaNoojuoQgdewAMOCBQO4R6OOsWSXbYnIvPGyUwJMlU7xa92N2JJwENkkmrdcuwY3ZQqFEzyUaGdaB5TNqA93jI2pAHXbjq5fUQOTNIlfqTMC5FM0t8TKQ20Hgae6Qwo9vVsbRz+V2sl6F+4qQjjBHnIpov8RBKMyBgE6QrFGcqhMZQpYW4lrE8VZWhwFQwEZ/bL86Z+XHbOyqc3J6XKZYYjD3uwb0A6cA4VuDaAa8DgEZ7gFd6skfVsvVsf09aclc3swh9Znz+6M5EE</latexit>

P (q)
q

q

0
>
q mp
ju

<latexit sha1_base64="MF2y1bujfBMH8UBCe9Rk+1r/w1g=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBA8hV3xkWPAi8cI5gHJEnpnJ8mQ2dllZlYJS/7DiwdFvfov3vwbJ8keNLGgoajqprsrSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqKGvQWMSqHaBmgkvWMNwI1k4UwygQrBWMbqZ+64EpzWN5b8YJ8yMcSN7nFI2VBvAOAhAiCCAE7JXKbsWdgSwTLydlyFHvlb66YUzTiElDBWrd8dzE+Bkqw6lgk2I31SxBOsIB61gqMWLaz2ZXT8ipVULSj5UtachM/T2RYaT1OApsZ4RmqBe9qfif10lNv+pnXCapYZLOF/VTQUxMphGQkCtGjRhbglRxeyuhQ1RIjQ2qaEPwFl9eJs3zindVuby7KNeqeRwFOIYTOAMPrqEGt1CHBlBQ8AQv8Oo8Os/Om/Mxb11x8pkj+APn8wfINZAV</latexit><latexit sha1_base64="iVGWJjUY9T0HmMOpn/tMDZu36y8=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPRi0dI5JHAhswODYzMzq4zsyZkwxd48aAxXv0kb/6NA+xBwUo6qVR1p7sriAXXxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRDOssEpFqBVSj4BLrhhuBrVghDQOBzWB0O/WbT6g0j+S9Gcfoh3QgeZ8zaqxUe+wWS27ZnYEsEy8jJchQ7Ra/Or2IJSFKwwTVuu25sfFTqgxnAieFTqIxpmxEB9i2VNIQtZ/ODp2QE6v0SD9StqQhM/X3REpDrcdhYDtDaoZ60ZuK/3ntxPSv/ZTLODEo2XxRPxHERGT6NelxhcyIsSWUKW5vJWxIFWXGZlOwIXiLLy+TxlnZuyxf1M5LlZssjjwcwTGcggdXUIE7qEIdGCA8wyu8OQ/Oi/PufMxbc042cwh/4Hz+AN7djP8=</latexit><latexit sha1_base64="iVGWJjUY9T0HmMOpn/tMDZu36y8=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPRi0dI5JHAhswODYzMzq4zsyZkwxd48aAxXv0kb/6NA+xBwUo6qVR1p7sriAXXxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRDOssEpFqBVSj4BLrhhuBrVghDQOBzWB0O/WbT6g0j+S9Gcfoh3QgeZ8zaqxUe+wWS27ZnYEsEy8jJchQ7Ra/Or2IJSFKwwTVuu25sfFTqgxnAieFTqIxpmxEB9i2VNIQtZ/ODp2QE6v0SD9StqQhM/X3REpDrcdhYDtDaoZ60ZuK/3ntxPSv/ZTLODEo2XxRPxHERGT6NelxhcyIsSWUKW5vJWxIFWXGZlOwIXiLLy+TxlnZuyxf1M5LlZssjjwcwTGcggdXUIE7qEIdGCA8wyu8OQ/Oi/PufMxbc042cwh/4Hz+AN7djP8=</latexit>fifi

the theorems can be proved for any classical spin glassmodel with short-range interaction and bounded spins3-replica free energy and the literalreplica symmetry breaking (RSB)
Guerra 2013
order parameter qrsb

three replicas with explicit couplings λ, λ′ ∈ ℝ
Hamiltonian
1<latexit sha1_base64="avxOveHdyPTZEeMpsmjZrrNDdDE=">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</latexit>

HL (

2

,

,

3

0

; ,

) = HL (1) + HL (<latexit sha1_base64="vPN72RetsIjyz4KAVxNm2gwzC18=">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</latexit>

free energy<latexit sha1_base64="+TKXomuwfy74fenkXNZID+O5pp0=">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</latexit>

f

(3)

( , ,

0

)=

lim

L"1

2
1

P
1
log
1, 2, 3 e
Ld

) + HL (
HL (

)
1
·

0

2

·

3

1,

2,

3
, 0)

3;

the measure of spontaneous breakdown of the permutation
symmetry of the three replicas
2<latexit sha1_base64="F8FkEfYKFbWhX6CeAoaLBDiUGi8=">AAAB6HicbZC7SgNBFIbPxluMt6ilIItBsAq7ATWdARvLBMwFkhBmJ2eTMbOzy8ysEJaUVjYWitj6FKl8CDufwZdwcik08YeBj/8/hznneBFnSjvOl5VaWV1b30hvZra2d3b3svsHNRXGkmKVhjyUDY8o5ExgVTPNsRFJJIHHse4Nrid5/R6lYqG41cMI2wHpCeYzSrSxKoVONufknansZXDnkLv6GFe+H47H5U72s9UNaRyg0JQTpZquE+l2QqRmlOMo04oVRoQOSA+bBgUJULWT6aAj+9Q4XdsPpXlC21P3d0dCAqWGgWcqA6L7ajGbmP9lzVj7xXbCRBRrFHT2kR9zW4f2ZGu7yyRSzYcGCJXMzGrTPpGEanObjDmCu7jyMtQKefcif15xcqUizJSGIziBM3DhEkpwA2WoAgWER3iGF+vOerJerbdZacqa9xzCH1nvP0OnkP4=</latexit><latexit sha1_base64="XmN+ZeYDWeh4o2ZDZmv26RJVgO8=">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</latexit>

qrsb =

lim
#0

@f

(3)

+0
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( , ,
@

)

1
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3<latexit sha1_base64="KDHue//9yManPttmBaUs3fpk6Q8=">AAAB6HicbZC7SgNBFIbPxluMt6ilIItBsAq7iprOgI1lAuYCyRJmJyfJmNnZZWZWCEtKKxsLRWx9ilQ+hJ3P4Es4uRSa+MPAx/+fw5xz/IgzpR3ny0otLa+srqXXMxubW9s72d29qgpjSbFCQx7Kuk8UciawopnmWI8kksDnWPP71+O8do9SsVDc6kGEXkC6gnUYJdpY5bNWNufknYnsRXBnkLv6GJW/Hw5HpVb2s9kOaRyg0JQTpRquE2kvIVIzynGYacYKI0L7pIsNg4IEqLxkMujQPjZO2+6E0jyh7Yn7uyMhgVKDwDeVAdE9NZ+Nzf+yRqw7BS9hIoo1Cjr9qBNzW4f2eGu7zSRSzQcGCJXMzGrTHpGEanObjDmCO7/yIlRP8+5F/rzs5IoFmCoNB3AEJ+DCJRThBkpQAQoIj/AML9ad9WS9Wm/T0pQ169mHP7LefwBFK5D/</latexit><latexit sha1_base64="388RS/RuKWBdDecyA+HyE1xtFFw=">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</latexit>￼<latexit sha1_base64="xeNRz3KNObhmstAvgI8O1KjSZY8=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoPgxbArqDkGvAheopgHJEuYnfTGIbOzy8ysEEL+wIsHRbz6B36K5OKHeHfyOGhiQUNR1U13V5AIro3rfjmZpeWV1bXsem5jc2t7J7+7V9NxqhhWWSxi1QioRsElVg03AhuJQhoFAutB73Ls1x9QaR7LO9NP0I9oV/KQM2qsdHvitvMFt+hOQBaJNyOFMnx/jEaZ60o7/9nqxCyNUBomqNZNz02MP6DKcCZwmGulGhPKerSLTUsljVD7g8mlQ3JklQ4JY2VLGjJRf08MaKR1PwpsZ0TNvZ73xuJ/XjM1YckfcJmkBiWbLgpTQUxMxm+TDlfIjOhbQpni9lbC7qmizNhwcjYEb/7lRVI7LXrnxbMbm0YJpsjCARzCMXhwAWW4ggpUgUEIj/AML07PeXJenbdpa8aZzezDHzjvPxM5kMM=</latexit>= hR12 ihR13 i0the third theorem
THEOREM 3: in almost any spin glass model(including long-range models), one has 2 qjump ≤ qrsb
non-differentiability of the t wo-replica free energy
implies literal replica symmetry breaking<latexit sha1_base64="xnPa8O65Z2wR+DmUzGV9+K2BcEw=">AAACFXicbVDLSsNAFL2pr1pfUcGNm2ARLEhJio8uC25cVrAPaGOZTCbt0EkmzEyEEvoTbvwC/8GNC0XcCu78G6dtFtp6YTiHc+7lzj1ezKhUtv1t5JaWV1bX8uuFjc2t7R1zd68peSIwaWDOuGh7SBJGI9JQVDHSjgVBocdIyxteTfzWPRGS8uhWjWLihqgf0YBipLTEzQMI4A5SOIEKlGCs8Qk8IKAAwanmTGOoFV9jqWcW7bI9LWuROBkpQlb1nvnV9TlOQhIpzJCUHceOlZsioShmZFzoJpLECA9Rn3Q0jVBIpJtOrxpbx1rxrYAL/SJlTdXfEykKpRyFnu4MkRrIeW8i/ud1EhVU3ZRGcaJIhGeLgoRZiluTiCyfCoIVG2mCsKD6rxYeIIGw0kEWdAjO/MmLpFkpOxfl85uzYq2axZGHQzjSATtwCTW4hjo0AMMDPMMrvBmPxovxbnzMWnNGNrMPf8r4/AGmEpZg</latexit>f (2) ( , )

0
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2
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0
>
qrsb

3
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0that qjump > 0 has been proved (only) in long-range spin glassmodels such as the Sherrington-Kirkpatrick (SK) model (withor without a magnetic eld) and the random energy model
Talagrand 2003, 2011, Guerra 2013

fi

Theorem 3 establishes that these models exhibit literal replica
symmetry breaking

introduction
definitions and main results
discussion

limitation of the theory

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P (q)qbr > 0

0
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q mp
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0
>
qrsb
0

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none of the conditions have been proved rigorously forshort-range spin glass models
these conditions do not always imply spin glass order.
in the ferromagnetic Ising model, μSM > 0 implies qbr > 0,
qjump > 0, and qrsb > 02<latexit sha1_base64="F8FkEfYKFbWhX6CeAoaLBDiUGi8=">AAAB6HicbZC7SgNBFIbPxluMt6ilIItBsAq7ATWdARvLBMwFkhBmJ2eTMbOzy8ysEJaUVjYWitj6FKl8CDufwZdwcik08YeBj/8/hznneBFnSjvOl5VaWV1b30hvZra2d3b3svsHNRXGkmKVhjyUDY8o5ExgVTPNsRFJJIHHse4Nrid5/R6lYqG41cMI2wHpCeYzSrSxKoVONufknansZXDnkLv6GFe+H47H5U72s9UNaRyg0JQTpZquE+l2QqRmlOMo04oVRoQOSA+bBgUJULWT6aAj+9Q4XdsPpXlC21P3d0dCAqWGgWcqA6L7ajGbmP9lzVj7xXbCRBRrFHT2kR9zW4f2ZGu7yyRSzYcGCJXMzGrTPpGEanObjDmCu7jyMtQKefcif15xcqUizJSGIziBM3DhEkpwA2WoAgWER3iGF+vOerJerbdZacqa9xzCH1nvP0OnkP4=</latexit>+0<latexit sha1_base64="SvofpUwiUNbp8sHlQWStUDsAqE8=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoMgCGFXUHMMeBG8RDEPSJYwO+mNQ2Znl5lZIYT8gRcPinj1D/wUycUP8e7kcdDEgoaiqpvuriARXBvX/XIyS8srq2vZ9dzG5tb2Tn53r6bjVDGssljEqhFQjYJLrBpuBDYShTQKBNaD3uXYrz+g0jyWd6afoB/RruQhZ9RY6fbEbecLbtGdgCwSb0YKZfj+GI0y15V2/rPViVkaoTRMUK2bnpsYf0CV4UzgMNdKNSaU9WgXm5ZKGqH2B5NLh+TIKh0SxsqWNGSi/p4Y0EjrfhTYzoiaez3vjcX/vGZqwpI/4DJJDUo2XRSmgpiYjN8mHa6QGdG3hDLF7a2E3VNFmbHh5GwI3vzLi6R2WvTOi2c3No0STJGFAziEY/DgAspwBRWoAoMQHuEZXpye8+S8Om/T1owzm9mHP3DefwAQL5DB</latexit>

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0

0
>
qrsb

3
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spin glass models under amagnetic field
our theory is probably most meaningful for (short-range)
spin glass models under a (random or non-random)magnetic eld, which have no obvious symmetry
3-replica system

2-replica
systemsingle system
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s
s
a
l
g
spi n
o rde rq br > 0<latexit sha1_base64="xeNRz3KNObhmstAvgI8O1KjSZY8=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoPgxbArqDkGvAheopgHJEuYnfTGIbOzy8ysEEL+wIsHRbz6B36K5OKHeHfyOGhiQUNR1U13V5AIro3rfjmZpeWV1bXsem5jc2t7J7+7V9NxqhhWWSxi1QioRsElVg03AhuJQhoFAutB73Ls1x9QaR7LO9NP0I9oV/KQM2qsdHvitvMFt+hOQBaJNyOFMnx/jEaZ60o7/9nqxCyNUBomqNZNz02MP6DKcCZwmGulGhPKerSLTUsljVD7g8mlQ3JklQ4JY2VLGjJRf08MaKR1PwpsZ0TNvZ73xuJ/XjM1YckfcJmkBiWbLgpTQUxMxm+TDlfIjOhbQpni9lbC7qmizNhwcjYEb/7lRVI7LXrnxbMbm0YJpsjCARzCMXhwAWW4ggpUgUEIj/AML07PeXJenbdpa8aZzezDHzjvPxM5kMM=</latexit>

0
>
q mp
ju

no symmetry breaking(in the standard sense)

0

0
>
qrsb

manifest spontaneous
symmetry breaking

it may be that there is no spin glass order in d = 3
fi

Sasaki, Hukushima, Yoshino, Takayama 2007, Baity-Jesi et al. 2014

summary
we proved inequalities for general short-range spinglass models that clarify the relations bet ween
different characterizations of “spin glass order”
we proved that qbr > 0 implies qjump > 0, and thenqjump > 0 implies qrsb > 0fi

the theory has an interesting implication in amodel with a magnetic eld. such a model does notexhibit a symmetry breaking by itself but may exhibit
spontaneous breakdown of replica permutation
symmetry in the 3-replica system
our inequality may be used to prove the absence ofspin glass order in some short-range models