Florent Dubois

Interpretation or "the Thing Itself"? The Art of Performance in Lucy Gayheart [START_REF]Light on Adobe Walls[END_REF] With Lucy Gayheart Willa Cather returned to the genre of the Künstlerroman she had first explored in The Song of the Lark. Interestingly, while there was one artist in the 1915 novel, there are two in Lucy Gayheart and both meet a tragic death, giving this coming-of-age story a markedly different resonance. Whereas The Song of the Lark depicted Thea Kronborg's ascent to stardom, Lucy Gayheart tells the story of a moderately ambitious piano student who is cut off in her prime before she can prove her worth. In this later novel, Cather seems to display the same aesthetic ideals as in her early works. And yet, as a brief look at the synopsis of both novels quickly shows, it would be quite hasty to regard Lucy Gayheart as a mere rewriting of The Song of the Lark. Although many expressions and turns of phrase regarding music are very reminiscent of Cather's previous novels and stories, it is hard to ignore the much gloomier atmosphere that pervades the novel, which according to Susan Rosowski even carries Gothic undertones (219-231). One other major difference between the two novels is the perspective from which art is described in each : whereas The Song of the Lark takes us to the heart of the creative process by showing us how someone becomes a great artist, in Lucy Gayheart the protagonist most of the time is not the object of study. On the contrary, she is in the position of the spectator who looks up to the great artist (Clement Sebastian but also the soprano the Gayhearts admire in a performance of The Bohemian Girl in Haverford [bk. 2, ch. 7]) and tries to understand where his or her mysterious power comes from. This change of perspective is not merely a ploy to make yet another novel about a subject that Cather had already dealt with many times in her fiction. In the twenty years that separate the two novels I would argue that Cather's views on life and the arts have shifted from optimism to skepticism. At the beginning of her writing career, when she was still mostly a journalist, she stated that the hardest thing for an artist was to "express" his or her ideas and perceptions. However "perilous" that "voyage" from the brain to the instrument of one's art, she did not doubt that a few managed to journey safely to "the truth." 2 Nineteen years later, the virtual apotheosis of Thea as Sieglinde 3 seems to be proof that Cather still maintained a rather optimistic view of the powers of the artist. However, The Song of the Lark is probably her most teleological piece of fiction and following novels such as The Professor's House or Death Comes for the Archbishop show much more circumspection toward the possibility of delineating any straightforward voyage, be it in life [START_REF]Light on Adobe Walls[END_REF] I would like to thank Sherrill Harbison and the late David Porter for their very kind, constructive feedback on the first version of this paper. 2 "To keep an idea living, intact, tinged with all its original feeling, […] to keep it so all the way from the brain to the hand […], that is what art means […]. And that is the voyage perilous, and between those two ports more has been lost than all the yawning caverns of the sea have ever swallowed. […] In the kingdom of art there is no God, but one God, and his service is so exacting that there are few men born of woman who are strong enough to take the vows. There is no paradise offered for a reward to the faithful, no celestial bowers, no houris, no scented wines; only death and the truth." ("A Mighty Craft" 417) 3 The Song of the Lark 478; pt. 6 ch. 11.

Florent Dubois

2/11 or in art. At sixty one Cather offers a much more problematic account of artistic success. Describing Lucy Gayheart as "Cather's most fashionable text in its self-deconstruction" (260, note 5), Blanche Gelfant shows how Cather's novel constantly betrays her anxieties regarding the power of words and her own ability to make the reader feel what is not directly there on the page. Following a similar line of argument, I will examine how she very subtly conveys her views on the art of performance, in a way that confirms how much of a master in concision she was at the end of her career, while letting a persistent skepticism creep into any statement the narrator or the characters seem to make, revealing her gnawing doubts about the meaning and purpose of any artistic endeavor. In doing so, I hope to demonstrate that the musical scenes in Lucy Gayheart-and indeed the whole novel-need to be approached as original statements in their own right rather than mere rewritings of previous scenes from Cather's fiction.

The second time Lucy goes to hear Clement Sebastian in recital, Cather describes Lucy's reaction to Die Winterreise.

She had never heard Die Winterreise sung straight through as an integral work. For her it was being sung the first time, something newly created, and she attributed to the artist much that belonged to the composer. She kept feeling that this was not an interpretation, this was the thing itself, with one man and one nature behind every song. The singing was not dramatic, in any way she knew. Sebastian did not identify himself with this melancholy youth; he presented him as if he were a memory, not to be brought too near into the present. One felt a long distance between the singer and the scenes he was recalling, a long perspective. (38; bk. 1 ch. 5) Parallels have been drawn between the "long perspective" Lucy notices here and the distance that separates the narrative and the events described in Lucy Gayheart, a story that is presented from the start as a memory ("In Haverford on the Platte the townspeople still talk of Lucy Gayheart" [3; bk. 1 ch. 1]). 4 Not much thought has been given, however, to the blurring of the distinction between composer and singer which Cather describes. Obviously it is a compliment to the singer and it is something Cather seems to value as the hardest thing to achieve in art. In an essay on Sarah Orne Jewett she writes that the "sketches" in The Country of the Pointed Firs "melt into the land and the life of the land until they are not stories at all, but life itself." (849) Using a similar formula about a minor story of Jewett's, she writes, "It simply is the look-shy, kind, a little wistful-which shines out at one from good country faces on remote farms; it is the look itself" (Cather's italics). She calls this accomplishment "a little miracle," adding, "To have got it down upon the printed page is like bringing the tenderest of early spring flowers from the deep wood into the hot light of noon without bruising its petals." (851) Yet although Cather assigns a similar goal to literature and music-to be the "thing itself"-we must not forget that literature and music use completely different media. While the reader has a first-hand knowledge of the text of the novel the listener needs the mediation of the musicians to get access to the piece. The novel is its text while the musical piece exists only in performance, the score being but a convenient way of keeping a tangible trace of it and preserving its unity through its various performances. The role of the performer is therefore essential because he is as necessary to the existence of the musical piece as the composer is. With this in mind, the fusion of performer and composer Lucy experiences can Interpretation or "the Thing Itself"? The Art of Performance in Lucy Gayheart Florent Dubois 3/11 literally mean that the spectators forget that there is an intermediary between Schubert and themselves, as if Sebastian were the composer. In other words, the performance has a certain immediacy which makes the listener forget that the singer is following a score written a hundred years or so before. Yet Cather does not talk of a feeling of improvisation either, which would be the contrary of following a score. She-or rather Lucy-attributes this impression to the sense of unity that the whole song cycle displays, "with one man and one nature behind every song." This means that if there is a distance between Sebastian and the "melancholy youth" it is only a distance in time. The singer truly seems to have been this youth: the situations depicted in the songs have been experienced and are now remembered ("the scenes he was recalling"). The idea that "this was not an interpretation," therefore, must come not from a sense of improvisation or freedom on the part of the singer but from his deep involvement in the words and notes he has to sing. And although Sebastian does not "identify himself with this melancholy youth" (38), that is with the speaker of the poems, Lucy cannot help identifying him with the music he sings. The first time Lucy hears him in recital Cather writes, After this invocation came five more Schubert songs, all melancholy. They made Lucy feel that there was something profoundly tragic about this man. (30; bk. 1 ch. 4) For Lucy there is no distinction between the emotion conveyed by the music and the emotion the singer must feel. Sebastian's life must be as tragic as the songs he sings. In other words, the successful rendering of the score becomes very much a question of acting rather than musical craftsmanship. Sebastian presents the speaker of the poems "as if he were a memory." Similarly, the tenor in Lohengrin is good because "he believes in it" as Lucy tells Harry Gordon (104; bk. 1 ch. 16). This reminds us of the very similar words used to describe Thea Kronborg's art in The Song of the Lark. Fred Ottenburg says, "Every phrase she sang was basic. She simply was the idea of the Rhine music." (396; pt. 6 ch. 2) When listening to Thea, the spectator does not have the sensation he is hearing some very sophisticated interpretation, in which every phrase has been very meticulously planned in advance. What matters is that the identification of the singer with the meaning of the piece is simply complete. However, to say that Thea was "the idea of the Rhine music" is not merely saying that she was a good actress. It supposes an intimate understanding of the spirit animating the music, the vital force that is not written in it but is at its origin and shapes it. In The Song of the Lark and for Lucy as well, artistry relies on the ability to convey something that is not in the music but is at the same time its raison d'être. This principle of unity can only be found if the artist both has an excellent knowledge of the piece he or she performs5 and has an ability to put things into a larger perspective. Here Lucy attributes this "long perspective" to Sebastian's age. His age is supposed to give him a certain wisdom that Lucy calls "simplicity" ("He had a simplicity that must come from having lived a great deal and mastered a great deal" 46; bk. 1 ch. 6). As for Thea, she also gains a "long perspective" when she spends several months in Panther Canyon, where she discovers she is able to have intuitions about the lives of the Pueblo people.

Florent Dubois 4/11

On the first day that Thea climbed the water trail she began to have intuitions about the women who had worn the path, and who had spent so great a part of their lives going up and down it.

[…] It seemed to Thea that a certain understanding of those old people came up to her out of the rock shelf on which she lay; that certain feelings were transmitted to her, suggestions that were simple, insistent, and monotonous, like the beating of Indian drums. They were not expressible in words, but seemed rather to translate themselves into attitudes of body, into degrees of muscular tension or relaxation; .... (302-303; pt. 4 ch. 3) Through these bodily sensations, Thea travels the very long distance separating the Pueblo people from her own time. These intuitions are "simple, insistent, and monotonous" just as there are certain truths about life that are so "simple, insistent, and monotonous" that they may have become clichés but remain nonetheless true. As Carl says in O Pioneers!, "there are only two or three human stories, and they go on repeating themselves as fiercely as if they had never happened before" (119; pt. 2 ch. 4). Thus, for Cather simplicity in no way means banality. It means the artist has done his or her job, which consists in "selecting the eternal material of art," because "the higher processes of art are all processes of simplification" ("Novel Démeublé" 836). The ability of the singer to simplify leads to a performance in which everything seems natural because everything is subordinated to one higher goal, as composition assigns every element of a painting its place so as to create a coherent whole. While listening Lucy is not assailed by a profusion of images; instead, she progressively composes one picture corresponding to the mood of the performance. When Sebastian sings "Der Doppelgänger," she sees "moonlight pouring down on the narrow street of an old German town" and then she progressively adds elements to the picture ("old human houses," "a lonely black cloud") without these additions ever breaking the unity of the whole. Indeed, these are not additions but the deepening of one vision ("with every phrase that picture deepened" 30; bk. 1 ch. 4). In the same manner, Thea's successful performance as Sieglinde at the end of The Song of the Lark is only possible because of the complete submission of her body to her conception of the role6 , with everything she does subordinated to a unique principle, like the branches and flowers of "a tree bursting into bloom" (478; pt. 6 ch. 11).

Whether it is an "idea" or "one nature" presiding over the whole piece, this principle of unity can of course be identified with what Cather famously called "the inexplicable presence of the thing not named, of the over-tone divined by the ear but not heard by it, ... the emotional aura of the fact or the thing or the deed" ("Novel Démeublé" 837). Lucy, in one of the numerous passages of the novel in free indirect style, uses a similar image to describe Sebastian.

He had a simplicity that must come from having lived a great deal and mastered a great deal. If you brushed against his life ever so lightly it was like tapping on a deep bell; you felt all that you could not hear. (46; bk. 1 ch. 6)

The image of the bell is an excellent rephrasing of the "thing not named" as its very rich sound produces so many harmonics-or overtones-that it is often possible to hear other notes quite clearly besides its fundamental note. Moreover, an attentive reader of Cather's works might remember how, in Death Comes for the Archbishop, the bell has other ways of suggesting much more than the eye can see. When hearing the angelus in Santa Fe, Bishop Latour first thinks he is in Rome and then Jerusalem, "though he had never been there" (42; bk. 1 ch. 4), a sort of Florent Dubois 5/11 hallucination which he explains by the presence of silver in the bell, a metal the Spaniards learned to work from the Moors (45). Thus, the sound of the bell is able to suggest faraway places and summarize the history of civilizations.

A great artist, therefore, is someone that makes you feel what is not there. During the first recital Lucy attends, she is transported outside the concert hall to imaginary places. She sees "the narrow street of an old German town" (30; bk. 1 ch. 4). So real is the illusion that she no longer sees what is truly happening in front of her and misses Sebastian's exit from the stage.

The moon was gone, and the silent street.-And Sebastian was gone, though Lucy had not been aware of his exit. The black cloud that had passed over the moon and the song had obliterated him, too. (30; bk. 1 ch. 4)

The dash underlines the moment when Lucy is brutally brought back to reality. The last sentence shows how the fictional world evoked by the song is, as it were, superimposed on reality in Lucy's mind: the cloud has passed both over the fictional moon mentioned in Heine's poem and over the song itself. Moreover, the presence of the "red-haired accompanist, a lame boy, who dragged one foot as he went across the stage" (30; bk. 1 ch. 4) adds to the violence of this sudden confrontation to a much more prosaic reality.

Lucy's growing dislike of Mockford also indicates how much vision matters during a performance, even when it consists only of songs and not of operatic excerpts. Although Lucy realizes that "it was contemptible to hold a man's infirmity against him" (39; bk. 1 ch. 5), she cannot help noticing how his attitude and his whole body spoil a part of her pleasure.

She admired him, but she didn't like him. Was she jealous, already? No, something in his physical personality set her on edge a little. He was picturesque-too picturesque. (38; bk. 1 ch. 5) By contrast, Sebastian immediately makes a good impression when he steps on the stage.

Sebastian's personality had aroused her, even before he began to sing, the moment he came upon the stage. ... She said to herself immediately: "Yes, a great artist should look like that." (28-29; bk. 1 ch. 4) Thus, elements entirely foreign to the musical interpretation proper can have a significant impact on the way the spectator perceives the piece. Vision, however, seems to matter only if it has an emotional impact: Mockford "sets Lucy on edge" and "Sebastian's personality has aroused her." As we saw, Cather also describes the "thing not named" as an "emotional aura." I would argue that it is precisely emotion which brings the "thing not named" into existence. Emotion opens a new world to the listener, a world that is not in itself in the musical piece but rather is the product of the encounter between the piece and the listener. In the impressionable Lucy the "emotional aura" of the Schubert songs Sebastian and Mockford interpret is so strong that it upsets her deeply, turning her conception of the world upside down.

It was a discovery about life, a revelation of love as a tragic force, not a melting mood, of passion that drowns like black water. As she sat listening to this man the outside world seemed to her dark and terrifying, full of fears and dangers that had never come close to her until now. (31; bk. 1 ch. 4) Florent Dubois 6/11 Like Thea's intuitions, Lucy's emotions have both a physical and an intellectual meaning. She discovers new emotions she had never felt before ("love as a tragic force," "passion that drowns like black water") and it also changes the way she understands the world (it is a "discovery about life, a revelation"). The "thing itself," therefore, is not only the musical piece given intelligently with a sense of coherence but the result of a physical and emotional transaction between artists and listeners. The piece becomes a "thing" because the listener is deeply moved. Even though Lucy is able to understand German and does not "miss a word" (29; bk. 1 ch. 4), words in themselves are not what causes such an intense feeling. When the narrator-or Lucy in free indirect speechquotes a few words from a song, it is not so much to convey to the reader the atmosphere of the songs as to insist on the perfect adequacy of their musical setting.

In your light I stand without fear, O august stars! I salute your eternity. That was the feeling. Lucy had never heard anything sung with such elevation of style. In its calmness and serenity there was a kind of large enlightenment, like daybreak. (29-30; bk. 1 ch. 4)

Lucy is amazed at the capacity of the music to make one feel the exact emotions suggested by the words. The "calmness" and "serenity" of the voice have the power to communicate the exact emotion of the mariner saluting the stars. The successful interpretation, therefore, is the one that manages to breathe life into the piece. It must enact the emotional content of the music and convey all the while the sense of coherence and simplicity we talked about. In other words, the performer is constantly torn between, on the one hand, a desire to create a living thing that can encompass the infinite diversity of human experience and, on the other, the respect he owes to the score, a dead object composed of dead signs that break up the infinitely varied world of sound into a finite set of discreet unities. The need for the singer to remain simple is a direct result of the desire to remain close to the score. It explains, for instance, why the emotional involvement of the artist cannot be manifested through extremely passionate singing. Immediately after noting how well the feeling is expressed by Sebastian, Cather adds, "Lucy had never heard anything sung with such elevation of style." (29-30; bk. 1 ch. 4) Indeed, I would argue, simplicity implies "elevation of style." If we tried to define simplicity in singing, we would probably think of it as the elimination of personal trademarks or quirks so as to privilege the pure musical line and the perfect enunciation of the text without unnecessary emphasis on particular syllables or words. On a purely musical level, excessive emotional involvement usually leads to breaks from the musical line: sobs and glottal attacks-both characteristic of verismo singing, for instance-but also inaccuracies of pitch. 7 If they want to preserve the integrity of the musical line-and thus stay close to the score-singers must therefore find other devices than these showy effects. They will most probably use the technique of coloring, altering the timbre of the voice (its color) on certain vowels in order to create certain effects. This technique leaves the musical line intact and can even participate in making the text more understandable, as the words are no longer just a pretext for writing beautiful music. With coloring, words are invested with meaning; the way the singer colors each syllable is extremely suggestive and can radically alter the signification of a word (if he sings it with a smile or a smirk, for instance). As a matter of fact, Sebastian's diction is the first thing Lucy notices (29; bk. 1 ch. 4). Lilli Lehmann, Olive Florent Dubois 7/11

Fremstad's and Thea's teacher in Germany, was a stern stylist and a great advocate of coloring. In her book on singing she explains, Every vowel, every word, every tone, can be colored as by magic in all sorts of ways by the well-controlled play of the lips; can, as it were, be imbued with life, as the lips open or close more or less in different positions. ( 212)

In a surprisingly Catherian way she also writes, 8

A word is an idea; and not only the idea, but how that idea in color and connection is related to the whole, must be expressed. ( 264)

"Word," "idea," "color," and relation to "the whole" are the cornerstones of classical singing. It is a subtle art which, when successfully done, seems so natural that it is hard to tell exactly what the singer does that makes him superior to the others, as Lucy quickly notices.

The dark beauty of the songs seemed to her a quality in the voice itself, as kindness can be in the touch of a hand. It was as simple as that-like light changing on the water. (30; bk. 1 ch. 4)

The credibility of the interpretation relies on a particular "quality" in the singer's voice, something again "not expressible in words" and yet perceptible, like the intention behind a gesture. The image of "light changing on the water" adequately expresses the technique of coloring: the musical line remains untouched, like the smooth waters of a lake, while only its color changes just as the lake takes on various hues depending on the weather and the time of day.

Thus, Cather manages to suggest a very elaborate interpretation technique without ever being technical. She uses comparisons ("as if he were a memory," "like light changing on the water") or simple notions ("distance," "he believes in it") to convey the impact of the type of singing she favors. What changed since The Song of the Lark, which was also far more loquacious concerning musical performance, is that Cather's musical scenes are now intricately woven into the larger pattern of the novel. As we are going to see, the scenes of musical performance are both undermined by and in part responsible for the atmosphere of uncertainty that pervades the whole novel.

The complex structure of the novel and the difficulty of locating its narrator compel us to view every statement with caution. For instance, it would be hasty to consider that Cather thinks Sebastian's art superior because he manages to express "the thing itself." First, the words are attributed to Lucy ("she kept feeling that this was not an interpretation" [the emphasis is mine). Then it is obvious that the narrator considers Lucy's opinion naive: "she attributed to the artist much that belonged to the composer." 9 In this sort of conflict, of course, the narrator usually has 8 As Sherrill Harbison remarked to me (personal communication), it might just be the other way round (Cather being influenced by Lehmann) as many critics have underlined that Cather's article on "Three American Singers" reveals how much Cather made Fremstad's (and thereby Lehmann's) views on art her own. See, for instance, David Porter's "Historical Essay" (268-270). 9 The narrator's comment reflects Cather's more mature point of view. Now 60, she finds it difficult to identify herself with a young girl. In a letter to Zoë Akins dated August 1933 she writes, "I'm working on a book about a silly young girl, and I lose patience with her. Perhaps I am too old for that sort of thing." (Selected Letters 488-489) Florent Dubois 8/11 the upper hand, but the reader, who cannot help identifying himself or herself with the protagonist, may-consciously or unconsciously-ignore this patronizing voice. It is even characteristic of this novel that it is impossible for the reader to find a reassuring omniscient authority presiding over the narrative. In an article published in the second issue of Cather Studies, Linda Chown shows that all of Lucy's story can be seen as an embedded narrative whose narrator-a "persona narrator" as she prefers to call it-is Harry Gordon. Such a reading would necessarily lead us to consider the narrator's statements with caution. Whether or not we accept this interpretation, it remains undeniable that Cather keeps emphasizing the difficulty of assigning identity in Lucy Gayheart.

Whereas identities were clearly defined in The Song of the Lark through a set of Manichaean oppositions (between Thea's Friends, the musical Mexicans or Ray Kennedy on the one hand, and "Thea's natural enemies" [59] on the other, the "stupid faces" that give its name to the third part of the novel), nothing is as definite here. There is another, hidden side to everything and everyone.

The image of the lake used to describe Sebastian's voice, for instance, finds a troubling echo a few chapters later in a description of Lake Michigan.

There was very little ice in the water that January, and the blue floor of the Lake, wrinkled with gold, seemed to be the day itself, stretching before her unspent and beautiful. (47; bk. 1 ch. 7 my emphasis)

Lucy projects her personal mood onto the surface of the Lake, to the point that the Lake seems to become her own prospects for the day and embody them. The lake is both a lake and something more. But although Lucy tends to choose feelings and imagination over facts-her "city of feeling" is "beautiful because the rest was blotted out" (24; bk. 1 ch. 3)-the knowing reader will not fail to notice how ominous the image is, the dual nature of Lake Michigan foreshadowing two other, grimmer duplicities. Firstly, it is on the deceptively calm waters of another beautiful lake, Lake Como, a famous tourist attraction, that Sebastian and Mockford will die because of an unexpected, "sudden storm" (137; bk. 1 ch. 21). Secondly, the "ice in the water" necessarily reminds us of the Platte River, which is the place of joyful, carefree skating at the beginning of the novel, but which eventually proves treacherous, when the ice hides the true course of the river at the end of book 2. Nothing is ever only what it first seems to be in Lucy Gayheart: there is always another possible reading, always another layer under appearances, and the musical scenes are no exception to the rule. When Sebastian sings "Vision fugitive," an aria from Massenet's opera Hérodiade, the title reflects Lucy's awakening to the beauties of true art. After his death, she realizes that "Clement Sebastian had made the fugitive gleam an actual possession. With him she had learned that those flashes of promise could come true, that they could be the important things in one's life." (183-184; bk. 2 ch. 8) However, Lucy will never truly possess the ability to create this "gleam" because she will never become a professional pianist and this dream remains an unfulfilled "promise" and a truly "fugitive" vision. She dies a tragic death, profoundly unjust because utterly arbitrary, like Sebastian's. It is hard, therefore, to overlook the central theme of the novel: death eventually conquers all, from great artists to humble watchmakers, and no artistic beauty, however otherworldly it may seem, can save us from the unexpected twists of fate. The overarching pessimistic mood of the novel is constantly counterbalancing the enthusiasm of the naïve Lucy for the redeeming power of art.

More radically, the selection of songs mentioned in the novel also points to another central theme that is at odds with the supposed capacity of art to become the "thing itself": the omnipresence of masks, which makes the definition of the "thing itself" problematic. The whole novel is haunted by the disquieting presence of "doubles," to take up the word Hermione Lee uses Florent Dubois 9/11 to describe the "split personalities" of many characters in the novel (345), and the songs Sebastian sings develop this theme. Schubert's "Doppelgänger" echoes the dual nature that most characters conceal in the novel. Harry Gordon not only wears the "jocular masks" of the successful country businessman-which he takes off only with Lucy, "showing her a naked man who had perhaps never been exposed to any eye before" (109-110; bk. 1 ch. 16)-but he also represses feelings that he cannot accept ("underneath his resentment and his determination to punish, there was a contrary conviction lying very deep, so deep that he held no communication with it" 216-217; bk. 3 ch. 3).

Sebastian has a similar persona that he adopts with people he is not intimate with, "a manner so perfected that it could go on representing him when he himself was either lethargic of altogether absent." (49; bk. 1 ch. 7) It is as if there were two different Sebastians with the same body, like the twins Castor and Pollux in one Schubert song (29; bk. 1 ch. 4). In the same way, the person Lucy's sister, Pauline, shows to the people of Haverford is only a "mannikin" (168; bk. 2 ch. 4) and she hides her jealousy for Lucy so well that Lucy does not even suspect it ("Lucy had never been aware of these hidden feelings in her sister." 171). This omnipresence of doubles in daily life and in the songs Cather chooses to mention necessarily casts doubt on the possibility of ever reaching the "thing itself," in life as in art.

To conclude, Lucy Gayheart shows the extraordinary coherence of Cather's aesthetic conceptions, from the beginning up to the end of her career. And yet, without altering them in anything, she seems to look upon these beliefs from an almost ironic distance. Now a mature writer, Cather depicts how the illusions of youth are shattered by tragedies and the final acceptance of death as an inevitable part of life. For Cather as for Lucy, these illusions concern first and foremost the redeeming power of art, its capacity to create "an invisible, inviolable world" (Lucy Gayheart 104; bk. 1 ch. 16). In Lucy Gayheart there is no "inviolable world." Meaning is constantly shifting and the characters are fragmented into various masks and memories which forever preclude the possibility of a direct knowledge or perception of anything and anyone. The artist, then, cannot naively hope to "express" the truth of something in words, as the young Cather used to advocate.10 Even the truth of such a supposedly powerful thing as the "thing not named" is questioned in the novel. Lucy's footsteps in the concrete do suggest her presence, but it is a thirteen-year-old Lucy they conjure up (226; bk. 3 ch. 4) and she is "running away" (231; bk. 3 ch. 4). Art, therefore, is as elusive as life is and Lucy Gayheart, with its entire narration presented as a memory composed of memories, mirrors that constantly failing pursuit of life. To the melancholy of the Schubert songs answers the melancholy tone of the novel, arising in both cases from a tragic subject as well as from metaphysical doubt. The characters are constantly facing doubles and constantly missing the events they live; consequently, their lives are a constellation of fading memories rather than an addition of truly lived experiences. 11 In Lucy Gayheart one feels the anxiety Cather expressed in "Light on Adobe Walls," her last, unfinished essay: what if art were a mere "game of makebelieve," (976), a word Lucy also uses to describe her relationship with Sebastian?12 Although it remains, she tells us, "very exciting and delightful to people who have an ear for it or an eye for it"

It usually takes several weeks' work for Thea to find the idea that will inform her interpretation. Landry tells Ottenburg, "When she begins with a part she's hard to work with: so slow you'd think she was stupid if you didn't know her. ... All at once, she got her line-it usually comes suddenly, after stretches of not getting anywhere at alland after that it kept changing and clearing." (448; pt

. 6 ch. 8)

"While she was on the stage she was conscious that every movement was the right movement, that her body was absolutely the instrument of her idea." (478; pt. 6 ch. 11)

A recent study showed that singers who studied acting tend to show more inaccuracies of pitch than classically trained singers. It only confirms something that music lovers already knew: sometimes a slightly wrong note can be more moving than the perfectly executed one (Linvingstone, Choi, and Russo).

"Art is not thought or emotions, but expression, expression, always expression." ("A Mighty Craft" 417)

Lucy is so excited when she is in Sebastian's presence that she enjoys the time she spends with him better in memory: "It was at night, when she was quiet and alone, that she got the greatest happiness out of each day-after it had passed!" (93; bk. 1 ch.[START_REF]A few.' / 'Make it as many as you can, Lucy. Nothing really matters but living. Get all you can out of it. I'm an old woman, and I know. Accomplishments are the ornaments of life, they come second. Sometimes people disappoint us, and sometimes we disappoint ourselves; but the thing is[END_REF]

"Her playing for him was nothing but make-believe; and his friendliness was make-believe, perhaps." (61; bk 1 ch. 8)


L D C Jaubert

Swann Piatecki

Masudul Haque

R Moessner

Itinerant electrons in the Coulomb phase

We study the interplay between magnetic frustration and itinerant electrons. For example, how does the coupling to mobile charges modify the properties of a spin liquid, and does the underlying frustration favor insulating or conducting states? Supported by Monte Carlo simulations, our goal is in particular to provide an analytical picture of the mechanisms involved. The models under considerations exhibit Coulomb phases in two and three dimensions, where the itinerant electrons are coupled to the localized spins via double exchange interactions. Because of the Hund coupling, magnetic loops naturally emerge from the Coulomb phase and serve as conducting channels for the mobile electrons, leading to doping-dependent rearrangements of the loop ensemble in order to minimize the electronic kinetic energy. At low electron density ρ, the double exchange coupling mainly tends to segment the very long loops winding around the system into smaller ones while it gradually lifts the extensive degeneracy of the Coulomb phase with increasing ρ. For higher doping, the results are strongly lattice dependent, displaying loop crystals with a given loop length for some specific values of ρ, which can melt into another loop crystal by varying ρ. Finally, we contrast this to the qualitatively different behavior of analogous models on kagome or triangular lattices.

I. INTRODUCTION

The combination of magnetism and itinerant electrons is a multi-faceted field in the physics of correlated electrons, where our understanding is still remarkably patchy: even in the case of a square lattice Hubbard model, we lack consensus on a detailed phase diagram in the doping-temperature plane.

Besides the cuprate superconductors, there are plenty of other settings in which interesting questions arise, not least popularized of late by questions raised by the pnictide superconductors, where magnetic frustration and accidental degeneracies have started to be considered.

More broadly, there has been increased interest in the interaction of frustrated magnetism with itinerant electrons [1][2][3][4][5][6][7][8][9][10][11] . Here, we take up the spirit of this thread of work and study itinerant electrons on a highly frustrated lattice, the pyrochlore lattice. We consider both three and two dimensions, the latter case also being known as the square lattice with crossings, planar pyrochlore, or checkerboard lattice.

We start with an exotic frustrated phase of a magnetic insulator, the Coulomb phase, which has been extensively studied recently [12][13][14][15][START_REF]For a review of the Coulomb phase, see C. L. Henley[END_REF][17] . This phase has a number of unusual properties, including algebraic spin correlations and the emergence of extended one-dimensional degrees of freedom [18][19][20] , the nature of which is an independently interesting problem 21 .

Especially the latter will play an important role in the following analysis, given the ability of electrons to provide evidence for non-local structures through properties related to transport phenomena. Indeed, it is the marriage of the local constraints imposed by frustration with the 'non-local' physics describing mobile particles which makes up for much of the interest in this field.

The Ising-double exchange model which we study here has many parameters: electron density, ρ, temperature T , Ising anisotropy, Hund's coupling, J H , magnetic exchange, J, and electron hopping integral, t. A full study of general parameter choices is well-nigh impossible in any detail analytically.

Our approach to the problem considers a regime where the effects of frustration are particularly strong but where considerable progress towards a detailed description is nonetheless possible by analytical (or simple numerical) means. That a (non-trivial) regime where this is possible exists at all is a priori not obvious, and we find that we need to restrict a number of parameters to limiting values: we study the limit where the magnetic energy scales are much larger than the hopping integral t, so that the resulting problem is one of electrons hopping on a classical background spin configuration. We do not need to restrict the electron density to be small, although for that case, we have the most detailed set of results.

We find that the low-density behaviour can be mapped onto a study of a classical loop model with non-trivial weights arising from the addition of electrons Hund's coupled to the spins in the Coulomb phase. This results in phenomena such as a transition from a (in two dimensions, critical) percolation situation to one in which the loops acquire an exponential length distribution, thereby removing all conducting paths across the sample. As the doping is increased further, we find a sequence of density-dependent preferred loop lengths, which lead to a tendency to form loop crystals which may, however, be frustrated by the lattice geometry.

The organization of this article, and our main results, are summarized below.

A. Summary and Overview

We restrict ourselves to the limit of large exchange coupling interactions so that the ice-rules (see figure 1) themselves are never compromised. We make extensive use of the loop picture encoding the ice rules. 20 The loops serve as 1D channels for the electrons. The problem is thus transformed to entropy and energy considerations of possible loop coverings, with loops supporting varying numbers of electron. This "loop framework" for describing conduction electrons is in-troduced in Section II.

Ref. 20 has described in some detail the loop distributions for both 2D and 3D in the absence of electrons. Once electrons are added, within each loop electrons can occupy states whose energy is given by a 1D dispersion. The dispersion minimum is independent of loop length, so the first electron in a loop has the same energy in all loops. As a result, there is a low doping regime where it is possible to fit at most one electron per loop. All such configurations have the same energy. This is the entropic regime, because entropic arguments determine favorable configurations within an equal-energy manifold.

When the density of electrons ρ is larger, a sub-extensive number of states, possibly even a unique one, tend to be favored, because they manage to minimize the kinetic energy; this we term the energetic regime.

In Section III, we present entropic considerations relevant to the regime of low doping. The total entropy contribution comes from both the loops (with parameters extracted numerically), and the electrons (derived analytically). In particular, the presence of electrons acts like a cutoff on the total number of loops in the system. In 3D, forbidding configurations with less loops than electrons suppresses the formation of extensive loops (present in the peak of the probability distribution function (PDF), see e.g. figure 5), but does not modify the exponents of the PDF. On the other hand in 2D, there is evidence for a variation in the exponent of the power law of the loop distribution.

Once the electronic density gets too large for loops to be restricted to at most one electron, we need to consider energetics. Section IV presents these energy considerations and the phase diagram obtained thereby. Ignoring lattice constraints on loop coverings, we use energy calculations and a Maxwell construction to obtain the phase diagram of Figure 9. We verify some of these results through Monte Carlo simulations. We also present constraints imposed by the lattices under considerations and identify loop crystals arising as the doping is varied.

With periodic boundary conditions, loops spanning the system can be divided into segments connecting "opposite" faces, which we call filaments. Because the transmission of electrons through the system can only occur via these conducting channels, section V is dedicated to their statistics, as a function of doping ρ and dimension. In absence of itinerant electrons, the number of filaments grows linearly with (cubic) system size in 3D but remains constant and of O(1) in 2D. While these behaviors are qualitatively not modified at low doping, the conducting channels turn out to vanish at intermediate values of ρ.

A separate final section is devoted to an outlook which also contains some words on the behaviour of analogous models on the triangular and kagome lattices, which turn out to exhibit qualitatively different properties both from the pyrochlores and from each other: we find a magnetic conducting solid as well as an insulating cooperative paramagnet.

The numerical component of our work involves Monte Carlo simulations of several types; some details are provided in the Appendix.

II. THE SYSTEM AND THE LOOP FRAMEWORK

A. The model

We will focus primarily on the checkerboard and pyrochlore lattices, which are two lattices where a Coulomb phase can appear (see figure 1). The localized magnetic moments are Ising spins S i all parallel to a global axis, while itinerant electrons can hop on the lattice sites. The Hamiltonian is

H = J i,j S i • S j - i,j ,α t (c † i,α c j,α + c † j,α c i,α ) -J h i,α,β c † i,α (σ α,β • S i ) c i,β (1)
where t is the hopping integral between two neighboring sites, c † i,α (c i,α ) are creation (annihilation) operators of itinerant electrons of spin α on site i, and σ α,β are the Pauli matrices. In order of appearance, the terms in equation ( 1) are the antiferromagnetic nearest-neighbour exchange between the localized spins incorporating magnetic frustration, the hopping term allowing movement of itinerant electrons, whose spins interact with the localized magnetic moments through ferromagnetic Hund coupling (last term).

In this work, we focus on the limit t ≪ J h ≪ J. In this limit, the highly degenerate ground state of the frustrated system serves as background for the motion of the electrons. Magnetic excitations (violations of ice rules) are not present in this limit. Electrons can only hop between nearest neighbour spins having the same orientation.

At zero temperature, a Néel or ferromagnetic order would give rise to an insulating or metallic state respectively, but a spin liquid provides a network of conducting paths for itinerant electrons. The present work unveils the geometry of this network, as it is influenced by both the lattice and the minimization of the hopping energy.

B. Emergence of loops

The pyrochlore and checkerboard lattices are made of corner-sharing units with four spins; respectively the tetrahedron and the square with crossings. The antiferromagnetic couplings impose the so-called "ice-rules" with zero magnetization per unit, obtained with two spins pointing up and two spins pointing down 22 . In absence of itinerant electrons, this ground state is highly degenerate with 6 possible configurations per unit. It corresponds to the 6-vertex model in 2 dimensions [START_REF] Baxter | Exactly solved models in statistical mechanics[END_REF] and can be mapped onto the nearest neighbor spin ice model in 3 dimensions [START_REF] Harris | [END_REF]25 . These spin systems serve as background for the emergent physics of the so-called Coulomb phase [START_REF]For a review of the Coulomb phase, see C. L. Henley[END_REF] , a gauge theory where the discrete ice-rules under coarse-grained lead to the emergence of a divergence free flux.

Joining spins of the same orientation in every unit, one obtains loops of up spins and loops of down spins (see Fig. 1). The Coulomb phase can thus be described as an ensemble of possible loop coverings. The resulting loop model possesses two flavors (loops of up and down spins), where every site of the premedial lattice [START_REF]For a review of the Coulomb phase, see C. L. Henley[END_REF] is occupied by two loops, one of each flavor, and every bond is visited by one loop only. In previous work by some of the authors 20 , a detailed account has been given of the statistics and distribution of these loops, both for the 2D and 3D cases.

In the limit of large J h , all up (down) electrons are only allowed to hop along an up (down) loop, and are constrained to remain within this loop. This reduces the electron dynamics to be one-dimensional whatever the dimension of the lattice.

The 1D hopping restriction allows us to describe electron dynamics in terms of the dispersion of a 1D tight-binding problem, E k = -2t cos k, with k the 1D momentum along the loop. The 1D momentum along the loop, k = 2πq/ℓ, is discrete for a loop of finite length ℓ (q = -ℓ/2, ..., ℓ/2 -1). An up (down) loop of length ℓ can contain between 0 to ℓ up (down) electrons. Note that double occupancy does not occur in the limit we are considering.

The lowest and highest single-particle levels in the dispersion have energy ±2t, independent of the loop length (figure 2).

Our system is thus described by

• the number of lattice sites N , equal to 4 L 2 and 16 L 3 for the checkerboard and pyrochlore lattices respectively, where L is the linear number of unit cells;

• the total electron number N e , and the electron density ρ = N e /N ;

• the loop histogram of a given configuration, i.e. the number h i of loops of length ℓ i , ℓ i being necessarily an even number on a bipartite lattice;

• the type of lattice, which will among other things determine the smallest possible loop in the system ℓ min (4 for checkerboard and 6 for pyrochlore); the longest possible loop length is always ∼ N/2.

We define a few additional relevant observables:

-π π k -2t 2t energies, E k -π π k -2t 2t
FIG. 2: Single-particle energy levels of itinerant electrons confined to loops of length ℓ = 8 (left) and ℓ = 12 (right), due to the hopping term in Hamiltonian (1). There are nondegenerate levels at the highest and lowest energies, ±2t, independent of loop length; the other levels are doubly degenerate.

• the total number of loops in a given configuration N ℓ = i h i ; • the average loop length for a given configuration

ℓ = i h i ℓ i i h i = N N ℓ ;
• the statistical average loop length ℓ over all loop configurations;

• the statistical average number of loops N ℓ ;

• the number of filaments (see section V).

III. LOW DOPING REGIME

In this section we consider the low doping regime where N e is small enough to have loop configurations with more loops than electrons, N ℓ > N e . Since the lowest single-electron energy level in any loop is -2t, the minimum accessible energy is the same (-2tN e ) for all such configurations. Therefore, the ground state manifold consists of all such loop coverings with the same energy -2tN e . The free energy within this manifold is then only determined by entropics. At zero temperature, entropy is understood in the sense that all configurations have an equal probability to occur.

In the first subsection below, we present some entropy calculations, combining loop and electronic contributions to the entropy, and show how this determines the average loop length at nonzero electronic density ρ. In the second subsection we present numerical results on the effect of electrons on the entire loop length distribution (loop PDF). The effect on the loop PDF is a natural way to characterize the influence of electrons in the magnetic Coulomb-phase system. 2) collapsing all system sizes onto the same curve (L = 100 to 600 in 2D and L = 4 to 60 in 3D). In 2D, the two Gaussians have the same width and corresponds to even and odd N ℓ (upper and lower curve). In 3D, finite size corrections are visible for L = 4 (red crosses).

A. Entropy

The total entropy consists of loop and electronic contributions. The loop distribution is of course itself affected by the itinerant electrons. However, in the limit of small ρ, the change in loop entropy is small. Below we combine the ρ = 0 loop entropy with the finite-ρ electronic entropy to approximate the total entropy at small ρ.

Loop entropy. Figure 3 shows the distribution of the number of loops N ℓ having length ℓ, in the absence of electrons (ρ = 0), obtained from Monte Carlo simulations (Appendix A). The distribution has Gaussian form:

P (N ℓ ) ∼ 1 √ N exp - (N ℓ -N ℓ ) 2 2κN (2)
where κ 2d = 0.0384 and κ 3d = 0.00423. In 2D, there are two Gaussians, corresponding to even and odd N ℓ .

For a system with short-distance correlations, Gaussian distributions are natural to expect from the Central Limit Theorem, since by dividing the system into small mesoscopic segments the total distribution can be recast into a sum of many random variables. In our case, however, we have a system with algebraic correlations and extended objects (loops), so finding Gaussian distributions is not a priori trivial.

Equation 2 can be expressed in terms of the average loop length ℓ = N/N ℓ instead of N ℓ :

P (ℓ) ≡ P (N ℓ ) dN ℓ dℓ ∼ √ N ℓ 2 exp - N 2κ 1 ℓ - 1 ℓ 2 ∼ √ N ℓ 2 exp - N 2κ ℓ -ℓ ℓ 2 2 (3)
Here we have used 1/ℓ ℓ ≈ 1/ ℓ 2 , which is valid for large N in the region where the Gaussian is appreciable. Thus the loop contribution to the entropy (∼ ln P ) is

S loop = S 1 - N 2κ ℓ 4 ℓ -ℓ 2 , (4)
where S 1 is a constant. Electronic entropy. Since we have at most one electron per loop, the number of possible combinations to put N e electrons in N ℓ loops is the binomial (Pascal) coefficient

N l !/[N e !(N l -N e )!]
. The logarithm then gives the electronic contribution to the entropy. Using Stirling's approximation for the thermodynamic limit (N l ≫ 1, N e ≫ 1), we get as per usual

S elec = N ℓ [x ln x + (1 -x) ln(1 -x)] (5)
where x = N e /N l = ρℓ. Total Entropy. For small electron densities, we can approximate the loop entropy by the ρ = 0 expression calculated above. The total entropy of electrons and loops is then

S tot = - N ℓ ρℓ ln ρℓ + (1 -ρℓ) ln(1 -ρℓ) + S 1 - N 2κ ℓ 4 0 ℓ -ℓ 0 2 . ( 6
)
The subscript 0 represents the ensemble average at ρ = 0. The value of ℓ that maximizes S tot is found (in the limit ρℓ ≪ 1) to be

ℓ opt ≈ ℓ 0 -ρκ ℓ 3 0 . (7)
For ρ ℓ 0 ≪ 1, the distribution stays almost Gaussian, so this most probable value of ℓ is approximately the mean value of the distribution ℓ ρ . In Figure 4, we compare this prediction For both panels, the blue triangles data points are very close to thermodynamic limit.

to numerical data. For each system size L, we can extract ℓ 0 from the numerical ℓ at ρ = 0. Eq. ( 7) then gives a linear prediction (dashed straight lines in Figure 4), which works well for small ρ.

B. Loop length distribution

We next examine the effect of electrons on the entire loop probability distribution function (PDF). We denote by τ the power-law exponent, when the PDF has form ℓ -τ . Without electrons (ρ = 0), the PDF follows P 2D ∼L 2 /ℓ τ in 2D with τ = 2 + 1/7. In 3D, the ρ = 0 PDF displays a crossover around ℓ ∼ L 2 between two power laws, from L 3 /ℓ 5/2 to 1/ℓ. (For details, see Ref. 20.) At finite electron densities, loop configurations with N ℓ < N e are rejected due to energetics, as explained previously. Thus electron doping acts not entirely unlike a "chemical potential" for loops, favoring configurations with more loops, and thus a priori shorter ones. In both 2D and 3D, this implies a disappearance of longer loops, as can be seen in the calculated PDF's of Figures 5 and6, where the large-ℓ parts of the PDF are progressively decimated for increasing ρ. In 3D the form of the PDF is otherwise unchanged (Figure 5). In 2D the effect seems to be more drastic; the inset of figure 6 suggests that the entire power-law behavior of the PDF is modified.

To quantify how the 2D loop PDF changes qualitatively at finite ρ, we define and compute a local exponent in ℓ: τ local (ℓ, ρ) = log (P 2D (ℓ, ρ)/P 2D (2ℓ, ρ)) / log 2. This is displayed in the upper right inset to Figure 6. Our results suggest a trend toward increasing τ local as a function of ρ, consistent with the idea that itinerant electrons favor small loops.

This outcome deserves a few comments. In Ref. 20, the loop statistics of the 2D Coulomb phase (zero doping) has been shown to be analogous to the Stochastic Loewner evolution process SLE κ=6 with fractal dimension D f = 1 + κ/8 = 7/4. The SLE κ can be identified to various realizations of the O(n) model through the relation n = -2 cos(4π/κ). 26 The O(n) model is often used to describe fully packed loop Higher values of the fugacities favor configurations with more loops and tend to increase the value of τ 27 , in a way reminiscent of the influence of doping here. The addition of itinerant electrons remains a non-trivial problem and is not exactly the same as a fugacity for a loop, but at small and finite doping ρ, some features could be captured by O(n(ρ) > 1) models or SLE κ(ρ)<6 processes.

IV. LARGE DENSITIES; PHASE DIAGRAMS

In this section, we present results relevant for higher densities, where entropic considerations are no longer sufficient, and non-trivial electronic hopping energies need to be considered. We continue to describe the system in terms of loop coverings.

We first provide an analysis based on calculations for loop coverings of equal-length loops. Using a Maxwell construction, we can use this information to predict ranges of electron density where the ground state manifold consists of coverings by loops of two different lengths.

These considerations, described in the first two subsections below, do not take into account any lattice constraints other than the fact that the minimum loop length is ℓ min = 4(6) for the checkerboard (pyrochlore). Lattice constraints, disallowing some loop coverings, are difficult to enumerate or list comprehensively on account of their non-local nature. In the final subsection, we present the lattice constraints that we have identified, and their implications.

A. Equal-length loop configurations

In this subsection and the next, we imagine that coverings with any unique loop length ℓ ℓ min are possible. Later, we will show that at certain fillings, the ground states exhibit a unique loop length, while at others, the behaviour can be understood through a Maxwell construction based on the singlelength results.

For an odd number of electrons ρℓ = 2n o + 1 in a loop of length ℓ, the energy is (see figure 2)

E(ρ, ℓ) = -2t no n=-no cos 2πn ℓ = -2t sin (πρ) sin (π/ℓ) (8)
For an even number of electrons 2n o , this expression becomes

E(ρ, ℓ) = -2t sin π (ρ -1/ℓ) sin (π/ℓ) -2t cos(πρ) (9)
More generally, if ρℓ is not an integer, we define the highest odd integer below ρℓ as

η = 2E ρ ℓ -1 2 + 1 (10)
where E(.) is the floor function. Each loop is filled with at least η electrons up to the energy levels at k = ±π(η -1)/ℓ, while the remaining N (ρ -η/ℓ) electrons in the system are distributed in the partially filled level at k = ±π(η + 1)/ℓ. The total energy is

E(ρ, ℓ) = -2tN sin (πη/ℓ) ℓ sin (π/ℓ) + ρ - η ℓ cos π η + 1 ℓ (11)
These expressions are electron-hole symmetric, i.e. invariant under ρ ↔ (1 -ρ).

Figure 7 displays the loop length ℓ(ρ) that minimizes the energy (11) as a function of ρ, and thus corresponds to the ground state if we impose a unique loop length in the system. (The electron-hole symmetry shows up clearly through the mirror symmetry on either side of ρ = 0.5. Therefore, we shall from now on only consider densities below 1/2.)

Since the lowest possible energy for an electron is -2t and is only accessible for one electron per loop, loops of length 4 are favored up to ρ = 1/4. Of course for lower densities than 1/4, other configurations may be possible as long as the number of loops is larger than the number of electrons, but a system with only loops of length 4 will always be part of the ground state manifold for ρ 1/4.

Remarkably, for most densities, a finite loop length with discrete energy levels is preferred compared to infinite loops, except for ρ = 1/3 and 2/5.

We plot on figure 8 the minimum energy corresponding to the loop length ℓ(ρ) of figure 7: E min (ρ) = E(ρ, ℓ(ρ)).

B. Maxwell construction; lattice-independent phase diagram

We now move beyond configurations with unique loop length. We need to consider mixtures of electron densities in the different loops. This is done through a Maxwell construction, similar to the physics of a liquid-gas first-order phase transition.

A system of N sites and density ρ can be divided into two subsets of sites N 1 and N 2 , with density ρ 1 and ρ 2 of electrons and loop length ℓ 1 and ℓ 2 respectively, with

N = N 1 + N 2 N ρ = N 1 ρ 1 + N 2 ρ 2 . ( 12
(ρ) > N 1 E min (ρ 1 ) + N 2 E min (ρ 2 ) , (13)
then the mixture of two densities is more stable than a unique density and "phase separation" occurs. We thus construct a phase diagram, separating regions of different loop length combinations.

On figure 9, the dots correspond to densities where a unique loop length is favored, whereas the zones between them are "phase mixed", defined by the loop lengths and electron densities of the surrounding dots. The ratio of one phase compared to the other is given by equations (12). For example for ρ = 0.4, 80% of the sites belong to loops of length ℓ 1 = 8 with electron density ρ 1 = 3/8, while the remaining 20% belong to loops of length ℓ 2 = 6 with density ρ 2 = 1/2. As shown on figure 7, loops of length 4 are particularly robust over a wide range of ρ; preventing their formation, e.g. in the pyrochlore lattice in d = 3 which permits loops of minimal length 6, thus strongly modifies the phase diagram.

These results indicate that itinerant electrons tend to favor relatively small loops and to prevent the formation of infinite ones at zero temperature.

At low electron densities (up to 1/ℓ min ), the minimum energy for a given ρ is degenerate: e.g. for ρ = 1/12, having only loops of length 4, 6, 8, 10 or 12 gives the same energy, as the number of electrons is smaller than the number of loops on the system, and every electron can fill the lowest energy level. This explains why E min (ρ) is a straight line in this region (Figure 8): as ρ decreases from 1/ℓ min to 0, the degeneracy of the ground state increases until one recovers naturally the full degeneracy for ρ = 0. This of course corresponds to the entropic regime mentioned above.

We stress again that these results have been obtained independently of the lattice (except for the value of ℓ min ), and that lattice constraints (next subsection) will modify some of the phase diagram.

C. Lattice constraints on loop coverings

We now consider effects of the lattice in disallowing some of the configurations predicted by our analysis above. For the specific densities where it is possible to cover the lattice with loops of a unique length, the result is an ordered loop crystal. In some cases, however, a covering by a unique length or by a combination of loops of two lengths, is not possible. We point out some such cases below.

Checkerboard

We first focus on the regime ρ ∈ [1/4; 3/8] favoring loops of length 4 or 8 according to the Maxwell construction. As illustrated in figure 10, not only can the lattice be covered by these loops, but the transformation from 4 loops of length 4 to 2 of length 8 is also purely local and allows all possible ratios between these two phases in the thermodynamic limit. A pair of loops of length 8 (one made of up spins, the other of down ones) cannot be separated if there are no other lengths than 4 and 8 in the system. We shall call such pair a defect. The defect concentration is determined by ρ. Defects are not topological in the sense that they can be created and annihilated locally, they can be placed anywhere in the background made of loops of length 4.

At quarter filling, there is one electron per loop of length 4, its energy is -2t and their total number is N/4. Each additional pair of electrons fills the first excited level of the newly created defect ( trons added with respect to the loop crystal at quarter filling is (ρN -N/4) = (4ρ -1)L 2 . The total energy of the system between ρ = 1/4 and 3/8 is then

E(ρ) = -2t L 2 1 + (4ρ -1)( √ 2 -1) (14)
Once normalized per number of sites 4L 2 , this expression is the dashed line plotted on figure 8. We have verified this analytical result with finite-temperature Monte Carlo simulations; some details of the method are in Appendix A. Extensive degeneracy is recovered in this region between 1/4 and 3/8.

At higher doping, above ρ = 3/8, the Maxwell argument predicts a mixture of loops of length 8 and 6. However, as we can see in figure 10, a single loop of length 6 imposes the presence of loops of length 10 at least. This implies that this region of the phase diagram (obtained without accounting for such lattice constraints) is further modified, in an as yet unknown manner.

Pyrochlore

Analogous modification of the phase diagram of figure 9 is more severe for the pyrochlore, and shows up already at ρ = 1/ℓ min = 1/6. We have found that it is impossible to cover the pyrochlore lattice with loops of length 6 only. However, it is possible to do it for half of the system, as explained next.

The pyrochlore lattice can be seen as an alternative stack of kagome and triangular layers orthogonal to one of the global [111] axes. As depicted on figure 13(d), 2/3 of each kagome layer can be filled with loops of length 6 (blue hexagons), while the other 1/3 of the kagome sites forms extensive winding loops along the [111]-axis, crossing alternatively the kagome and triangular layers. Since all blue sites form loops of length 6, putting one electron per loop provides a ground state configuration up to ρ = 1/12 at least. This is a priori not the only one, but this proves its existence.

Indeed, despite an intensive search by complete enumeration of configurations respecting the ice-rules on the pyrochlore lattice, we have not detected any relevant loop crystal or mixture of them for systems up to 128 sites. A system of size 192 has been partially investigated, with the same outcome. The smallest occurrence of a single-length loop covering is for 8 loops of length 16 in a system of 128 sites; however this is not a relevant length according to our phase diagram of Figure 9. Thus, the loop coverings on an actual pyrochlore are modified from Figure 9 in most or all of the density range ρ ∈ [1/6, 1/2]. The details of this modification remains an important open problem.

V. CONDUCTING CHANNELS OR "FILAMENTS"

In this section, we discuss the effect of the Coulomb phase and loop structure on the conductivity of mobile electrons. Since the electrons are confined to loops, they can conduct only if a loop connects one edge of the sample to the opposite end. Therefore, we study the number of such sample-spanning loops, which, following Ref. 28, we refer to as "filaments".

When there are no filaments (e.g. in loop crystals with only finite-length loops), the system is unambiguously an insulator. When there are filaments spanning the system, the system cannot be immediately called a conductor, because the actual conductivity will depend on scattering mechanisms exterior to our model.

In the first subsection below, we consider small dopings, where we present Monte Carlo results for the average number of filaments as a function of system size. In the second subsection, we comment on the consequences of our phase diagram.

A. Filaments at low densities

We first consider very low densities that the loop distribution can be assumed to be largely unchanged from the ρ = 0 case. 11: Average number of filaments, i.e. those segments of loops spanning the entire system from one border to the opposite one, as a function of system size L. We use two definitions of filaments; they can either cross the orthogonal borders thanks to the periodic boundary conditions (•) or not ( ). The cartoon on the right shows such a boundary-crossing filament, which would be excluded in the data. Dashed lines are guides to the eye for the linear behavior with L. Both x-and y-axes are on a logarithmic scale.

In 2D, there is a small but constant number (≈ 1.86) of winding loops on average in the thermodynamic limit 20 . There is thus some probability to have filaments in the checkerboard case, but the number of filaments does not grow with system size.

In 3D, the background of winding loops ensures that there are filaments whose number grows with system size. The data in Figure 11 shows that the number of filaments increases linearly with the linear size of the cubic sample.

Since our simulations and loop counting are performed with periodic boundary conditions, some of the filaments, while spanning the sample in one direction, also cross one of the orthogonal boundaries. (An example is shown in the cartoon to the right of Figure 11.) One can argue that this type of filament would not contribute to conduction in a real-life cubic pyrochlore sample. Therefore, we show data both excluding and including this type of filament, and they are seen to have the same power-law behavior.

Since the data including these filaments have better statistics, for finite ρ we display the inclusive data, with the expectation that there is no qualitative difference.

According to Fig. 5, itinerant electrons tend to make disappear extensive loops of length L 3 in 3D. Whether or not this prevents the formation of filaments is not as straightforward as it seems. Indeed, one could naively assume that the number of conducting channels will decrease and maybe even vanish in the thermodynamic limit. However the number of filaments remains approximately constant as plotted on figure 12. This means that the average number of loops increases with doping via dividing the very long ones into smaller but nonetheless extensive loops spanning the system. In the low doping regime in 3D, the number of conducting channels is approximately independent of ρ.

By contrast, in 2D, with relatively large error-bars, the number of filaments decreases but remains of O(1) as electrons are added.

B. Conduction channels at larger fillings

In Section IV, we identified densities (for both 2D and 3D) where the system is a loop crystal or is covered by loops of two finite lengths only. In such cases, there are no filaments, and the system is truly insulating: lifting the frustration-induced degeneracy removes the non-locality of the loops. How the energetics (and resulting degeneracies) imposed by lattice constraints (see section IV C) manifest themselves in transport properties is an intriguing separate question.

For some ranges of ρ, especially in 3D, the loop coverings indicated by the Maxwell construction are disallowed by the lattice geometry (section IV C). In such cases it remains an open question whether or not the lattice constraints result in loop coverings including infinite (sample-spanning) loops. Unfortunately, the issue of conducting channels depends on the answer to this generally unresolved question.

VI. SUMMARY AND OUTLOOK

We have analyzed the double exchange model on the pyrochlore lattice in two and three dimensions. We have chosen to consider a parameter range for which (i) magnetic frustration is known to give rise to unconventional ground state ensembles and (ii) where it is possible to make considerable analytical progress by mapping the system onto an ensemble of loops, the statistical properties of which are influenced by the addition of the electrons.

We have identified a number of phenomena which depend on features such as dimensionality, which determines whether or not there exist loop segments winding around the system; or lattice structure, which may frustrate the geometric packing of preferred loop lengths.

The model studied here leaves unanswered a number of questions and immediately suggests many generalizations and extensions. We have worked in a limit of parameters such that the magnetic exchange and anisotropy dominate over the Hund coupling which in turn dominates the hopping integral. Our analysis applies to zero temperature.

It would be interesting to relax any of these choices, although technically this may not be easy. In particular, given the presence of gapless excitations on long loops, interesting low-temperature physics may appear. Canting can give rise to non-trivial Berry phase physics, and finite Hund's coupling will enable electrons to hop between loops. For example, recent work at quarter filling on the pyrochlore Kondo lattice has shown the emergence of a chiral magnetic order in the weak-coupling regime 29 .

Finally, even in the parameter range discussed here, it will be interesting to ask how different frustrated lattices shape up compared to the pyrochlores. We devote the final paragraphs of this paper to discussing this question for the case of triangle-based lattices, the triangular Bravais lattice and the Archimedean kagome lattice.

A. Other lattices: triangular and kagome

The Ising ground states observed for the pyrochlore lattice have vanishing total spin on each tetrahedron. These 'ice rules' ensure the existence of the Coulomb phase and states obeying them amount to a moderate zero-point entropy of less than a third of that of a free spin. By contrast, the zero-point entropy of the triangular Ising magnet is not far from half of that of a free spin, while that of the kagome magnet is over 70% of log 2.

Most fundamentally, the single triangle is relatively much more degenerate than a tetrahedron, with 6 out of 8 (rather than out of 16) states being ground states. The triangle states have varying magnetizations of ±1 (whereas states obeying the ice rule have a unique magnetization, unless one tunes a field to a transition between magnetization plateaux 30 ).

To move towards the full triangle-based lattices, it is worth noting that for any Ising antiferromagnet, the 'hopping network' formed by neighbouring aligned spins can have a coordination of at most half that of z, the coordination of the underlying lattice -otherwise it would be energetically favorable to flip the highly coordinated spin. The concept of the hopping network generalizes the loops on which the electrons hop on the planar and three dimensional pyrochlores.

It now turns out that triangle and kagome lattices behave entirely differently from pyrochlore in both two and three dimensions, on account of the nature of their frustrated ground state ensembles.

Triangular lattice

The ground-state degeneracy of the triangular lattice is immediately lifted by the addition of even a single hole. This result is entirely analogous to the frustrated Nagaoka theorem presented in Ref. 31, in the context of the magnetic supersolid discussed there, and it is also connected to the triangular Bosonic supersolids [32][33][34][35] .

As z = 6 for the triangular lattice, the hopping network of aligned spins no longer has coordination two as in the loops of the pyrochlore lattice. In fact, the coordination of a site of the hopping network no longer even needs to be uniform, so that there may be dangling or even isolated sites, as on Fig. 13.(a). The coordination can range all the way from zero (for a spin surrounded by a hexagon of oppositely aligned spins) to maximally z/2 = 3.

The latter happens when the hole sits on a site experiencing zero net exchange field from its six neighbors. There exists a unique state (pictured in Fig. 13.(b)) in which there is a network of three-fold coordinated sites. This state breaks translational symmetry by tripling the unit cell as well as timereversal symmetry as it corresponds to a state with a magnetization of a third of the saturated value. We have not studied what happens to a finite doping but a Fermi liquid regime on the hexagonal backbone at low doping looks likely. This would imply a conducting state, with the possibility of additional low-energy excitations in the form of defects of the hexagonal backbone.

Kagome lattice

The situation on the kagome lattice is different still. Its magnetic ground state ensemble is very short-range correlated, unlike that of the triangular lattice, which has algebraic correlations. With z = 4 for the kagome lattice, the hopping network can no longer branch but it can now have dangling links or isolated spins (coordination 1 and 0, respectively, as illustrated on Fig. 13.(c)).

The hopping networks thus consist of loop segments which need no longer close on themselves, minimally containing only one spin but not bounded above in the thermodynamic limit. The length distribution is, however, unlikely to contain long loops as this would only happen if each the magnetization of all the triangles along the loop segment has the same sign, at considerable cost in entropy.

Energetically, it is of course again most advantageous to have closed loops, as electrons on them gain hopping energy -2|t|. On the kagome lattice, such loops are readily constructed. The shortest ones are obtained by arranging spins to be aligned around a hexagon. Their densest packing is obtained for a state which breaks translational symmetry, tripling the unit cell and incorporating a 1/3 magnetization as was the case of the triangular lattice above, see Fig. 13.(d). However, this state -which is the unique ground state at electron density ρ = 1/9 is now an insulating one -all hopping paths for the electrons close back on themselves after six steps. energetically equivalent and eventually hit its initial position; the worm is then closed. Reversing all spins in the worm gives way to a new configuration in the Coulomb phase (figure 14). This method ensures both ergodicity and detailed balance to the algorithm in absence of electrons 36 .

Worms, as defined above, are not self-avoiding, i.e. they can erase their own path. If a worm goes an odd number of times through the same site, the corresponding spin will be flipped during the Monte Carlo update; if it happens an even number of times, the spin shall not be flipped. b. Decorrelation. Since we are dealing with Ising spins, two random configurations roughly differ by one half of the system sites on average. Hence two configurations separated by k worm updates can be considered decorrelated if close to one half of the system sites are different: we arbitrarily chose 45%. Previous work 20 has shown that a finite constant number of worm updates (∼ 10) was enough to decorrelate the system in 3D, but an increasing number with system size L was necessary in 2D. This is related to Pólya's theorem stating that a random walk (similar to the worm here) in 3D is transient (finite probability never to come back to the origin), while it is recurrent in 2D (it always comes back) [START_REF] Hughes | Random walks and random environments[END_REF] .

c. Worm algorithm with electrons -entropic regime. In the entropic regime, i.e. at low doping, there is at most one electron per loop. A correct sampling can then be done by using the above worm algorithm free of electrons, and rejecting all configurations with more electrons N e than loops N ℓ . However, the distribution of number of loops per configuration being Gaussian (see figure 3), the density of electrons acts as a cutoff and above a certain threshold of the order of 1/ ℓ , almost all configurations are rejected and it becomes impossible to get good statistics.

In order to take into account the influence of itinerant electrons in the entropy, we chose to weigh any loop configuration by the number of possible combinations to distribute N e electrons in N ℓ loops, namely the binomial coefficient C Ne N ℓ . d. Worm algorithm with electrons -energetic regime. In the energy regime, i.e. at intermediate or high doping, there is more than one electron per loop. The above method based on the worm algorithm without electrons become inefficient and finite temperature Monte Carlo simulations are necessary.

Starting from a given configuration, a worm update is proposed. The loop histogram of the states before and after the proposed update are computed, and the corresponding electron energies are calculated after filling the energy levels with N e electrons. The worm update is then accepted or not via a Metropolis argument.

FIG. 1 :

 1 FIG.1:In the Coulomb phase, each frustrated unit (crossed squares in 2D checkerboard, left, and tetrahedra in 3D pyrochlore, right) possesses two up and two down spins, respectively colored in blue and red. Those are the so-called ice-rules or divergence free conditions. Connecting spins of the same color forms a network of loops, as illustrated for the checkerboard. This model is equivalent to the nearest neighbour spin ice model.

FIG. 3 :

 3 FIG.3: Histogram of distribution of the number of loops N ℓ per configuration in 2D (left) and 3D (right), in the absence of electrons. The distribution is scaled according to the Gaussian expression of equation (2) collapsing all system sizes onto the same curve (L = 100 to 600 in 2D and L = 4 to 60 in 3D). In 2D, the two Gaussians have the same width and corresponds to even and odd N ℓ (upper and lower curve). In 3D, finite size corrections are visible for L = 4 (red crosses).

FIG. 4 :

 4 FIG.4: Mean value of the loop length ℓ as a function of doping. Left: checkerboard, L = 10 (red dots) and 60 (blue triangles). Right: pyrochlore, L = 6 (red dots) and 20 (blue triangles). The dashed straight lines are predictions from Equation (7), without any fitting parameters. For both panels, the blue triangles data points are very close to thermodynamic limit.

FIG. 5 :

 5 FIG.5: Loop length distribution P3D(ℓ, ρ) in 3D for different values of doping (ρ = {0, 0.00075, 0.0015, 0.002}]) and system size L = 10. The distribution is normalized such that ℓP3D(ℓ, ρ)dℓ = 16L 3 = N . The dashed line indicates the power law fit ℓ -1 at ρ = 0 and the arrow shows the shifting of the distribution for increasing ρ. The exponents of the two power law regions (before and after ℓ ≈ L 2 = 100) do not vary, but the peak for long winding loops ℓ ∼ 8L 3 = N/2 gets smaller with increasing ρ.

FIG. 6 :

 6 FIG.6: Main: Loop length distribution P2D(ℓ, ρ) in 2D for different values of doping (ρ = {0, 0.0025, 0.0056, 0.0087}]) and system size L = 40. The distribution is normalized such that ℓP2D(ℓ, ρ)dℓ = 4L 2 = N . The dashed line indicates power law L 2 /ℓ 15/7 and the arrow shows the shifting of the distribution for increasing ρ. On this scale, it is not obvious whether the exponent of the power law varies or not. Inset: The power-law exponent defined locally on ℓ, τ local for ℓ = 8( ), 16( ), 32(•) for L = 40 (similar behavior obtained for L = 20 and 60).

FIG. 8 :FIG. 9 :FIG. 10 :

 8910 FIG.8: Minimum energy E(ρ) corresponding to the value of ℓ(ρ) plotted on figure7, for a system filled with loops of length ℓ only, assuming no lattice constraints other than lmin = 4 (left, checkerboard) or 6 (right, pyrochlore). ρ takes all rational values p/q, with q ∈ {0, 1, 2, ..., 1000} and p ∈ {0, 1, 2, ..., q}. We consider all loop length from ℓmin to ℓmax = 1000. We chose arbitrarily t = 1/2. A Maxwell construction can be visualized from this figure. The red dots are a set of points, such that the dashed lines connected them are always below the red curve E(ρ). Hence a mixture of two phases corresponding to two consecutive red dots (ℓ1, ρ1) and (ℓ2, ρ2) has a lower energy than a single phase with a unique density of electrons ρ on a unique type of loop length ℓ. For increasing values of ρ, the red dots correspond to the most favored loop length ℓ = {4, 8, 6, 8, 4} (left panel) and ℓ = {6, 12, 10, 8, 6, 8, 10, 12, 6} (right panel), as can be read from figure7.

  FIG.11: Average number of filaments, i.e. those segments of loops spanning the entire system from one border to the opposite one, as a function of system size L. We use two definitions of filaments; they can either cross the orthogonal borders thanks to the periodic boundary conditions (•) or not ( ). The cartoon on the right shows such a boundary-crossing filament, which would be excluded in the data. Dashed lines are guides to the eye for the linear behavior with L. Both x-and y-axes are on a logarithmic scale.

FIG. 12 :FIG. 13 :

 1213 FIG.12: Average number of filaments in 2D (left) and 3D (right), corresponding respectively to systems of sizes L = 10, 20, 40 and L = 6, 12, 18 (red squares, green dots, blue triangles) as a function of doping ρ.

FIG. 14 :

 14 FIG.14:On the left, the same loop configuration as in figure1. The thick dashed green line represents a possible worm, as built by our algorithm; flipping the spins along this worm does not break the ice-rules (see on the right) and allows us to visit the configurational space of the Coulomb phase, ensuring ergodicity.

  Loop length ℓ(ρ) minimizing the energy(11) for a system filled with loops of length ℓ only, assuming no lattice constraints other than lmin = 4 (left panel, checkerboard) or 6 (right panel, pyrochlore). ρ takes all rational values p/q, with q ∈ {0, 1, 2, ..., 1000} and p ∈ {0, 1, 2, ..., q}, and we consider all loop lengths from ℓmin to ℓmax = 100 (red crosses) and 1000 (blue squares): plotting two different values of ℓmax provides a graphical way to visualize the two values of ρ where the most favorable loop length is infinite, namely ρ = 1/3 and 2/5 (and their symmetric images with respect to ρ = 1/2). If several loop lengths give the same energy, we plot the smallest one.

	1000					1000
	100					100
	l(ρ)					l(ρ)
	10					10
	4					6
	1					1
	0	0.2	0.4	0.6	0.8	1	0	0.2	0.4	0.6	0.8	1
			ρ						ρ
	FIG. 7:
												)

Now if a straight line between E min (ρ 1 ) and E min (ρ 2 ) remains below the curve E min (ρ) on figure

8

, i.e. if

N E min

Acknowledgments

We thank J. Chalker, C. Henley, and M. Viret for useful discussions.

Appendix A: Worm algorithm a. Worm algorithm without electrons. In our definition, a loop is made of nearest neighbour spins pointing in the same direction: it is uniquely defined and possesses an up or down flavor. On the other hand, a worm consists of alternating up/down/up/down/... spins; through each vertex of the premedial lattice (square in 2D or diamond in 3D); see figure 14. The worm can randomly choose between two paths