# The Structure of State Transition Graphs in Systems with Return Point Memory: I. General Theory

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## Abstract

We consider the athermal quasi-static dynamics (AQS) of disordered systems driven by an external field. Our interest is in an automaton description (AQS-A) that represents the AQS dynamics via a graph of state transitions triggered by the field and in the presence of return-point-memory (RPM) property, a tendency for the system to return to the same microstate upon cycling driving. The existence of three conditions, (1) a partial order on the set of configuration; (2) a no-passing property; and (3) an adiabatic response to monotonously changing fields, implies RPM. When periodically driven, such systems settle into a cyclic response after a transient of at most one period. While sufficient, conditions (1)–(3) are not necessary. We show that the AQS dynamics provides a more selective partial order which, due to its explicit connection to hysteresis loops, is a natural choice for establishing the RPM property. This enables us to consider AQS-A exhibiting RPM without necessarily possessing the no-passing property. We call such automata \(\ell \)AQS-A and work out the structure of their state transition graphs. We find that the RPM property constrains the *intra-loop* structure of hysteresis loops, namely its hierarchical organization into sub-loops, but not the *inter-loop* structure. We prove that the topology of the intra-loop structure can be represented as an ordered tree and show that the corresponding state transition graph is planar. On the other hand, the RPM property does not significantly restrict the inter-loop transitions. A system exhibiting RPM and subject to periodic forcing can undergo a large number of transient cycles before settling into a periodic response. Such systems can even exhibit subharmonic response.

## Mathematics Subject Classification

Primary 82D30 Secondary 82C44## Notes

### Acknowledgements

The authors would like to thank M. Işeri and D.C. Kaspar for many fruitful exchanges. They also acknowledge discussions with B. Behringer, A. Bovier, K. Dahmen, N. Keim, J. Krug, C. Maloney, A. A. Middleton, S. Nagel, A. Rosso, S. Sastry, K. Sekimoto, J. Sethna, D. Vandembroucq, T. A. Witten, and A. Yılmaz. Many of these took place during the Memory Formation in Matter program of KITP, and the authors thank KITP for its kind hospitality.

## References

- 1.Fleming, R.M., Schneemeyer, L.F.: Observation of a pulse-duration memory effect in \({\rm K}_{0.30}{\rm MoO}_{3}\). Phys. Rev. B
**33**, 2930–2932 (1986)ADSCrossRefGoogle Scholar - 2.Pierce, M.S., Moore, R.G., Sorensen, L.B., Kevan, S.D., Hellwig, O., Fullerton, E.E., Kortright, J.B.: Quasistatic X-ray speckle metrology of microscopic magnetic return-point memory. Phys. Rev. Lett.
**90**, 175502 (2003)ADSCrossRefGoogle Scholar - 3.Pierce, M.S., Buechler, C.R., Sorensen, L.B., Turner, J.J., Kevan, S.D., Jagla, E.A., Deutsch, J.M., Mai, T., Narayan, O., Davies, J.E., Liu, K., Dunn, J.H., Chesnel, K.M., Kortright, J.B., Hellwig, O., Fullerton, E.E.: Disorder-induced microscopic magnetic memory. Phys. Rev. Lett.
**94**, 017202 (2005)ADSCrossRefGoogle Scholar - 4.Pierce, M.S., Buechler, C.R., Sorensen, L.B., Kevan, S.D., Jagla, E.A., Deutsch, J.M., Mai, T., Narayan, O., Davies, J.E., Liu, Kai, Zimanyi, G.T., Katzgraber, H.G., Hellwig, O., Fullerton, E.E., Fischer, P., Kortright, J.B.: Disorder-induced magnetic memory: experiments and theories. Phys. Rev. B
**75**, 144406 (2007)ADSCrossRefGoogle Scholar - 5.Hauet, T., Guenther, C.M., Pfau, B., Schabes, M.E., Thiele, J.U., Rick, R.L., Fischer, P., Eisebitt, S., Hellwig, O.: Direct observation of field and temperature induced domain replication in dipolar coupled perpendicular anisotropy films. Phys. Rev. B
**77**, 184421 (2008)ADSCrossRefGoogle Scholar - 6.Pine, D.J., Gollub, J.P., Brady, J.F., Leshansky, A.M.: Chaos and threshold for irreversibility in sheared suspensions. Nature
**438**, 997–1000 (2005)ADSCrossRefGoogle Scholar - 7.Corte, L., Chaikin, P.M., Gollub, J.P., Pine, D.J.: Random organization in periodically driven systems. Nat. Phys.
**4**(5), 420 (2008)CrossRefGoogle Scholar - 8.Paulsen, J.D., Keim, N.C., Nagel, S.R.: Multiple transient memories in experiments on sheared non-Brownian suspensions. Phys. Rev. Lett
**113**, 068301 (2014)ADSCrossRefGoogle Scholar - 9.Toiya, M., Stambaugh, J., Losert, W.: Transient and oscillatory granular shear flow. Phys. Rev. Lett.
**93**(8), 088001 (2004)ADSCrossRefGoogle Scholar - 10.Mueggenburg, N.W.: Behavior of granular materials under cyclic shear. Phys. Rev. E
**71**(3), 031301 (2005)ADSCrossRefGoogle Scholar - 11.Ren, J., Dijksman, J.A., Behringer, R.P.: Reynolds pressure and relaxation in a sheared granular system. Phys. Rev. Lett.
**110**(1), 018302 (2013)ADSCrossRefGoogle Scholar - 12.Haw, M.D., Poon, W.C.K., Pusey, P.N., Hebraud, P., Lequeux, F.: Colloidal glasses under shear strain. Phys. Rev. E
**58**(4), 4673 (1998)ADSCrossRefGoogle Scholar - 13.Petekidis, G., Moussaid, A., Pusey, P.N.: Rearrangements in hard-sphere glasses under oscillatory shear strain. Phys. Rev. E
**66**(5), 051402 (2002)ADSCrossRefGoogle Scholar - 14.Lundberg, M., Krishan, K., Xu, N., O’Hern, C.S., Dennin, M.: Reversible plastic events in amorphous materials. Phys. Rev. E
**77**, 041505 (2008)ADSCrossRefGoogle Scholar - 15.Tang, C., Wiesenfeld, K., Bak, P., Coppersmith, S., Littlewood, P.: Phase organization. Phys. Rev. Lett.
**58**, 1161–1164 (1987)ADSMathSciNetCrossRefGoogle Scholar - 16.Coppersmith, S.N., Littlewood, P.B.: Pulse-duration memory effect and deformable charge-density waves. Phys. Rev. B
**36**, 311–317 (1987)ADSCrossRefGoogle Scholar - 17.Dahmen, K.A., Ben-Zion, Y., Uhl, J.T.: Micromechanical model for deformation in solids with universal predictions for stress–strain curves and slip avalanches. Phys. Rev. Lett.
**102**, 175501 (2009)ADSCrossRefGoogle Scholar - 18.Keim, N.C., Nagel, S.R.: Generic transient memory formation in disordered systems with noise. Phys. Rev. Lett.
**107**, 010603 (2011)ADSCrossRefGoogle Scholar - 19.Libal, A., Reichhardt, C., Olson Reichhardt, C.J.: Hysteresis and return-point memory in colloidal artificial spin ice systems. Phys. Rev. E
**86**, 021406 (2012)ADSCrossRefGoogle Scholar - 20.Keim, N.C., Paulsen, J.D., Nagel, S.R.: Multiple transient memories in sheared suspensions: robustness, structure, and routes to plasticity. Phys. Rev. E
**88**, 032306 (2013)ADSCrossRefGoogle Scholar - 21.Fiocco, D., Foffi, G., Sastry, S.: Encoding of memory in sheared amorphous solids. Phys. Rev. Lett.
**112**, 025702 (2014)ADSCrossRefGoogle Scholar - 22.Fiocco, D., Foffi, G., Sastry, S.: Memory effects in schematic models of glasses subjected to oscillatory deformation. J. Phys. Condens. Matter
**27**, 194130 (2015)ADSCrossRefGoogle Scholar - 23.Regev, I., Weber, J., Reichhardt, C., Dahmen, K.A., Lookman, T.: Reversibility and criticality in amorphous solids. Nat. Commun.
**6**, 8805 (2015)ADSCrossRefGoogle Scholar - 24.Lin, J., Gueudré, T., Rosso, A., Wyart, M.: Criticality in the approach to failure in amorphous solids. Phys. Rev. Lett.
**115**, 168001 (2015)ADSCrossRefGoogle Scholar - 25.Leishangthem, P., Parmar, A.D.S., Sastry, S.: The yielding transition in amorphous solids under oscillatory shear deformation. Nat. Commun.
**8**, 14653 (2017)ADSCrossRefGoogle Scholar - 26.Keim, N.C., Paulsen, J., Zeravcic, Z., Sastry, S., Nagel, S.R.: Memory formation in matter (2018). arXiv preprint arXiv:1810.08587
- 27.Barker, J.A., Schreiber, D.E., Huth, B.G., Everett, D.H.: Magnetic hysteresis and minor loops: models and experiments. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci.
**386**, 251–261 (1983)ADSCrossRefGoogle Scholar - 28.Sethna, J.P., Dahmen, K., Kartha, S., Krumhansl, J.A., Roberts, B.W., Shore, J.D.: Hysteresis and hierarchies: dynamics of disorder-driven first-order phase transformations. Phys. Rev. Lett.
**70**, 3347 (1993)ADSCrossRefGoogle Scholar - 29.Sethna, J.P., Dahmen, K.A., Myers, C.R.: Crackling noise. Nature
**410**, 242–250 (2001)ADSCrossRefGoogle Scholar - 30.Sethna, J.P., Dahmen, K.A., Perkovic, O.: Random field ising models of hysteresis. In: Bertotti, G., Mayergoyz, I. (eds.) The Science of Hysteresis, vol. 2, pp. 107–179. Academic Press, New York (2006)CrossRefGoogle Scholar
- 31.Middleton, A.A.: Asymptotic uniqueness of the sliding state for charge-density waves. Phys. Rev. Lett.
**68**, 670–673 (1992)ADSCrossRefGoogle Scholar - 32.Malandro, D.L., Lacks, D.J.: Relationships of shear-induced changes in the potential energy landscape to the mechanical properties of ductile glasses. J. Chem. Phys.
**110**, 4593–4601 (1999)ADSCrossRefGoogle Scholar - 33.Maloney, C.E., Lemaître, A.: Amorphous systems in athermal, quasistatic shear. Phys. Rev. E
**74**, 016118 (2006)ADSCrossRefGoogle Scholar - 34.Deutsch, J.M., Dhar, A., Narayan, O.: Return to return point memory. Phys. Rev. Lett.
**92**, (2004)Google Scholar - 35.Katzgraber, H.G., Zimanyi, G.T.: Hysteretic memory effects in disordered magnets. Phys. Rev. B
**74**, 020405 (2006)ADSCrossRefGoogle Scholar - 36.Han, Y., Shokef, Y., Alsayed, A.M., Yunker, P., Lubensky, T.C., Yodh, A.G.: Geometric frustration in buckled colloidal monolayers. Nature
**456**, 898–903 (2008)ADSCrossRefGoogle Scholar - 37.Gilbert, I., Chern, G.-W., Fore, B., Lao, Y., Zhang, S., Nisoli, C., Schiffer, P.: Direct visualization of memory effects in artificial spin ice. Phys. Rev. B
**92**, 104417 (2015)ADSCrossRefGoogle Scholar - 38.Keim, N.C., Hass, J., Kroger, B., Wieker, D.: Return-point memory in an amorphous solid (2018). arXiv preprint arXiv:1809.08505
- 39.Deutsch, J.M., Narayan, O.: Subharmonics and aperiodicity in hysteresis loops. Phys. Rev. Lett.
**91**, 200601 (2003)ADSCrossRefGoogle Scholar - 40.Lavrentovich, M.O., Liu, A.J., Nagel, S.R.: Period proliferation in periodic states in cyclically sheared jammed solids. Phys. Rev. E
**96**, 020101 (2017)ADSCrossRefGoogle Scholar - 41.Fukuyama, H., Lee, P.A.: Dynamics of the charge-density wave. I. impurity pinning in a single chain. Phys. Rev. B
**17**, 535 (1978)ADSCrossRefGoogle Scholar - 42.Lee, P.A., Rice, T.M.: Electric field depinning of charge density waves. Phys. Rev. B
**19**, 3970 (1979)ADSCrossRefGoogle Scholar - 43.Kaspar, D.C., Mungan, M.: Subthreshold behavior and avalanches in an exactly solvable charge density wave system. EPL
**103**, 46002 (2013)ADSCrossRefGoogle Scholar - 44.Kaspar, D.C., Mungan, M.: Exact results for a toy model exhibiting dynamic criticality. Ann. Henri Poincaré
**16**, 2837–2879 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 45.Terzi, M.M., Mungan, M.: The structure of state transition graphs in hysteresis models with return point memory. II. Applications (
**in preparation**) (2018)Google Scholar - 46.Magni, A., Basso, V.: Study of metastable states in the random-field ising model. J. Mag. Mag. Mat.
**290**, 460–463 (2005)ADSCrossRefGoogle Scholar - 47.Bertotti, G., Bortolotti, P., Magni, A., Basso, V.: Topological and energetic aspects of the random-field Ising model. J. Appl. Phys.
**101**, 09D508 (2007)CrossRefGoogle Scholar - 48.Bortolotti, P., Basso, V., Magni, A., Bertotti, G.: Oriented graph structure of local energy minima in the random-field ising model. Physica B
**403**, 398–401 (2008)ADSCrossRefGoogle Scholar