Many Sepharadim have a custom to take a slight bow when saying Barechu both during Shacharit and Arvit. Others insist on standing. Is there any source for such a minhag in the sifrei Halacha (or elsewhere)? Do Ashkenazim have a similar custom?
Biur Halacha 113 (hakoreah) says it is based on Divrei haYamim I 29:20:
וַיֹּאמֶר דָּוִיד לְכָל-הַקָּהָל, בָּרְכוּ-נָא אֶת-יְהוָה אֱלֹהֵיכֶם; וַיְבָרְכוּ כָל-הַקָּהָל, לַיהוָה אֱלֹהֵי אֲבֹתֵיהֶם, וַיִּקְּדוּ וַיִּשְׁתַּחֲווּ לַיהוָה, וְלַמֶּלֶךְ.
And David said to all the congregation: 'Now bless the LORD your God.' And all the congregation blessed the LORD, the God of their fathers, and bowed down their heads, and prostrated themselves before the LORD, and before the king.
I recall that because it is questionable, the bow should be slight, not like an avos/hoda'ah bow. I cannot recall the place i saw it.
Ashkenazim practice this widely by the barchu's preceding birchos krias shema (as well as the make-up barchu's). Strangely, I don't see it widely practiced by the barch's at torah reading, though there is no reason to distinguish.
Actually, Or Letzion is against it (although some question if this is really the opinion of Rav BenTzion Abba Shaul, see comments), and Yalkut Yosef (57:9 O"H) writes that those that do it have sources to rely on. I believe this custom has a source in the Ben Ish Hai though. HaRav Musafi Shelit"a (Q&A 3176) says because we can only bow where the Hachamim decreed to do so.
Here's an excerpt of a teshuva of Rav Ovadia Yosef that I came across that addresses this issue of bowing during Barechu:
יחוה דעת ח"ג סי' ד
ועיין למרן הגאון רבי יוסף חיים בספר עוד יוסף חי (פרשת ויחי אות י') שכתב, שבעירו בגדאד היו נוהגים רוב המון העם לקום מעט כשאומר השליח צבור ברכו את ה' המבורך, ולא מצא מנהג זה בשום ספר ע"כ. והגאון משאוול בשו"ת זכר יהוסף סימן צ' סוף ד"ה ויתכן, כתב שמנהג העולם לכרוע בברכו, ומכיון שכולם עושים כן, אף על פי שאינו מן הדין, אין בזה משום יוהרא ע"ש. ובמגן גבורים סימן קי"ג סק"ו כתב, שהכורעים בברכו אין להם סמך.
Yechave Daat 3:4 (Loose translation- my own)
...Rav Yosef Haim wrote in his work Od Yosef Chai that in Baghdad most of the people would rise slightly when the sheliach tzibbur would say Barechu, but he could not find this custom recorded in any book. In Shu"t Zecher Yehosef siman 90 s.v. veyitachen the author wrote that the practice of the general population is to bow at Barechu, and since it is a widespread custom, even though there is no halachic source for it, one does not need to worry about looking arrogant when practicing it. In the work Magen Giborim (113:6) he wrote that those who bow during Barechu do not have any source to rely on....
(I am surprised that Rav Ovadia did not quote the Biur Halacha which YDK posted in the answer above, which is written as a direct response to the Magen Giborim quoted in his teshuva. However, it seems that most acharonim were unable to find a source for this practice.)
Rav Ovadia discusses the issue at length in this teshuva, and concludes that it is unnecessary for Sepharadim to stand during Kadish or Barechu, unless they were already standing when the sheliach tzibbur began kaddish (in which case it is proper to remain standing until the end of "Amen yehe shme rabba..."). It is proper to stand during the kaddish and Barechu before Arvit of Shabbat Eve, to accept Shabbat.