say, $r_{m}(n,1.9)>e^{cn}$ for some constant $c$.

In some mixing systems, the dependence of $r_{m}(n,\delta)$ on $n$ is known.

######  Example 19.1 (The contact process.).

The contact process is a one-dimensional CCA as defined in Subsection 2.4. Let
us note that this this process is not noisy: not all local transition rates
are positive. The process has states 0,1. In trajectory $\eta(x,t)$, state
$\eta(x,t)=1$ turns into 0 with rate 1. State $\eta(x,t)=0$ turns into 1 with
rate $\lambda(\eta(x-1,t)+\eta(x+1,t))$. It is known that this process has a
critical rate $\lambda_{c}\in[0+,\infty-]$ with the following properties.

If $\lambda\leq\lambda_{c}$ then the process is mixing with the invariant
measure concentrated on the configuration $\xi$ with $\xi(x)=0$ for all $x$.
If $\lambda>\lambda_{c}$ then the process is non-ergodic.

If $\lambda<\lambda_{c}$ then it is known (see Theorem 3.4 in Chapter VI of
[20]) that the order of magnitude of $r_{\infty}(n,\delta)$ is $\log n$.

If $\lambda=\lambda_{c}$ then the convergence is much slower, with an order of
magnitude that is a power of $n$ (see Theorem 3.10 in Chapter VI of [20] and
[6]). ∎

We believe it possible to construct a medium that is mixing for $m=\infty$ but
loses information arbitrary slowly: for any computable function $f(n)$, and a
constant $c$ there is a medium with

$r_{m}(n,1.9)>f(n)\wedge e^{cm}$

for finite or infinite $m$. Notice that this includes functions $f(n)$ like
$e^{e^{e^{n}}}$. Such a result could be viewed as an argument against the
relevance of the non-ergodicity of the infinite medium for practical
information conservation in a finite medium. The construction could be based
on the ability to perform arbitrary computation reliably and therefore also to
destroy locally identifiable information arbitrarily slowly.

## References

  * [1] Michael Aizenman, Translation invariance and instability of phase coexistence in the two-dimensional Ising system, Comm. Math. Phys 73 (1980), no. 1, 83–94.
  * [2] Charles H. Bennett, G. Grinstein, Yu He, C. Jayaprakash, and David Mukamel, Stability of temporally periodic states of classical many-body systems, Physical Review A 41 (1990), 1932–1935.
  * [3] Charles H. Bennett and Geoffrey Grinstein, Role of irreversibility in stabilizing complex and nonergodic behavior in locally interacting discrete systems, Physical Review Letters 55 (1985), 657–660.
  * [4] E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning ways for your mathematical plays, Academic Press, New York, 1982.
  * [5] Piotr Berman and Janos Simon, Investigations of fault-tolerant networks of computers, Proc. of the 20-th Annual ACM Symp. on the Theory of Computing, 1988, pp. 66–77.
  * [6] C. Bezuidenhout and G. Grimmett, The critical contact process dies out, Annals of Probability 18 (1990), 1462–1482.
  * [7] R. E. Blahut, Theory and practice of error-control codes, Addison-Wesley, Reading, MA, 1983.
  * [8] Paula Gonzaga de Sá and Christian Maes, The Gács-Kurdyumov-Levin Automaton revisited, Journal of Statistical Physics 67 (1992), no. 3/4, 607–622.
  * [9] R. L. Dobrushin and S. I. Ortyukov, Upper bound on the redundancy of self-correcting arrangements of unreliable elements, Problems of Information Transmission 13 (1977), no. 3, 201–208.
  * [10] Peter Gács, Reliable computation with cellular automata, Journal of Computer System Science 32 (1986), no. 1, 15–78.
  * [11] by same author, Self-correcting two-dimensional arrays, Randomness in Computation (Greenwich, Conn.) (Silvio Micali, ed.), Advances in Computing Research (a scientific annual), vol. 5, JAI Press, Greenwich, Conn., 1989, pp. 223–326.
  * [12] by same author, Deterministic parallel computations whose history is independent of the order of updating, Tech. report, Boston University, Department of Computer Science, Boston, MA 02215, 1995.
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  * [17] by same author, The behavior of processes with statistical mechanical properties, Percolation Theory and Ergodic Theory of Infinite Particle Systems, Springer-Verlag, 1987, pp. 131–167.
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  * [19] G. L. Kurdyumov, An example of a nonergodic homogenous one-dimensional random medium with positive transition probabilities, Soviet Mathematical Doklady 19 (1978), no. 1, 211–214.
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  * [21] Jacques Neveu, Bases mathematiques du calcul des probabilités, Masson et Cie, Paris, 1964.
  * [22] Kihong Park, Ergodicity and mixing rate of one-dimensional cellular automata, Ph.D. thesis, Boston University, Boston, MA 02215, 1996.
  * [23] Nicholas Pippenger, On networks of noisy gates, Proc. of the 26-th IEEE FOCS Symposium, 1985, pp. 30–38.
  * [24] Charles Radin, Global order from local sources, Bull. Amer. Math. Soc. 25 (1991), 335–364.
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  * [26] Tommaso Toffoli and Norman Margolus, Cellular automata machines, (Cambridge, MA.), MIT Press, Cambridge, MA., 1987.
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## Notation

  * accepted by decoding §3.2.1
  * active level §4.2
  * address §3.1
  * affecting
    * directly §11.1
    * via neighbors §11.1
  * age
    * effective §7.1
  * aggregation Example 3.2
  * amplifier §3.6
    * broadcast §9.2
    * error-correcting §3.6.2
    * frame §9.1, §9.1.3
      * broadcast §9.1.2
    * initially stable Lemma 3.7
    * self-organizing §10.1
  * atomicity item e
  * attribution §14.2
  * Property
    * Computation §8.1
    * Restoration §8.1
  * bandwidth §2.2.1
  * birth §10.1
  * blue §10.1
    * trajectory §10.1
  * body §3.3.1
    * space-time §3.3.1
  * broadcast item d, §4.1.3
  * capacity §2.2
  * cell §3.3.1
    * channel §11.3
    * dead §11.3
    * doomed §11.4
    * expansion §12.1
    * exposed item Shrinking
    * father §12.5
    * frozen §11.6.2
    * germ §11.3
    * growth §11.3
    * inner §11.3
    * kind §11.3
    * latent §11.3
    * live §11.3
    * member §11.3
    * mother §12.3, §12.5
    * outer §11.3
    * strength §11.3
    * vacant §11.3
  * cellular automaton
    * deterministic §2.1, §3.3.1
    * probabilistic §2.3
    * totally asynchronous §7.1
  * code §3.2.1
    * bits
      * error check Example 4.5
      * information Example 4.5
    * block
      * overlap-free §3.3.2
    * composition §4.1.1
    * error-correcting Example 4.5
    * hierarchical §4.1.2
    * limit §4.1.4
      * approximation §4.1.4
    * linear Example 4.5
    * Reed-Solomon Example 4.6
  * colony §3.1, §7.3
    * base §3.1
    * endcell §12.1
    * full §11.6.1
    * in a cellular medium §3.3.2
  * computation
    * result §7.2
    * size §5.2
  * Condition
    * Address and Age Condition 11.4
    * Animation Condition 12.10
    * Cling to Life Condition 8.7
    * Creation Condition 8.6
    * Dooming Condition 12.2
    * Freeze Condition 12.3
    * Killing Condition 12.11
    * Latent Cells Condition 11.3
    * local §6.1.1
      * disjoint §6.1.1
      * random §6.1.2
      * random-stopping §6.1.2
    * Outer Info Condition 11.5
    * Time Marking Condition 8.5
    * Time Stability item Time Stability
    * type §6.1.1
    * Waiting Condition 11.2
  * configuration §2.1, §3.3.1
    * input §5.2
    * lattice §3.3.1
    * space-time §2.1, §3.3.1, §3.3.1
      * random §2.3
  * consistent
    * space- §11.6.1
  * contact process Example 19.1
  * control
    * delegation §10.1, Lemma 18.4
    * lasting §10.1, Lemma 18.2
  * controlling §10.1, §18.1, §4.1.3
  * correction
    * end §12.5
    * internal §12.5
    * near-end §12.5
  * creation §10.1
  * creator §8.2.2
    * potential §8.2.2
  * critical rate Example 19.1
  * cut item Finish
  * damage §8.1
    * rectangle §11.1
      * extended §11.1
      * healing wake §16.5
      * wake §13.1
  * damage-free §8.1
  * deviation §5.2
  * domain §11.6.1
    * multi- §12.1
  * don’t-care symbol §3.6.2
  * edge
    * exposed §12.1
      * weak §14.1
    * protected §12.1
  * ergodic §5.1.1
  * error §16.2
    * correction item Error-correction, §3.6.2
  * event function §2.3
  * field §2.2
    * broadcast §4.1.3
    * identification §4.1.3
    * locally maintained §11.4
    * rider §9.1.1
    * shared §4.1.4
      * guarded §4.2
      * primitive §4.1.3
  * fitted sequence §4.1.2
  * forgetful
    * strongly not §5.1.4
    * uniformly §5.1.3
  * Galois field Example 4.6
  * gap §13.2
    * bad §14.1
    * right-age §13.2
  * germ
    * active §18.2
    * computation §18.2
    * exposed §18.2
    * growth §18.2
    * leader §18.2
    * level §18.2
    * passive §18.2
  * hard-wiring §4.3
  * interacting particle system §2.4
  * invariant histories §7.1
  * Ising model §19.1
  * killing §11.3
  * Lemma
    * Age Check Lemma 17.2
    * Amplifier Lemma 9.4
    * Ancestor §11.7, Lemma 13.3
    * Animation Support Lemma 12.9
    * Bad Gap Inference 3rd item, Lemma 14.2
    * Bad Gap Opening Lemma 14.1
    * Birth Lemma 18.8
    * Cover Lemma 14.4
    * Creation Lemma 12.8
    * Crossing Lemma 13.2
    * Exposing Lemma 12.15
    * Germ Attribution Lemma 18.6
    * Glue Lemma 12.13
    * Growth Lemma 17.4
    * Healing Lemma 15.1
    * Initially Stable Amplifier Lemma 3.7
    * Large Progress Lemma 14.8
    * Legality 2nd item
    * Level Increase Lemma 18.7
    * Parent Lemma 12.12
    * Present Attribution 3rd item, Lemma 16.7
    * Refresh Lemma 16.1
    * Retrieval Lemma 17.3
    * Running Gap 2nd item
    * Self-Organization Lemma 10.2
    * Simulation Damage Probability Bound Lemma 8.4
    * Small Progress Lemma 14.3
  * link §13.1
  * location §7.3
  * Lower Bound
    * Bandwidth item Bandwidth Lower Bound, item Bandwidth Lower Bound
    * Capacity item Capacity Lower Bound, item Capacity Lower Bounds
    * Cell Capacity item Cell Capacity Lower Bound
    * Colony Size item Colony Size Lower Bound
    * Redundancy item Redundancy Lower Bound
    * Work Period item Work Period Lower Bound, item Work Period Lower Bound, item Work Period Lower Bound
  * marching soldiers §11.6.2, §7.1
  * Markov
    * chain §2.3
    * operator §5.1.1
    * process §2.4.1
  * measurable space §2.3
  * measure
    * invariant §5.1.1
  * medium §6.1.1
    * abstract §3.3
      * cellular §3.3.1
    * constant-period §3.3.1
    * robust §8
    * standard computing §7.2
    * variable-period §3.3.1
      * primitive §6.3
  * mixing §5.1.1
  * monotonic output §5.2
  * noisy §2.3
  * non-degenerate §4.1.2
  * organized into colonies §3.1
  * packet §16.1.1
  * partners §12.2
  * path
    * boundary §14.1
    * forward, backward §13.1
  * period
    * computing §16.2
    * damage-free §16.2
    * dwell §3.3.1
      * lower bound §6.3
    * expansion §12.1
    * growth item Shrinking
    * observation Figure 15
    * wait §13.2
    * work §11.6.1, §3.1
      * boundary §11.6.1
      * germ §18.2
      * size §3.4
      * transition §18.3
  * perturbation §2.5.1
  * program
    * rule §7.3
    * uniform §9.1.2
  * rectangle process §6.1.2
  * redundancy
    * space §4.1.4
    * time §5.2
  * refreshing step §11.4
  * relatives §11.6.1
  * relaxation time §19.2.2, §5.1.2
  * remembering §2.5.3
  * renormalization §3.6
  * rule §7.3
    * default §7.3
    * sub- §7.3
  * self
    * organization §5.2.2, §6.5
    * reference §7.3
    * stabilization 8th item
  * separability §2.2.1
  * sibling §11.6
    * strong §14.1
  * simulation item Simulation, §3.6
    * block code §3.4
    * canonical §6.2.1
      * deterministic §6.2.1
      * injective Lemma 18.7, §6.2.2
    * local §3.6
    * memoryless §3.6
    * non-anticipating §3.6
    * non-local §7.1
  * site §2.1
    * free §7.1
    * map §4.1.4
  * sparsity §5.2
  * spin system §19.1
  * standard
    * alphabet §3.6.2
    * system of shared fields §4.1.5
  * state
    * Gibbs §19.1
    * newborn §8.2.2
      * canonical §8.2.2
    * vacant §3.3.1
  * stopping time §6.1.2
  * successor §15.1
  * supercolony §4.1.1
  * support §5.2
    * $n$- §11.6.1
  * switch §2.1
    * time §6.2.2
  * temperature §19.1
  * Theorem
    * Asynchronous Simulation Theorem 7.1
    * Basic Block Simulation Theorem 7.13
    * Canonical Simulation Theorem 6.9
    * CCA-simulation Theorem 6.12
    * Reach Extension Theorem 8.10
    * Rule Language Theorem 7.8
  * Toom rule §19.2.2, §5.1.1
  * track §2.2
    * mail §7.4.2
  * trajectory §2.3, §3.3, §6.1.2
    * deterministic cellular automaton §2.1, §3.3.1
    * weak §6.1.1
  * transition function §2.1
    * aggregated Example 3.4
    * combined §9.1.1
    * commutative §7.1
    * program of §7.2.1
    * universal §3.4
      * efficiently §7.2.1
  * transition matrix §2.3
  * transition rate §2.4
  * translation-invariance §6.1.1
  * Turing machine §19.2.1, §2.2
  * update
    * age §11.4
    * interval §11.4
    * set §7.1
  * Upper Bound
    * Complexity item Complexity Upper Bounds, item Complexity Upper Bounds
    * Error item Error Upper Bound
  * vacant symbol §3.2.1
  * weak convergence §5.1.1

glo

## Index

  * accepted by decoding §3.2.1
  * active level §4.2
  * address §3.1
  * affecting
    * directly §11.1
    * via neighbors §11.1
  * age
    * effective §7.1
  * aggregation Example 3.2
  * amplifier §3.6
    * broadcast §9.2
    * error-correcting §3.6.2
    * frame §9.1, §9.1.3
      * broadcast §9.1.2
    * initially stable Lemma 3.7
    * self-organizing §10.1
  * atomicity item e
  * attribution §14.2
  * Property
    * Computation §8.1
    * Restoration §8.1
  * bandwidth §2.2.1
  * birth §10.1
  * blue §10.1
    * trajectory §10.1
  * body §3.3.1
    * space-time §3.3.1
  * broadcast item d, §4.1.3
  * capacity §2.2
  * cell §3.3.1
    * channel §11.3
    * dead §11.3
    * doomed §11.4
    * expansion §12.1
    * exposed item Shrinking
    * father §12.5
    * frozen §11.6.2
    * germ §11.3
    * growth §11.3
    * inner §11.3
    * kind §11.3
    * latent §11.3
    * live §11.3
    * member §11.3
    * mother §12.3, §12.5
    * outer §11.3
    * strength §11.3
    * vacant §11.3
  * cellular automaton
    * deterministic §2.1, §3.3.1
    * probabilistic §2.3
    * totally asynchronous §7.1
  * code §3.2.1
    * bits
      * error check Example 4.5
      * information Example 4.5
    * block
      * overlap-free §3.3.2
    * composition §4.1.1
    * error-correcting Example 4.5
    * hierarchical §4.1.2
    * limit §4.1.4
      * approximation §4.1.4
    * linear Example 4.5
    * Reed-Solomon Example 4.6
  * colony §3.1, §7.3
    * base §3.1
    * endcell §12.1
    * full §11.6.1
    * in a cellular medium §3.3.2
  * computation
    * result §7.2
    * size §5.2
  * Condition
    * Address and Age Condition 11.4
    * Animation Condition 12.10
    * Cling to Life Condition 8.7
    * Creation Condition 8.6
    * Dooming Condition 12.2
    * Freeze Condition 12.3
    * Killing Condition 12.11
    * Latent Cells Condition 11.3
    * local §6.1.1
      * disjoint §6.1.1
      * random §6.1.2
      * random-stopping §6.1.2
    * Outer Info Condition 11.5
    * Time Marking Condition 8.5
    * Time Stability item Time Stability
    * type §6.1.1
    * Waiting Condition 11.2
  * configuration §2.1, §3.3.1
    * input §5.2
    * lattice §3.3.1
    * space-time §2.1, §3.3.1, §3.3.1
      * random §2.3
  * consistent
    * space- §11.6.1
  * contact process Example 19.1
  * control
    * delegation §10.1, Lemma 18.4
    * lasting §10.1, Lemma 18.2
  * controlling §10.1, §18.1, §4.1.3
  * correction
    * end §12.5
    * internal §12.5
    * near-end §12.5
  * creation §10.1
  * creator §8.2.2
    * potential §8.2.2
  * critical rate Example 19.1
  * cut item Finish
  * damage §8.1
    * rectangle §11.1
      * extended §11.1
      * healing wake §16.5
      * wake §13.1
  * damage-free §8.1
  * deviation §5.2
  * domain §11.6.1
    * multi- §12.1
  * don’t-care symbol §3.6.2
  * edge
    * exposed §12.1
      * weak §14.1
    * protected §12.1
  * ergodic §5.1.1
  * error §16.2
    * correction item Error-correction, §3.6.2
  * event function §2.3
  * field §2.2
    * broadcast §4.1.3
    * identification §4.1.3
    * locally maintained §11.4
    * rider §9.1.1
    * shared §4.1.4
      * guarded §4.2
      * primitive §4.1.3
  * fitted sequence §4.1.2
  * forgetful
    * strongly not §5.1.4
    * uniformly §5.1.3
  * Galois field Example 4.6
  * gap §13.2
    * bad §14.1
    * right-age §13.2
  * germ
    * active §18.2
    * computation §18.2
    * exposed §18.2
    * growth §18.2
    * leader §18.2
    * level §18.2
    * passive §18.2
  * hard-wiring §4.3
  * interacting particle system §2.4
  * invariant histories §7.1
  * Ising model §19.1
  * killing §11.3
  * Lemma
    * Age Check Lemma 17.2
    * Amplifier Lemma 9.4
    * Ancestor §11.7, Lemma 13.3
    * Animation Support Lemma 12.9
    * Bad Gap Inference 3rd item, Lemma 14.2
    * Bad Gap Opening Lemma 14.1
    * Birth Lemma 18.8
    * Cover Lemma 14.4
    * Creation Lemma 12.8
    * Crossing Lemma 13.2
    * Exposing Lemma 12.15
    * Germ Attribution Lemma 18.6
    * Glue Lemma 12.13
    * Growth Lemma 17.4
    * Healing Lemma 15.1
    * Initially Stable Amplifier Lemma 3.7
    * Large Progress Lemma 14.8
    * Legality 2nd item
    * Level Increase Lemma 18.7
    * Parent Lemma 12.12
    * Present Attribution 3rd item, Lemma 16.7
    * Refresh Lemma 16.1
    * Retrieval Lemma 17.3
    * Running Gap 2nd item
    * Self-Organization Lemma 10.2
    * Simulation Damage Probability Bound Lemma 8.4
    * Small Progress Lemma 14.3
  * link §13.1
  * location §7.3
  * Lower Bound
    * Bandwidth item Bandwidth Lower Bound, item Bandwidth Lower Bound
    * Capacity item Capacity Lower Bound, item Capacity Lower Bounds
    * Cell Capacity item Cell Capacity Lower Bound
    * Colony Size item Colony Size Lower Bound
    * Redundancy item Redundancy Lower Bound
    * Work Period item Work Period Lower Bound, item Work Period Lower Bound, item Work Period Lower Bound
  * marching soldiers §11.6.2, §7.1
  * Markov
    * chain §2.3
    * operator §5.1.1
    * process §2.4.1
  * measurable space §2.3
  * measure
    * invariant §5.1.1
  * medium §6.1.1
    * abstract §3.3
      * cellular §3.3.1
    * constant-period §3.3.1
    * robust §8
    * standard computing §7.2
    * variable-period §3.3.1
      * primitive §6.3
  * mixing §5.1.1
  * monotonic output §5.2
  * noisy §2.3
  * non-degenerate §4.1.2
  * organized into colonies §3.1
  * packet §16.1.1
  * partners §12.2
  * path
    * boundary §14.1
    * forward, backward §13.1
  * period
    * computing §16.2
    * damage-free §16.2
    * dwell §3.3.1
      * lower bound §6.3
    * expansion §12.1
    * growth item Shrinking
    * observation Figure 15
    * wait §13.2
    * work §11.6.1, §3.1
      * boundary §11.6.1
      * germ §18.2
      * size §3.4
      * transition §18.3
  * perturbation §2.5.1
  * program
    * rule §7.3
    * uniform §9.1.2
  * rectangle process §6.1.2
  * redundancy
    * space §4.1.4
    * time §5.2
  * refreshing step §11.4
  * relatives §11.6.1
  * relaxation time §19.2.2, §5.1.2
  * remembering §2.5.3
  * renormalization §3.6
  * rule §7.3
    * default §7.3
    * sub- §7.3
  * self
    * organization §5.2.2, §6.5
    * reference §7.3
    * stabilization 8th item
  * separability §2.2.1
  * sibling §11.6
    * strong §14.1
  * simulation item Simulation, §3.6
    * block code §3.4
    * canonical §6.2.1
      * deterministic §6.2.1
      * injective Lemma 18.7, §6.2.2
    * local §3.6
    * memoryless §3.6
    * non-anticipating §3.6
    * non-local §7.1
  * site §2.1
    * free §7.1
    * map §4.1.4
  * sparsity §5.2
  * spin system §19.1
  * standard
    * alphabet §3.6.2
    * system of shared fields §4.1.5
  * state
    * Gibbs §19.1
    * newborn §8.2.2
      * canonical §8.2.2
    * vacant §3.3.1
  * stopping time §6.1.2
  * successor §15.1
  * supercolony §4.1.1
  * support §5.2
    * $n$- §11.6.1
  * switch §2.1
    * time §6.2.2
  * temperature §19.1
  * Theorem
    * Asynchronous Simulation Theorem 7.1
    * Basic Block Simulation Theorem 7.13
    * Canonical Simulation Theorem 6.9
    * CCA-simulation Theorem 6.12
    * Reach Extension Theorem 8.10
    * Rule Language Theorem 7.8
  * Toom rule §19.2.2, §5.1.1
  * track §2.2
    * mail §7.4.2
  * trajectory §2.3, §3.3, §6.1.2
    * deterministic cellular automaton §2.1, §3.3.1
    * weak §6.1.1
  * transition function §2.1
    * aggregated Example 3.4
    * combined §9.1.1
    * commutative §7.1
    * program of §7.2.1
    * universal §3.4
      * efficiently §7.2.1
  * transition matrix §2.3
  * transition rate §2.4
  * translation-invariance §6.1.1
  * Turing machine §19.2.1, §2.2
  * update
    * age §11.4
    * interval §11.4
    * set §7.1
  * Upper Bound
    * Complexity item Complexity Upper Bounds, item Complexity Upper Bounds
    * Error item Error Upper Bound
  * vacant symbol §3.2.1
  * weak convergence §5.1.1

ind


# Direct confirmation of long-range magnetic order and evidence for multipoles
in Ce2O3

Alexandra Cote  Department of Physics and Materials Research Laboratory,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  J.
Eddie Slimak  Department of Physics and Materials Research Laboratory,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  Astha
Sethi  Department of Physics and Materials Research Laboratory, University of
Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  Dalmau Reig-i-
Plessis  Department of Physics and Astronomy and Stewart Blusson Quantum
Matter Institute, University of British Columbia, Vancouver, Canada V6T 1Z4
Qiang Zhang  Neutron Scattering Division, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37831, USA  Yang Zhao  NIST Center for Neutron Research,
National Institute of Standards and Technology, Gaithersburg, Maryland 20899,
USA and Department of Materials Science and Engineering, University of
Maryland, College Park, Maryland 20742, USA  Devashibhai Adroja  ISIS Neutron
and Muon Facility, STFC, Rutherford Appleton Laboratory, Chilton, Didcot Oxon,
OX11 0QX, United Kingdom and Highly Correlated Matter Research Group, Physics
Department, University of Johannesburg, Auckland Park 2006, South Africa
Gerald Morris  TRIUMF Centre for Molecular and Materials Science, Vancouver,
BC V6T 2A3, Canada  Taras Kolodiazhnyi  National Institute for Materials
Science, 1-1 Namiki, Tsukuba, Ibaraki, 305-0044, Japan  Alannah M. Hallas
Department of Physics and Astronomy and Stewart Blusson Quantum Matter
Institute, University of British Columbia, Vancouver, Canada V6T 1Z4  Jeffrey
W. Lynn  NIST Center for Neutron Research, National Institute of Standards and
Technology, Gaithersburg, Maryland 20899, USA  S. Lance Cooper  Department of
Physics and Materials Research Laboratory, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA  Gregory J. MacDougall  Department of
Physics and Materials Research Laboratory, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA

###### Abstract

The sesquioxide, Ce2O3, has been a material of intense interest in recent
years due to reports of an anomalous giant magnetodielectric effect and
emergent mixed crystal field-phonon (vibronic) excitations below a putative
antiferromagnetic transition at TN = 6.2 K. The claim of long-range magnetic
order in this material is based on heat capacity and temperature-dependent
susceptibility measurements; however, multiple neutron diffraction studies
have been unable to distinguish any magnetic Bragg peaks. In this article, we
present the results of a comprehensive investigation of the low-temperature
phase in symmetry-broken polycrystalline Ce2O3 using a combination of magnetic
susceptibility, heat capacity, neutron diffraction, triple-axis and time-of-
flight (TOF) inelastic neutron scattering (INS), and muon spin rotation
($\mu$SR). Our measurements and subsequent analysis confirm that the
transition at TN can be associated with the ordering of moments on the Ce3+
site. Both a spontaneous magnetic order observed with $\mu$SR and a dispersive
spin-wave spectrum observed with inelastic neutron scattering suggest a model
wherein planar dipoles order antiferromagnetically. Notable inconsistencies
between $\mu$SR and neutron scattering data within the dipole picture provide
strong evidence for the ordering of higher-order moments.

##  I Introduction

Cerium-based compounds are an interesting subset of materials that possess
strongly correlated 4f electrons presenting both localized and itinerant
behavior [1, 2]. It is this property of the electrons that lends itself to a
broad display of exotic phenomena in these materials, fueling much of their
interest. Among these phenomena are strongly renormalized magnetic moments [3,
4], multipolar magnetic order [5, 6], the Kondo effect [7], heavy-fermion
behavior [8], and heavy-fermion superconductivity [9]. Additional emergent
phenomena have been reported in certain Ce-based systems, including
observation of composite ‘vibronic’ excitations arising from mixed phonon and
crystal field excitations [10, 11, 12, 13].

Here, we investigate the metastable oxide, Ce2O3. In this material the Ce3+
ion has C3v symmetry with 7-fold coordination to O2- ligands (Fig. 1). The
structure consists of bi-layer triangular nets of Ce atoms sandwiched by
triangular nets of oxygen atoms, which crystallize in a trigonal
$P\overline{3}m1$ phase. Lattice parameters are a = 3.88 Å and c = 6.05 Å
[14], making the in-plane Ce-Ce distance 3.88 Å, while Ce-Ce distances along c
are slightly smaller at 3.74 Å (intercell distance) and 3.80 Å (intracell
distance). Previous experiments examining the magnetic susceptibility and heat
capacity of Ce2O3 have identified a magnetic transition near TN = 6.2 K [15,
16], assumed to be associated with the onset of antiferromagnetic order.
Additional investigations have reported unusual behaviors setting in near this
transition. These include reports of a giant magnetodielectric effect near the
ordering transition temperature, despite the lack of an obvious structural
distortion [16]. Further, a Raman scattering study reported the emergence of
mixed vibronic modes in Ce2O3 as the system was cooled below TN [17], implying
strong mixing between lattice and magnetic degrees of freedom.

While these experimental signatures are quite suggestive of a magnetic
ordering transition in Ce2O3, conclusive evidence for the existence of such
order has not yet been documented. Multiple investigations using neutron
diffraction have failed to reveal any signature of magnetic Bragg peaks below
TN, evoking comparison to the hidden order system URu2Si2 [18, 16]. In one
study, neutron diffraction revealed a realignment of crystallites in a powder
sample with the application of a magnetic field, which the authors argued was
consistent with an easy-plane anisotropy for ordered antiferromagnetic moments
[16]. This was consistent with conclusions drawn from a modified crystal-field
theory analysis in the same paper. The authors further used the observation of
a linear magnetocapacitance and the lack of splitting between zero-field-
cooled (ZFC) and field-cooled (FC) temperature curves in magnetization to
argue for the existence of a low-temperature collinear antiferromagnetic phase
with a k=0 propagation vector [16]. Another ordering picture was put forth in
a recent study of the isostructural analogue Nd2O3, wherein magnetic Bragg
peaks observed with neutron diffraction were indexed to a k=(1/2,0,1/2)
propagation vector [19]. At this time, the exact nature of magnetic order in
Ce2O3 remains an open question.

As the exotic behavior seen in this material is inextricably linked to the
putative order, a more thorough examination is essential. In this article, we
present the results of a multifaceted approach to probe the low temperature
phase in Ce2O3. We investigate polycrystalline samples with multiple
techniques, including neutron diffraction, time-of-flight (TOF), triple-axis
inelastic neutron scattering (INS), and muon spin rotation ($\mu$SR). Our TOF
data allowed us to characterize the crystal electric field (CEF) spectra
associated with the Ce3+ cations, the analysis of which confirms the
conclusion of planar cerium spins. TOF and triple-axis data demonstrate a
shifting and broadening of the CEF line at E=27 meV, in line with previous
observations from Raman scattering of emergent vibronic modes. Significantly,
TOF also revealed the existence of dispersive magnon excitations at lowest
temperatures, where a precessive signal in $\mu$SR spectra implies the
existence of magnetic order. These observations constitute the first direct
evidence of long-range magnetic order in this material.

Figure 1: Crystal structure of Ce2O3.(a) Unit cell representation of
structure. Orange spheres represent Ce atoms and blue spheres represent oxygen
atoms. (b) Cross-section view illustrating alternating layers of O2- and Ce3+
atoms. (c) View along the c-axis showing ab-planes from above. Oxygen atoms
connected by black triangle are above the Ce-plane while those connected by
grey triangle are below the Ce-plane.

##  II Experimental Methods

Polycrystalline samples of Ce2O3 were prepared following the procedure laid
out in Ref. 16 and characterized first using heat capacity and magnetization.
Heat capacity measurements were taken using a Quantum Design PPMS Dynacool
instrument, and magnetization measurements were performed with a Quantum
Design MPMS3 Vibrating Sample Magnetometer. The PPMS and MPMS systems are both
housed in the Stewart Blusson Quantum Matter Institute at the University of
British Columbia (UBC).

Phase purity and characterization of the structure were determined using both
x-ray diffraction (XRD) and neutron powder diffraction (NPD) measurements.
Powder XRD was measured on a Bruker D8 Advance instruments both at UBC and in
the Illinois Materials Research Laboratory using a copper K-$\alpha$ source
and a Johansson monochromater that blocks any K-$\alpha_{2}$. NPD measurements
were taken using the POWGEN high-resolution neutron powder diffractometer at
the Spallation Neutron Source of Oak Ridge National Laboratory and the BT-1
the high-resolution neutron powder diffractometer at the NIST Center for
Neutron Research (NCNR). Measurements at POWGEN were performed using an Orange
cryostat at T = 2 K, 20 K, and 294 K with central wavelength 0.8 Å and at T =
2 K, 3 K, 4 K, 5 K, 6 K, 7 K, and 10 K using central wavelength 2.67 Å.
Measurements at BT-1 were taken at 2 K and 20 K using a Cu (311) monochromator
and a closed cycle refrigerator (CCR). Structural refinements were carried out
using the GSAS-II software [20].

Inelastic neutron scattering measurements were carried out using both the TOF
direct-geometry chopper spectrometer Merlin at the ISIS Muon and Neutron
Source [21, 22] and the triple-axis spectrometer (TAS) BT-7 at the NCNR [23].
TOF measurements on Merlin used the Gd chopper at 500 Hz to obtain neutrons of
incident energy Ei = 16 meV, 23 meV, 66 meV, and 160 meV and at 350 Hz for
neutrons of incident energy Ei = 12 meV, 20 meV, 38 meV, and 100 meV. Samples
on Merlin were measured at 1.8 K, 10 K, and 140 K. Resultant scans were
normalized using comparable measurements of a vanadium standard sample. TAS
measurements were taken with incident neutrons of energy 14.7 meV, selected
with a pyrolytic graphite monochromator, and collimations of
open-80’-80’-open. The sample was loaded in a He gas environment into an
aluminum can with an indium seal. Measurements carried out at temperatures T=
2.6 K, 4 K, 5 K, 5.5 K, 6 K, 7 K, 8 K, and 15 K using a CCR.

$\mu$SR measurements were performed at TRIUMF using the LAMPF spectrometer in
a low-background configuration on the M20 beamline in a helium flow cryostat.
Zero-field (ZF) $\mu$SR measurements were then taken at various temperatures
above and below the magnetic transition temperature. Longitudinal-field
measurements were also taken at base temperature (T = 1.9 K) and in applied
fields ranging from H = 100 Oe to 2000 Oe.

Figure 2: (a) Heat capacity of Ce2O3 vs. temperature (raw data). (b) Magnetic
contribution to heat capacity. Solid red line indicates fit to low-temperature
data. Inset: Entropy from magnetic contribution to heat capacity.

##  III Heat Capacity

Heat capacity measurements were taken from 1.8 K to 70 K, resulting in the
curve shown in Figure 2(a). The overall temperature dependence was consistent
with measurements published elsewhere [16] and contained contributions from
magnons, phonons and CEF excitations. To isolate the magnetic contribution,
the CEF and lattice contributions were estimated and subtracted. The lattice
contribution was estimated from measurements on the isomorphic La2O3 [24] and
polynomial extrapolations to low temperature, as outlined in Ref. 16. The CEF
contribution can be computed directly from the three Kramers doublets observed
with TOF neutron inelastic scattering, as described later in this paper. The
heat capacity arising from these modes is given by the expression:

$C_{v}=\frac{1}{k_{B}T^{2}}(\frac{1}{Z}\sum_{n}E_{n}^{2}exp(-\beta
E_{n})-[\frac{1}{Z}\sum_{n}E_{n}exp(-\beta E_{n})]^{2})$ (1)

where $k\textsubscript{B}$ is the Boltzmann constant, $Z$ is the partition
function, and $E_{n}$ are the CEF energy levels. A small contribution from a
CeO2 impurity identified with x-ray diffraction was also subtracted [25].

The resulting magnetic contribution is shown in Figure 2(b) and reveals most
prominently a lambda-anomaly at T = 6.2 K, indicating a second-order phase
transition. Integrating with temperature, we find that the entropy (inset of
Fig. 2(b)) approaches $Rln2$, as expected if we associate this transition with
the ordering of effective spin degrees-of-freedom within the ground state
doublet.

The heat capacity data at temperatures below 4 K fit well to an expression
expected for antiferromagnetic spin wave materials [26]:

$C_{AF}={B_{AF}}{T}^{n}exp(-\Delta_{AF}/T)$ (2)

where $n$ was empirically fit to 3.14(14). $\Delta_{AF}$ represents the
magnitude of the spin gap and fit to a value of 3.08(46) K while BAF fit to a
value of 0.14(4).

##  IV Magnetic Susceptibility

Magnetic susceptibility measurements were performed in the temperature range T
= 1.8 K - 400 K with an applied field H = 1000 Oe. Results taken using both
ZFC and FC protocols are plotted in Figure 3(a). High temperature behavior was
well-described by a Curie-Weiss function, $\chi$=$\chi_{0}$+C/(T-$\theta$), as
demonstrated by the plot of inverse susceptibility shown in Figure 3(b). Fits
of the data in the temperature region 100 K <T <400 K to this function yielded
an effective moment size $\mu_{eff}$ = 2.40(4) $\mu$B and Weiss constant
$\theta$ = -57.0(5) K. This moment size is slightly smaller than the free-ion
value of 2.54$\mu$B for Ce3+ with J=5/2. The Weiss constant is significantly
larger than the ordering of TN = 6.2 K, indicating a moderate level of
frustration. This may be a consequence of competition between the three
nearest-neighbor Ce3+ interactions described above, as seen in Nd2O3 [19]. The
expected susceptibility was calculated using the eigenvectors and
eigenenergies derived from the CEF analysis utilizing the MANTID package [27].
The data are reasonably well-described by the simulation at high temperatures,
but diverge at lower temperatures, likely the result of magnetic interactions
which are not accounted for in this model. At the ordering temperature, the
susceptibility reaches a cusp suddenly, consistent with a sharp onset of
antiferromagnetic correlations. No discernible splitting is seen between ZFC
and FC curves, arguing against any weak ferromagnetic moment or significant
disorder.

Figure 3: (a) Magnetic susceptibility of Ce2O3 vs temperature (red squares).
Blue dashed line corresponds to expected susceptibility with calculation as
described in the text. Inset: Susceptibility near TN. (b) Inverse
susceptibility with solid line representing fit to data as discussed in text.

##  V Powder Diffraction

Table 1: Results of x-ray and neutron refinements on Ce2O3 in space group
$P\overline{3}m1$

. Uncertainties are statistical in origin and represent one standard
deviation.  Instrument Lattice Parameters Atom Positions (Fractional
Coordinates) Rw a (Å) c (Å) Ce O1 O2 Bruker D8 at 293 K 3.8906(2) 6.0658(1)
1/3, 2/3, 0.2478(19) 0, 0, 0 1/3, 2/3, 0.661(10) 5.158 POWGEN at 300 K
3.8917(1) 6.0585(1) 1/3, 2/3, 0.2451(2) 0, 0, 0 1/3, 2/3, 0.6474(1) 2.054
POWGEN at 25 3.8833(1) 6.0459(1) 1/3, 2/3, 0.2448(1) 0, 0, 0 1/3, 2/3,
0.6472(1) 2.412 POWGEN at 2 K 3.8835(1) 6.0463(1) 1/3, 2/3, 0.2451(1) 0, 0, 0
1/3, 2/3, 0.6474(1) 0.948 BT-1 at 20 K 3.8833(1) 6.0526(1) 1/3, 2/3, 0.2437(2)
0, 0, 0 1/3, 2/3, 0.6486(2) 6.73 BT-1 at 2 K 3.8825(1) 6.0511(1) 1/3, 2/3,
0.2440(2) 0, 0, 0 1/3, 2/3, 0.6482(2) 6.794

As detailed above, diffraction measurements were performed using both XRD at
room temperature and NPD at multiple lower temperatures. We did not find, in
the diffraction patterns, any sign of impurities other than a persistent
fraction of $CeO_{2}$, which can form from metastable $Ce_{2}O_{3}$ at a rate
set by storage conditions. $CeO_{2}$ has no identifiable magnetic transitions,
and this impurity did not affect the conclusions of this paper. Nonetheless,
XRD was performed at room temperature using lab-based sources before every
inelastic neutron, muon or thermodynamic measurement to ascertain the impurity
fraction. The POWGEN TOF NPD instrument at the SNS has an extended range of
coverage in reciprocal space (Q) and was used to determine the structure of
$Ce_{2}O_{3}$ with a high degree of precision over an extended temperature
region. Additional NPD measurements were performed at lowest temperatures
using the BT-1 instrument at the NCNR to more carefully look for magnetic
Bragg peaks at low scattering angles.

Example patterns obtained using the POWGEN and BT-1 instruments can be found
in Fig. 4. It was found that all diffraction data sets are well described by
the previously reported $P\overline{3}m1$ structure for $Ce_{2}O_{3}$, and the
results of Rietveld refinements on several typical data sets are given in
Table 1. Sample parameters are found to be consistent both with the literature
[16] and between fits performed on different instruments at comparable
temperatures. Notably, the quality of fits to $P\overline{3}m1$ also did not
worsen considerably for NPD measurements performed at the lowest temperatures
at either BT-1 or POWGEN, which limits the size of any structural distortion
which might exist at the ordering transition at TN = 6.2 K. As with previous
attempts to investigate Ce2O3 with NPD [16, 18], our data revealed no evidence
for magnetic Bragg peaks below TN. Assuming a traditional $Ce^{3+}$ form
factor, this allows us to simulate the magnetic contribution to the powder
diffraction pattern and determine an upper limit on the ordered magnetic
moment. In doing so we calculate a maximum magnetic moment of $\mu$ =
0.32(11)$\mu_{B}$ associated with a predicted k=0 ordered state [16] or $\mu$
= 0.026(12)$\mu_{B}$ for a state with k=(1/2 0 1/2) as seen recently in Nd2O3
[19].

Figure 4: Neutron powder diffraction data from (a) POWGEN measured at T = 2 K
with center wavelength $\lambda$ = 0.8 $\AA^{-1}$ and (b) BT-1 measured at 2K
with wavelength $\lambda$ = 1.54 $\AA^{-1}$. Solid black lines depict fits to
the $P\overline{3}m1$ structural model appropriate for Ce2O3. Additional peaks
in (b) arise from Al can. Inset shows enlarged region at low Q where magnetic
Bragg peaks are expected.

##  VI TOF Inelastic Neutron Scattering

Figure 5: Time-of-flight inelastic neutron scattering measurements on Ce2O3
with neutrons at Ei = 66 meV (a,d), 100 meV (b) and 160 meV (c). Constant-Q
cuts of data with scaling as described in text (e). Constant-Q cuts of data
with fits represented by solid black line (f). Solid blue line depicts results
from crystal field analysis performed by Gruber et al. [28]

.

The main results of our TOF inelastic neutron scattering measurements are
summarized in Figs. 5 and 6. In Figs. 5 (a), (b) and (c), we show the
normalized scattering data taken using neutrons with incident energies $E_{i}$
= 66 meV, 100 meV, and 160 meV, respectively, and temperature T = 10 K.
Consistent across all spectra, several phonon excitations with a $I(Q)\propto$
Q2 momentum transfer dependence are apparent at energies $E\sim$ 9.4, 13.5,
19.4, 27, 43.5, 52.2, and 63 meV. These are equivalent to the phonon
excitations reported for the structural analogue La2O3 [29, 30], and are as
expected for this material. In addition, magnetic excitations were observed
near $E\sim$ 27 and 98 meV, which we associate with CEF excitations of the
Ce3+ cations.

Notably, we see scattering from both phonons and CEF modes at the singular
energy $E$ = 27 meV. This is in line with the Raman scattering measurements of
Sethi et al. [17], where emergent vibronic excitations were also seen at
temperatures below $T=T_{N}$. The current measurements did not have sufficient
energy resolution to confirm the existence of mode splitting, but the data at
the lowest temperatures did reveal a shift in spectral weight consistent with
this previous study [17]. We demonstrate this shift in Figs. 5 (d) and (e) and
explore it in greater detail using neutron triple-axis spectroscopy, discussed
in a later section.

Table 2: Calculated wavefunctions for Ce2O3 determined using fits to the time-of-flight inelastic neutron scattering data. Crystal field parameters are also included in the first line of the table. |  $B_{2}^{0}$=4.26 meV |  |  $B_{4}^{0}$ = 0.0339 meV |  |  $B_{4}^{3}$ =-3.3516 meV |
---|---|---|---|---|---|---
E (meV) | $|-\frac{5}{2}\rangle$ | $|-\frac{3}{2}\rangle$ | $|-\frac{1}{2}\rangle$ | $|\frac{1}{2}\rangle$ | $|\frac{3}{2}\rangle$ | $|\frac{5}{2}\rangle$
0.000 | 0.3455 | 0.0 | 0.0 | -0.9384 | 0.0 | 0.0
0.000 | 0.0 | 0.0 | 0.9384 | 0.0 | 0.0 | 0.3455
27.107 | 0.0 | -1.0 | 0.0 | 0.0 | 0.0 | 0.0
27.107 | 0.0 | 0.0 | 0.0 | 0.0 | -1.0 | 0.0
98.063 | -0.9384 | 0.0 | 0.0 | -0.3455 | 0.0 | 0.0
98.063 | 0.0 | 0.0 | -0.3455 | 0.0 | 0.0 | 0.9384
Figure 6: Inelastic neutron scattering with Ei=12 meV at temperature above (a)
and below transition (b). Constant-Q cuts of INS data where
$\mid$Q$\mid$=0.8-1.2 $\AA^{-1}$ (d) Simulation of spin waves as described in
text.

To extract information about the nature of the local Ce3+ moments, we fit the
intensity profiles of the two magnetic modes observed in our multiple data
sets at $T$ = 10 K to a simple crystal field model. Magnetic scattering
intensity was first isolated by taking constant-$Q$ cuts of normalized
scattering data, integrating over the range $Q$ = [3.5,6] Å-1, and subtracting
the scattering contribution from phonon scattering as determined by scaling
data in the range $Q$ = [9.6,11.6] Å-1. The magnetic scattering near $E$ = 98
meV was characterized solely using neutrons with $E_{i}$ = 160 meV neutrons,
whereas the magnetic scattering near $E$ = 27 meV was characterized through
simultaneous consideration of $E_{i}$ = 100 meV and the $E_{i}$ = 66 meV data
sets. The $E_{i}$ = 160 meV data was not considered when characterizing the
$E$ = 27 meV region due to a high level of quasi-elastic scattering observed
in the lower-resolution measurement at energies up to $E$ = 35 meV,

Our diffraction results showed that the Ce3+ atoms possess C3v point group
symmetry, which implies non-zero values for $B_{2}^{0}$, $B_{4}^{0}$, and
$B_{4}^{3}$ CEF parameters. As a first estimate of these parameters, we
performed a simple point-charge model calculation, and then refined them
further through consideration of the magnetic scattering data. Fits assumed a
pseudo-Voigt energy profile for the CEF modes with varying widths and energies
and intensities calculated by the software PyCrystalField [31]. Best fit
crystal field parameters are presented alongside the energy eigenvalues and
wave functions in Table 2. The quality of the fit is demonstrated in Fig. 5,
which compares the predictions of the current work to those of the
thermodynamic analysis of Gruber et al. [28].

This CEF analysis reveals a ground state Kramer’s doublet, as expected for a
Ce3+ cation, with two doublets removed to higher energies. Using the ground
state wave functions and Equations 3 and 4, we are able to calculate
g$\perp$=2.26 and g$\parallel$=0.24:

$g\textsubscript{{\hbox{\perp}}}=2g_{J}\matrixelement{\Psi_{\downarrow}}{J\textsuperscript{+}}{\Psi_{\uparrow}}$
(3)
$g\textsubscript{{\hbox{\parallel}}}=2g_{J}\matrixelement{\Psi_{\uparrow}}{J_{z}}{\Psi_{\uparrow}}$
(4)

These values imply a total dipole moment of 1.6 $\mu_{B}$ with a distinct
easy-plane spin anisotropy. This conclusion about anisotropy is in agreement
with the modified ligand field analysis of Kolodiazhnyi et al. [16], as well
as studies investigating other materials housing Ce3+ ions in trigonal
environments [32].

Figure 6 shows the results of our measurements with $E_{i}$ = 12 meV neutrons,
which allow for closer inspection of excitations near the elastic line. Upon
lowering the temperature to $T$ = 10 K, these data reveal a significant build-
up of paramagnetic scattering centered most strongly at $Q\approx$1 $\AA^{-1}$
(Fig. 6(a)), which evolves into a dispersive spin-wave spectrum on cooling to
$T$ = 2 K (Fig. 6(b)). The magnon density of states is centered at $E\sim$ 1.5
meV and has bandwidth of $\Delta E\sim$ 1 meV (Fig. 6(c)), consistent in scale
with an ordering transition at $T_{N}$ = 6.2 K.

For comparison, we show in Fig. 6(d) a simulation of the expected scattering
pattern from a powder of Ce2O3 containing the $\mathbf{k}=0$ collinear
antiferromagnetism proposed by Kolodiazhnyi et al. [16]. Powder-averaged
simulations were performed using the SpinW software package [33] and assumed
dipole $Ce^{3+}$ moments, the g-tensor calculated above, an exchange
coefficient J= 1 meV and a spin gap determined from fits of our heat capacity
data ($\Delta$ =3.08 K). The simulation successfully reproduces several
features of the observed scattering pattern, including the ridge of scattering
near 1.5 meV and the positions in $Q$ of several local scattering maxima. The
measured spin-wave scattering seems to fall away more rapidly than the dipole
simulation, probably reflecting the significant multipolar character of the
$Ce^{3+}$ moments which can be inferred from the wavefunctions for the
electrons in the Kramers doublet determined above and the associated
distribution of electron density in the unit cell. The apparent gap of $\sim$
1 meV in the scattering data is understood as an effect of the powder density-
of-states. Overall, the appearance of collective spin-wave excitations is a
significant observation, which constitutes the strongest evidence to date of
long-ranged spin order in Ce2O3.

##  VII Triple-Axis Neutron Scattering

Figure 7: (a) Inelastic neutron scattering in the region around expected
vibronic modes. Solid lines represent fits to data as described in text. (b)
Constant-Q cuts at 2.6 K. (c) Temperature dependence of the integrated
intensity of the peaks at approximately 26 meV (Mode 1) and 31 meV (Mode 2).

We followed our TOF inelastic studies with a more detailed study of the 27 meV
mode with triple-axis neutron scattering. The underlying motivation was to
confirm and expand upon recent Raman spectroscopy results which reported
emergent vibronic excitations near this energy [17]. In particular, the major
features observed in the Raman data were two new modes near a purported two-
phonon band at 24 meV and the CEF excitation at 27 meV as the sample was
cooled below TN. These emergent modes appeared at 22 meV and 31 meV and
appeared to have phonon-like and electron-like behavior, respectively.
Notably, a clear shift in spectral weight from the CEF mode into the 31 meV
mode was also observed as temperature was decreased.

Our TOF data reveal a similar shift in spectral weight as the sample is cooled
below TN, and this is reaffirmed by temperature-dependent TAS scans at Q = 2.4
Å-1 (Figure 7(a)). In examining constant-Q scans at base temperature, one can
see that much of the scattering intensity is magnetic, as evidenced by the
reduced intensity with increasing momentum transfer (Figure 7(b)). Phonon
scattering is relatively weak in this region of momentum space, but there may
also be a phonon-like component, seen as a subtle inflection in scattering
intensity at higher Q. This Q-dependence is similar to that seen in the TOF
data for this region. Notably, the peak in this region remains broad, despite
the increase in energy resolution of our TAS measurements over our TOF data.
This is actually fairly consistent with the Raman scattering study, where the
CEF modes were broad in energy and the phonon-like modes much sharper. The
Q-range of our TAS measurements strongly suppressed the phonon scattering in
favor of scattering from the magnetic modes. The fact that the width is much
wider than instrument resolution implies that the magnetic modes have a finite
lifetime, possibly due to scattering from phonons or delocalized electrons. We
do not observe more than a single peak, but the observed mode has a distinct
asymmetry. This might imply a profile consisting of at least two unresolved
peaks, though best fits to these data are achieved using a sum of three
Gaussian functions. Based on the Q-dependence of these data and the TOF data
above, we associate these three peaks with one phonon at 28 meV and two
magnetic excitations at 26 and 31 meV, respectively.

To characterize the temperature evolution of this scattering, the constant-Q
scans were fit to the three Gaussian profile at all temperatures, and the best
fits are shown as solid lines in Figs. 7(a) and (b). As no significant
temperature-dependence of the phonon mode at 28 meV was observed in the TOF
data, fits for the phonon peak were made for the lowest temperatures only and
then held constant for higher temperatures. Figure 7(c) shows the fitted
intensities of the two temperature-dependent modes, and implies an apparent
transfer of intensity from the 26 meV mode (Mode 1) into the mode at 31 meV
(Mode 2). This is consistent with the previous Raman study, though we do not
see the additional peak at approximately 22 meV, likely due to its heavily
phonon-like nature. This limitation and the large intrinsic width prevents us
from commenting more explicitly on the presence or nature of possible vibronic
modes, which were clearly observed in CeCuAl3 [12]. A future investigation
using polarized neutrons on single crystal samples is suggested to better
separate phonon and magnetic contributions and track dispersion of these modes
in three dimensional reciprocal space.

##  VIII Muon Spin Rotation

Finally, we performed a series of $\mu$SR measurements to directly couple to
the magnetic order parameter and glean further information about the nature of
the low-temperature state. $\mu$SR is exquisitely sensitive to small magnetic
fields inside materials and has a long, established history of detecting small
moment spin order in f-electron systems. It is sensitive to both dipole and
multipolar magnetism, and is largely unaffected by absorption, disorder,
delocalization and other complicating factors which can obscure Bragg peaks in
a neutron scattering experiment.

Figure 8 (a) shows spectra from several zero-field (ZF) $\mu$SR measurements
taken at temperatures above and below the transition temperature at $T_{N}$ =
6.2 K. Well above $T_{N}$, the ZF spectra exhibit a weak Gaussian relaxation,
as expected for a system of randomly oriented dipoles. Upon cooling, the most
salient feature of the data is the emergence of clear oscillations below TN
with a frequency that increases with decreasing temperature to a value of
about $f$ = 14 MHz. This reflects the Larmor precession of muon spins around
an emergent local field of Hlocal = 0.1 T and provides direct and unambiguous
evidence for long-ranged magnetic order. Longitudinal-field (LF-$\mu$SR)
measurements performed at base temperature (T = 1.9 K) show that the spectra
decouple in comparably sized fields (Figure [8] (b)), indicating that the
order is static on the $\mu$SR timescale.

Zero-field spectra were fit to the muon spin depolarization function shown in
Eq. 5. In this equation, A0 is the total muon asymmetry, which is a property
of the spectrometer and determined by an independent calibration measurement
performed in the paramagnetic state at T = 20 K. A1 is the fraction of the
initial asymmetry representing the magnetically ordered volume and is itself
separated into a 2/3 oscillating component and a 1/3 non-oscillating
component, corresponding to mean field components perpendicular and parallel
to the initial muon spin direction. A2 and A3 are non-oscillating components,
with and without discernible time relaxation, respectively. Inclusion of these
two non-oscillating components was determined empirically to be the minimal
model required to explain the spectra at the lowest temperatures. All fits
were performed under the constraint A1 \+ A2 \+ A3 = 1, and the weight of the
various fractions were found to be temperature independent well below the
transition with approximate values A1 = 0.51, A2 = 0.34, and A3 = 0.15. These
values were held constant in fits close to the transition for stability. The
size of the time-independent portion, A3, is consistent with the nonmagnetic
CeO2 impurity fraction determined using x-ray diffraction, whereas the non-
oscillating component A2 may represent either muons landing in a non-magnetic
volume inside our Ce2O3 sample or muons residing at a symmetric location in
the unit cell with zero mean field (where the likely muon stopping site is
discussed below).

$\displaystyle
A_{0}P(t)=A_{0}A_{1}(\frac{1}{3}exp^{-(\lambda_{1a}t)}+\frac{2}{3}\cos(2\pi
ft+\phi)\exp^{-(\lambda_{1b}t)})$ (5)
$\displaystyle+A_{2}A_{0}\exp^{-(\lambda_{2}t)}+A_{3}A_{0}$

Figure 8: (a) Raw ZF-$\mu$SR spectra at various temperatures, where
independent scans are offset by increments of 0.1 along the y-axis for data
taken at temperatures between 1.9 K and 5.7 K and by increments of 0.05
thereafter. Solid black lines represent fits using Eq. 5. (b) Raw LF-$\mu$SR
spectra taken at T = 1.9 K in several different applied fields. Solid black
lines represent similar fits without constraints on the ratio between
oscillating and non-oscillating fractions. (c)-(e) Parameters extracted from
fits of ZF-$\mu$SR spectra, plotted as a function of temperature. (f) A
schematic of the atoms in a single unit cell, with the location of the
predicted muon site shown as a grey sphere and the dipole spin direction in
the k = 0 ordered state shown as red arrows.

As seen by the black lines in Fig. 8(a), fits using this equation were largely
successful in describing the data at all temperatures, with the possible
exception of near the transition where it misses at early times. Parameters
extracted from these fits are shown in Figs. 8(c)-(e) and show clear and
smooth evolution with temperature. The oscillation frequency in the ordered
volume is plotted in Fig. 8(c) and shows a clear order-parameter temperature
dependence, consistent with our interpretation of a second order phase
transition. In this case thermal dependence of the oscillation frequency in
the ordered phase can be described by:

$f(T)=f_{0}(1-\frac{T}{T_{N}})^{\beta}$ (6)

Fits to this formula give a critical temperature of $T_{N}$ = 6.1 K,
consistent with the value we determined from heat capacity, and the exponent
$\beta$ fits to a value of 0.46, close to the value of $\beta=1/2$ expected
from mean field theory. Fig. 8(d) shows the exponential relaxation rates
associated with oscillating and non-oscillating components of the ordered
fraction. Here, $\lambda$1b represents the relaxation of the cosine component
and should be associated with inhomogeneous broadening of the local field
distribution at the muon site; relaxation here is proportional to the second
moment of the distribution and has similar temperature dependence as the mean
field, which is common. The parameter $\lambda$1a represents the relaxation of
the non-oscillating component in the ordered volume and likely reflects a weak
dynamic relaxation mechanism; it seems to diverge at the ordering temperature,
consistent with critical fluctuations near a second order phase transition.
Fig. 8(e) shows the exponential relaxation of the A2 non-oscillatory fraction,
which surprisingly has a similar temperature dependence and magnitude of
frequency in the A1 fraction. This implies that the local field distribution
at the site of muons in the A2 comprises stray fields from the ordered
fraction. In fact, given the lack of net magnetization in the low-temperature
region, the magnitude of the fields contributing to the A2 strongly implies
these muons reside within the ordered volume, but are located at symmetric
positions with respect to the magnetic cations.

Beyond these details, it important to note that the size of the mean field
implied by the precession frequency of A1 muons is quite significant and
provides great contrast to the tight upper limits on the possible ordered
moment inferred from neutron diffraction data. To put this statement on more
quantitative footing, we used the MuFinder software suite [34] to find the
most likely muon stopping site via advanced DFT analysis. This was found to be
at the fractional coordinates x = 0.000, y = 0.003, z = 0.458 in hexagonal
coordinates as shown by the grey sphere in (Figure 8 (f)), favored over other
stopping sites by an energy of greater than 300 meV. Assuming this site, we
were able to calculate the mean dipole field likely felt by muons as a result
of the ordered states suggested in the previous literature. For example, if we
assumed the k = (1/2, 0, 1/2) ordered state with planar spins seen in
isostructural Nd2O3 [19], we infer from the oscillation frequency in the
ZF-$\mu$SR data that the ordered moment would have to be approximately 0.4
$\mu$B. Assuming the k = 0 with easy-plane spins requires a moment size of
2.45-2.65 $\mu$B, depending on the angle of rotation in the planes. Within the
context of the dipole moment picture, neither of these calculated moment sizes
are compatible with the restrictions imposed by our neutron diffraction data,
which imply ordered moments smaller than 0.3 and 0.03 $\mu$B for the k = 0 and
k = (1/2, 0, 1/2) structures, respectively. This paradox is easily resolved if
one considers a potential role for higher order magnetic multipoles, the
ordering of which would contribute to the local field distribution experienced
by muons which would not contribute to a neutron diffraction pattern. The
existence of multipolar order in Ce2O3 would put this material in line with a
long list of other f-electron magnets [35, 36].

##  IX Conclusions

Altogether, the data presented in this article reveal several new properties
and characteristics that resolve some of the outstanding mysteries surrounding
the material Ce2O3 that are essential for a full understanding of the low-
temperature magnetic properties. Most revealing, the emergence of clear
oscillations in our $\mu$SR data demonstrating the development of magnetic
order and the associated spin wave excitations observed in our TOF inelastic
neutron scattering data provide the first direct confirmation of long-ranged
magnetic order in this material below 6.2 K. While the absence of observable
magnetic Bragg peaks in our neutron powder diffraction patterns puts tight
limits on the maximum size of ordered dipole moments, the presence of
dispersive magnon excitations confirm that ordered moments must exist in some
form. The success of our spin-wave simulations provides support for the k = 0
ordered state suggested by previous studies, with spins aligned in the easy
planes. However, the interpretation of the $\mu$SR data utilizing this
magnetic structure suggests an ordered moment size of approximately 2.5 $\mu$B
within the dipole picture, impossible to reconcile with the conclusions from
neutron diffraction. Similar inconsistencies have been seen previously in
cases of hidden order emerging in f-electron magnets [37, 38]. Perhaps the
most notable example is that of URu2Si2, where the nature of the hidden order
remains unresolved after decades of study [39]. It seems an inescapable
conclusion then that higher-rank multipolar moments may provide the answer to
these seemingly contradictory observations. This further helps explain the
more rapid than expected suppression of magnetic scattering with increasing Q.
We note in passing that the involvement of higher-rank multipoles in the low
temperature region may also help explain reports of a giant magnetodielectric
effect and the appearance of vibronic excitations in Raman scattering data,
since multipoles are tied to electron orbitals and thus distribution of
electron charge density within the crystallographic unit cell. To illuminate
this physics further, we suggest a combined theory and resonant x-ray
scattering to determine the nature of the multipolar order in Ce2O3 and its
coupling to other material properties.

###### Acknowledgements.

We thank ISIS Facility for providing the beam time RB2000208 (Gregory
MacDougall et al; (2021): Investigation of Vibronic Excitations in Ce2O3, STFC
ISIS Neutron and Muon Source, https://doi.org/10.5286/ISIS.E.RB2000208-1).
This work was supported by the the Natural Sciences and Engineering Research
Council of Canada and the CIFAR Azrieli Global Scholars program. This research
was undertaken thanks in part to funding from the Canada First Research
Excellence Fund, Quantum Materials and Future Technologies Program. S.L.C. and
J.E.S. were supported by the National Science Foundation under Grant No. NSF
DMR 1800982. The identification of any commercial product or trade name does
not imply endorsement or recommendation by the National Institute of Standards
and Technology.

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