New Bleach readers might ask, "Why does Byakuya hate Rukia?"—especially considering that the head of the Kuchiki clan adopted her and made her his adoptive sister. However, the reason behind Byakuya's harsh treatment of Rukia during the initial portion of the story is related to his responsibilities and the character connected to the two of them, Hisana, as explained by author Tite Kubo in the manga.
Byakuya married Rukia's older sister, Hisana, a few years before the events of Bleach, going against his parents' wishes. He also honored his wife's dying wish to adopt her younger sibling as his own. Because of this, he swore on his parents' grave that he would never break the rules again. This vow came in full force when Rukia was going to be executed in the Soul Society arc, thus explaining his treatment of her, including his cold demeanor.
Disclaimer: This article contains spoilers from the manga/anime and reflects the opinions of the writer.
How Hisana's death strained Byakuya and Rukia's relationship, explained

Many years before the beginning of the original story, Hisana Kuchiki lived in the 78th district of the Rukongai in Soul Society, struggling to survive alongside her baby sister, Rukia. One day, she decided to abandon her and move on her own, much to her hatred of herself. Eventually, she met Byakuya Kuchiki and fell in love.
They spent five years of marriage, and Hisana fell ill in the final portion of their time together, dying because of said disease. Before her death, she begged Byakuya to adopt Rukia as his younger sister but asked that he never tell her about their sibling connection, she felt unworthy after having abandoned her.
Byakuya found Rukia in the Shinigami Academy a year after Hisana's death and adopted her, although his demeanor was quite cold and distant. The death of his wife, plus the fact he went against his parents' orders and role as head of the Kuchiki clan, made him very withdrawn, which resulted in her adoptive sister believing that he hated her.
Does Byakuya actually hate Rukia in Bleach?

Author Tite Kubo answers the question "Why does Byakuya hate Rukia?" at the end of the Soul Society arc in Bleach when Byakuya protects her from Sosuke Aizen. He reveals to Ichigo Kurosaki that he never hated Rukia but had to put his feelings aside to uphold the law and his duty, which included executing her for breaking the rules of the Soul Society, since he broke his oath several times out of love.
However, after the resolution of that arc, Byakuya's care for Rukia becomes much more apparent. He calls her "her pride" when she fights against Zommari in the Hueco Mundo arc. Furthermore, when fighting As Nodt in the Thousand-Year Blood War arc, it is shown that his greatest fear is Rukia dying, which is Kubo's way of showing how much he truly cares for her.
Final thoughts
The real answer to the question, "Why does Byakuya hate Rukia?" in Bleach is that he never hated her and wanted to put an emotional wall between them so he could fulfill his duty after Hisana's death. This came in full force during the Soul Society, but, thanks to Ichigo Kurosaki, he learned the error of his ways and began to become a much more protective brother ever since.
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