Jeremy Cohen, who coined the term "Hermeneutical Jew," writes the following about Christian conceptions of the Jewish people:
"The Christian idea of Jewish identity crystallized around the theological purpose the Jew served in Christendom; Christians perceived the Jews to be who they were supposed to be, not who they actually were, and related to them accordingly." (Jeremy Cohen, Living Letters of the Law, 2)
Part 1: The Creation of the Hermeneutical Jew
The term "Hermeneutical Jew" is meant to show how Jews were viewed in the Middle Ages. It is at the end of the Patristic (Church Fathers) period that this understanding of the Jew is developed by Augustine of Hippo, considered by many to be the greatest theologian since Paul.
Dr. Jen Rosner points out that "Augustine and many of the other Christian theologians who will increasingly build upon a Christian theology of Judaism, or more properly: anti-Judaism, never encountered actual Jews. They were responding to an image, a hermeneutical Jew, a Jew that was conjured up in their minds based upon primarily their reading of Scripture."
We see two aspects here. 1) It is a Jew that they never met. But more than that, 2) it is a theological construct of a Jew based on Scripture. In other words, it is not typical racism: "fill-in-blank people are greedy, smelly, stupid, etc." This is a specific racism based on the Jewish rejection of Yeshua and the role of the Jew (who has rejected Yeshua) in a post-New Testament era—or else, it could be called the hypothetical Jew. The use of the term hermeneutical implies the interpretation of texts.
Church fathers before Augustine saw the Jewish people as lost, rejected, and cursed. Justin Martyr stated in the second century, regarding the Church, "We are the true spiritual Israel, and the descendants of Judah, Jacob, Isaac, and Abraham." And if the Church is the new Israel, what happens to the old Israel? H. H. Rowley can tell us. He "argues that because of their sin and disobedience, Jews have been disinherited and their title and blessing have passed on to the Church to fulfill their task."
But Augustine sees a role for the Jew. The Jewish people are now a witness people.
"By the evidence of their own scriptures, they bear witness for us that we have not fabricated the prophecies about Christ…It follows that when the Jews do not believe in our scriptures, their scriptures are fulfilled in them while they read them with blind eyes…It is in order to give this testimony which, in spite of themselves, they supply for our benefit by their possession and preservation of those books [of the Old Testament] that they are themselves dispersed among all nations, wherever the Christian Church spreads…Hence the prophecy in the Book of Psalms: 'Slay them not, lest they forget your law; scatter them by your might.'" (City of God 18:46)
There is quite a bit here, so let's unpack it. Augustine argues:
1. The fact that the Jewish people stand by the Hebrew Scripture authenticates the prophecies of Yeshua.
2. He refers to the Hebrew Bible as their Scriptures and the New Testament as ours.
3. And he states that the fact that Jews do not follow Yeshua proves Scripture to be true. For it was Isaiah that predicted the blindness that would come to the Jewish people (Is. 6:9-10). Yeshua himself employs these verses in rebuking the Scribes and Pharisees (Matt. 15:8-9). Paul confirms it (Rom. 11:25)
4. The scattering of Israel proves the truth of their Scriptures which point to ours.
5. He closes by misapplying Psalm 59—maybe the greatest misuse of Scripture of all time. While the passage refers to Israel's enemies, Augustine applies it to Israel in saying, "slay them not…." While the Gentile Church was horribly anti-Jewish and antisemitic for the next 700 years, it did not resort to genocide on theological grounds until the Crusades, which began in 1099 CE.
"Augustine's understanding of the Jewish presence in society has become known as the doctrine of Jewish witness." To be clear, Augustine's advice, "slay them not," was not based on some deep love for Israel but the need for a "Jewish witness."
In essence, according to Augustine, the Jews survive as living testimony to the antiquity of the Christian promise. They will eventually convert at the end of time, but until that time comes, they should not be harmed or forced to convert because their social degradation and Diaspora serve to confirm the supremacy of Christianity.
Augustine compares the Jewish people with the murderer Cain. It would be easy to assume he will call the Jewish people Christ-killers—and he does—but his comparison actually had to do with protection. In Augustine and the Jews, Jewish scholar Paula Fredrikson seeks to recast Augustine as a defender of the Jewish people in the face of the anti-Jewish theology and rhetoric of his predecessors (Justin, Ignatius, Origen, etc.) and contemporaries like Jerome.
"While in modern colloquial usage, the mark of Cain designates shame or guilt, for Augustine, it signified the protection of God. Marked by God, the designee is protected from harm and acts as a witness to God's judgment and goodness. For Augustine, 'God sent the Jews out…so that they would continue their unique historical mission as witnesses to the message of the incarnate Son.'"
Augustine was fighting against two views regarding the Jews. The first is the historical replacement theology, as mentioned above, that taught that the Jews have been disinherited by God in favor of the Church. The second was the Manichaeism heretics, which he was part of for 12 years. They taught that "the God of the Jews was an evil God … (and) could not possibly be the father of Jesus Christ, who had to be a different God…using the Old Testament [they] constructed an extremely demeaning caricature of the Jews."
Part 2 but there's more! keep reading!
 Justin, Dialogue with Trypho, ed. Michael Slusser, trans. Thomas B. Falls, FC 3 (Washington, DC: Catholic University of America Press, 2003)
 Lindemann, Albert S.; Levy, Richard S.. Antisemitism (p. 64). OUP Oxford. Kindle Edition.
 Lindemann and Levy. Rabbi Michael Leo Samuel says, “Curiously, Augustine, said nothing about this mark serving as a protective device; instead, he (and his contemporary, Jerome) subverted what was originally an act of grace and mercy into a fiendish excuse to treat the Jews with cruelty.” https://www.rabbimichaelsamuel.com/augustine-and-the-mark-of-cain/
 Lest we go too far in recasting Augustine, here are his actual words: “Abel, the younger brother was killed by the elder; so too Jesus, head of the younger people, is killed by the elder people—the Jews. Just as Abel’s blood cursed Cain, so too does the blood of Jesus accuses the Jews. As Cain was cursed from by the earth, so too unbelieving Jews are cursed from the Holy Church. As Cain was punished to be a mourner and an abject on the earth, so too are the Jews.”
 Paula Fredrikson https://www.youtube.com/watch?v=s9NBIhdFkpU