Deeply flawed, Sydney Carton is, indeed, representative of the anti-hero as he is certainly not dynamic, brave, or charismatic. Instead, he is passive and withdrawn; uninspired, he allows C. J. Stryver to exploit his talents and take the credit for what his legal and logical acumen achieve. Rather than possessing virtue, Carton looks at Charles Darnay with envy, realizing that he represents what Carton himself could have become.
"Do you particularly like the man?" he muttered, at his own image; "why should you particularly like a man who resembles you! What a change you have made in yourself! A good reason for taking to a man, that he shows you what you have fallen away from, and what you might have been! Come on....You hate the fellow."
But, it is his love that transforms Carton into the hero who sacrifices himself in order to save the husband of the woman he loves more than himself. And, in so doing, he redeems himself.
Another source of an anti-hero is Alexandre Dumas's The Count of Monte Cristo. Naive and unworldly, Edmond Dantes finds himself victimized by rivals and spends fourteen years in prison (much like Dr. Manette). However, after he escapes, he vows revenge upon these foes and commits retalitory acts. However, he, too, is redeemed as he witnesses the harm of his actions upon an innocent child and as he observes the redemptive powers of love in a young couple as he prevents the young man from committing suicide.
A second source of an anti-hero is in the eighteenth-century political satire of John Gay, The Beggar's Opera, in which the anti-hero is a certain Macheath, a highwayman and philanderer who is representative of the Whig statesman Robert Walpole. Although he is firmly engrossed in crime, there are many women who love him. After several uproarious twists and turns of plot, he altruistically chooses to be hanged rather than cause all these women conflicts and hardship.
Here is an additional source for the Beggar's Opera the one below:
Noble, Yvonne, ed. Twentieth Century Interpretations of “The Beggar’s Opera.” Englewood Cliffs, N.J.: Prentice-Hall, 1975. Comprises a series of nine critical and informative essays on Gay’s masterpiece as well as an informative introduction by the editor.