We show that a simple two-dimensional model of spin crossover materials gives rise to spin-state smectic phases where the pattern of high-spin (HS) and low-spin (LS) metal centers spontaneously breaks rotational symmetry and translational symmetry in one direction only. The spin-state smectics are distinct thermodynamic phases and give rise to plateaus in the fraction of HS metal centers. Smectic order leads to lines of Bragg peaks in the x-ray and neutron scattering structure factors. We identify two smectic phases and show that both are ordered in one direction, but disordered in the other, and hence that their residual entropy scales with the linear dimension of the system. This is intermediate to spin-state ices (examples of “spin-state liquids”) where the residual entropy scales with the system volume, and antiferroelastic ordered phases (examples of “spin-state crystals”) where the residual entropy is independent of the size of the system.
REFERENCES
1.
2.
B. J.
Powell
, “Emergent particles and gauge fields in quantum matter
,” Contemp. Phys.
61
, 96
–131
(2020
). 3.
L.
Savary
and L.
Balents
, “Quantum spin liquids: A review
,” Rep. Prog. Phys.
80
, 016502
(2016
). 4.
T.
Giamarchi
, Quantum Physics in One Dimension
(Oxford University Press
, Oxford
, 2004
).5.
A.
Simonov
and A. L.
Goodwin
, “Designing disorder into crystalline materials
,” Nat. Rev. Chem.
4
, 657
–673
(2020
). 6.
L.
Pauling
, “The structure and entropy of ice and of other crystals with some randomness of atomic arrangement
,” J. Am. Chem. Soc.
57
, 2680
–2684
(1935
). 7.
C.
Castelnovo
, R.
Moessner
, and S.
Sondhi
, “Spin ice, fractionalization, and topological order
,” Annu. Rev. Condens. Matter Phys.
3
, 35
–55
(2012
). 8.
C. L.
Henley
, “The ‘coulomb phase’ in frustrated systems
,” Annu. Rev. Condens. Matter Phys.
1
, 179
–210
(2010
). 9.
J.
Cruddas
and B. J.
Powell
, “Spin-state ice in elastically frustrated spin-crossover materials
,” J. Am. Chem. Soc
141
, 19790
–19799
(2019
). 10.
J.
Cruddas
and B. J.
Powell
, “Multiple Coulomb phases with temperature tunable ice rules in pyrochlore spin crossover materials,” arXiv:2007.13983 (2020).11.
B. J.
Powell
, “The expanding materials multiverse
,” Science
360
, 1073
–1074
(2018
). 12.
J. D.
Bernal
and R. H.
Fowler
, “A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions
,” J. Chem. Phys.
1
, 515
–548
(1933
). 13.
T.
Fennell
, P. P.
Deen
, A. R.
Wildes
, K.
Schmalzl
, D.
Prabhakaran
, A. T.
Boothroyd
, R. J.
Aldus
, D. F.
McMorrow
, and S. T.
Bramwell
, “Magnetic coulomb phase in the spin ice
,” Science
326
, 415
–417
(2009
). 14.
D. J. P.
Morris
, D. A.
Tennant
, S. A.
Grigera
, B.
Klemke
, C.
Castelnovo
, R.
Moessner
, C.
Czternasty
, M.
Meissner
, K. C.
Rule
, J.-U.
Hoffmann
, K.
Kiefer
, S.
Gerischer
, D.
Slobinsky
, and R. S.
Perry
, “Dirac strings and magnetic monopoles in the spin ice
,” Science
326
, 411
–414
(2009
). 15.
P.
Chaikin
and T.
Lubensky
, Principles of Condensed Matter Physics
(Cambridge University Press
, 2000
).16.
17.
R. M.
Fernandes
, A. V.
Chubukov
, and J.
Schmalian
, “What drives nematic order in iron-based superconductors?
,” Nat. Phys.
10
, 97
–104
(2014
). 18.
S. A.
Kivelson
, E.
Fradkin
, and V. J.
Emery
, “Electronic liquid-crystal phases of a doped mott insulator
,” Nature
393
, 550
–553
(1998
). 19.
H.-H.
Kuo
, J.-H.
Chu
, J. C.
Palmstrom
, S. A.
Kivelson
, and I. R.
Fisher
, “Ubiquitous signatures of nematic quantum criticality in optimally doped fe-based superconductors
,” Science
352
, 958
–962
(2016
). 20.
J.-H.
Chu
, H.-H.
Kuo
, J. G.
Analytis
, and I. R.
Fisher
, “Divergent nematic susceptibility in an iron arsenide superconductor
,” Science
337
, 710
–712
(2012
). 21.
V. J.
Emery
, E.
Fradkin
, S. A.
Kivelson
, and T. C.
Lubensky
, “Quantum theory of the smectic metal state in stripe phases
,” Phys. Rev. Lett.
85
, 2160
–2163
(2000
). 22.
C. M.
Yim
, C.
Trainer
, R.
Aluru
, S.
Chi
, W. N.
Hardy
, R.
Liang
, D.
Bonn
, and P.
Wahl
, “Discovery of a strain-stabilised smectic electronic order in lifeas
,” Nat. Commun.
9
, 2602
(2018
). 23.
K.
Penc
and A. M.
Läuchli
, “Spin nematic phases in quantum spin systems,” in Introduction to Frustrated Magnetism: Materials, Experiments, Theory, edited by C. Lacroix, P. Mendels, and F. Mila (Springer, Berlin, 2011), pp. 331–362.24.
B. W.
Lebert
, T.
Gorni
, M.
Casula
, S.
Klotz
, F.
Baudelet
, J. M.
Ablett
, T. C.
Hansen
, A.
Juhin
, A.
Polian
, P.
Munsch
, G.
Le Marchand
, Z.
Zhang
, J.-P.
Rueff
, and M.
d’Astuto
, “Epsilon iron as a spin-smectic state
,” Proc. Natl. Acad. Sci. U.S.A.
116
, 20280
–20285
(2019
). 25.
J.
Cruddas
and B. J.
Powell
, “Structure–property relationships and the mechanisms of multistep transitions in spin crossover materials and frameworks
,” Inorg. Chem. Front.
7
, 4424
–4437
(2020
). 26.
N. F.
Sciortino
, K. A.
Zenere
, M. E.
Corrigan
, G. J.
Halder
, G.
Chastanet
, J.-F.
Létard
, C. J.
Kepert
, and S. M.
Neville
, “Four-step iron(II) spin state cascade driven by antagonistic solid state interactions
,” Chem. Sci.
8
, 701
–707
(2017
). 27.
Y.
Sekine
, M.
Nihei
, and H.
Oshio
, “Dimensionally controlled assembly of an external stimuli-responsive [] complex into supramolecular hydrogen-bonded networks
,” Chem. Eur. J.
23
, 5193
–5197
(2017
). 28.
G. S.
Matouzenko
, D.
Luneau
, G.
Molnár
, N.
Ould-Moussa
, S.
Zein
, S. A.
Borshch
, A.
Bousseksou
, and F.
Averseng
, “A two-step spin transition and order–disorder phenomena in the mononuclear compound
,” Eur. J. Inorg. Chem.
2006
, 2671
–2682
. 29.
M.
Ohlrich
and B. J.
Powell
, “Fast, accurate enthalpy differences in spin crossover crystals from DFT+U
,” J. Chem. Phys.
153
, 104107
(2020
). 30.
J.
Pavlik
and R.
Boča
, “Established static models of spin crossover
,” Eur. J. Inorg. Chem.
2013
, 697
–709
. 31.
P.
Gütlich
, Y.
Garcia
, and H. A.
Goodwin
, “Spin crossover phenomena in Fe(II) complexes
,” Chem. Soc. Rev.
29
, 419
(2000
). 32.
M.
Nishino
, K.
Boukheddaden
, and S.
Miyashita
, “Molecular dynamics study of thermal expansion and compression in spin-crossover solids using a microscopic model of elastic interactions
,” Phys. Rev. B
79
, 012409
(2009
). 33.
M.
Nishino
, K.
Boukheddaden
, Y.
Konishi
, and S.
Miyashita
, “Simple two-dimensional model for the elastic origin of cooperativity among spin states of spin-crossover complexes
,” Phys. Rev. Lett.
98
, 247203
(2007
). 34.
M.
Nishino
and S.
Miyashita
, “Termination of the Berezinskii-Kosterlitz-Thouless phase with a new critical universality in spin-crossover systems
,” Phys. Rev. B
92
, 184404
(2015
). 35.
M.
Nishino
, C.
Enachescu
, and S.
Miyashita
, “Multistep spin-crossover transitions induced by the interplay between short- and long-range interactions with frustration on a triangular lattice
,” Phys. Rev. B
100
, 134414
(2019
). 36.
S.
Miyashita
, Y.
Konishi
, M.
Nishino
, H.
Tokoro
, and P. A.
Rikvold
, “Realization of the mean-field universality class in spin-crossover materials
,” Phys. Rev. B
77
, 014105
(2008
). 37.
T.
Nakada
, P. A.
Rikvold
, T.
Mori
, M.
Nishino
, and S.
Miyashita
, “Crossover between a short-range and a long-range Ising model
,” Phys. Rev. B
84
, 054433
(2011
). 38.
T.
Nakada
, T.
Mori
, S.
Miyashita
, M.
Nishino
, S.
Todo
, W.
Nicolazzi
, and P. A.
Rikvold
, “Critical temperature and correlation length of an elastic interaction model for spin-crossover materials
,” Phys. Rev. B
85
, 054408
(2012
). 39.
M.
Nishino
and S.
Miyashita
, “Effect of the short-range interaction on critical phenomena in elastic interaction systems
,” Phys. Rev. B
88
, 014108
(2013
). 40.
H.
Watanabe
, K.
Tanaka
, N.
Bréfuel
, H.
Cailleau
, J.-F.
Létard
, S.
Ravy
, P.
Fertey
, M.
Nishino
, S.
Miyashita
, and E.
Collet
, “Ordering phenomena of high-spin/low-spin states in stepwise spin-crossover materials described by the ANNNI model
,” Phys. Rev. B
93
, 014419
(2016
). 41.
C.
Enachescu
, M.
Nishino
, S.
Miyashita
, A.
Hauser
, A.
Stancu
, and L.
Stoleriu
, “Cluster evolution in spin crossover systems observed in the frame of a mechano-elastic model
,” Europhys. Lett.
91
, 27003
(2010
). 42.
C.
Enachescu
, M.
Nishino
, S.
Miyashita
, L.
Stoleriu
, and A.
Stancu
, “Monte Carlo metropolis study of cluster evolution in spin-crossover solids within the framework of a mechanoelastic model
,” Phys. Rev. B
86
, 054114
(2012
). 43.
C.
Enachescu
, L.
Stoleriu
, A.
Stancu
, and A.
Hauser
, “Study of the relaxation in diluted spin crossover molecular magnets in the framework of the mechano-elastic model
,” J. Appl. Phys.
109
, 07B111
(2011
). 44.
C.
Enachescu
, L.
Stoleriu
, M.
Nishino
, S.
Miyashita
, A.
Stancu
, M.
Lorenc
, R.
Bertoni
, H.
Cailleau
, and E.
Collet
, “Theoretical approach for elastically driven cooperative switching of spin-crossover compounds impacted by an ultrashort laser pulse
,” Phys. Rev. B
95
, 224107
(2017
). 45.
M.
Paez-Espejo
, M.
Sy
, and K.
Boukheddaden
, “Elastic frustration causing two-step and multistep transitions in spin-crossover solids: Emergence of complex antiferroelastic structures
,” J. Am. Chem. Soc.
138
, 3202
–3210
(2016
). 46.
Y.
Konishi
, H.
Tokoro
, M.
Nishino
, and S.
Miyashita
, “Monte Carlo simulation of pressure-induced phase transitions in spin-crossover materials
,” Phys. Rev. Lett.
100
, 067206
(2008
). 47.
H.-Z.
Ye
, C.
Sun
, and H.
Jiang
, “Monte-Carlo simulations of spin-crossover phenomena based on a vibronic Ising-like model with realistic parameters
,” Phys. Chem. Chem. Phys.
17
, 6801
–6808
(2015
). 48.
L.
Stoleriu
, P.
Chakraborty
, A.
Hauser
, A.
Stancu
, and C.
Enachescu
, “Thermal hysteresis in spin-crossover compounds studied within the mechanoelastic model and its potential application to nanoparticles
,” Phys. Rev. B
84
, 134102
(2011
). 49.
G.
Ruzzi
, J.
Cruddas
, R. H.
McKenzie
, and B. J.
Powell
, “Equivalence of elastic and ising models for spin crossover materials,” arXiv:2008.08738 (2020).50.
L. B.
Frechette
, C.
Dellago
, and P. L.
Geissler
, “Consequences of lattice mismatch for phase equilibrium in heterostructured solids
,” Phys. Rev. Lett.
123
, 135701
(2019
). 51.
L. B.
Frechette
, C.
Dellago
, and P. L.
Geissler
, “Origin of mean-field behavior in an elastic ising model
,” Phys. Rev. B
102
, 024102
(2020
). 52.
M.
Newman
, G.
Barkema
, and I.
Barkema
, Monte Carlo Methods in Statistical Physics
(Clarendon Press
, 1999
).53.
F.
Kassan-Ogly
, A.
Murtazaev
, A.
Zhuravlev
, M.
Ramazanov
, and A.
Proshkin
, “Ising model on a square lattice with second-neighbor and third-neighbor interactions
,” J. Magn. Magn. Mater.
384
, 247
–254
(2015
). 54.
J.
Wajnflasz
and R.
Pick
, “Transitions ‘Low Spin’ – ‘High Spin’ dans les complexes de
,” J. Phys. Colloq.
32
, C1-91
–C1-92
(1971
). 55.
M.
Plischke
and B.
Bergersen
, Equilibrium Statistical Physics
(World Scientific
, 2006
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
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