What is the theme of the poem?
I believe the theme of this poem is about being trapped in a relationship that lacks love. It is not clear whether the two are married, but they might be. The title "Living in Sin" can refer to the woman's thoughts that are disloyal to her man and that may hint at an extramaritial affair. Or, it may refer to the fact that a loveless relationship or marriage is a sin. She expresses the idea that the love that she once had is either gone, or greatly reduced by the mundane chores a woman must do daily:
She had thought the studio would keep itself;
no dust upon the furniture of love
The religious imagery is continued - she expresses the idea that it might be "heresy" to wish the apartment would clean itself:
Half heresy, to wish the taps less vocal,
the panes relieved of grime
Her man gets up, plays the piano, complains it is out of tune and then goes outside for a smoke. She gets up and makes the bed. The daily grind. Then, by night time, she says she is in love again but....
though not so wholly but throughout the night
she woke sometimes to feel the daylight coming
like a relentless milkman up the stairs
This is the second mention of the milkman. Who is he? Her lover? Does he represent an imagined lover? She seems unable to escape her circumstances, so she may be dreaming of a different life - a life where the window panes would clean themselves, the apartment would dust itself and the leftover dishes from the night before would clean themselves. Her life is a mundane, repetitive existence of chores, not of love.
There is a great explanation of this poem here on enotes that goes into greater depth.
The theme is expansion, growth and the hope that life will improve and become refined over time.
The narrator describes a nautilus, the "ship of pearl, which, poets feign, Sail the unshadowed main." It is a strange creature, indeed, that year after year, builds its home, chamber after chamber, in a spiral of mathematical perfection. ("Year after year beheld the silent toil That spread his lustrous coil").
The inner creature eventually outgrows a chamber only to move, circle-like, into the next, more spacious chamber ("Still, as the spiral grew, He left the past year's dwelling for the new").
In the end, the narrator sees the life of the nautilus as a hidden message of human life, to ever build "more stately mansions" for the soul to inhabit, so that, at life's end it may be ready and worthy to inhabit heaven above.
Coincidentally, just last week I took a picture of a painted nautilus in a shop window. See the first link below.