# Boundary (Anti)Periodic conditions and fermion partition functions

+ 4 like - 0 dislike
115 views

The path integral with antiperiodic fermions (Neveu-Schwarz spin structure) on a circle of circumference $$\beta$$, in a theory with Hamiltonian $$H$$, has partition function $$\rm{Tr} \exp(−\beta H)$$ The path integral with periodic fermions (Ramond spin structure) has partition function $$\rm{Tr} (−1)^F \exp(−\beta H).$$

1. Why antiperiodic and periodic boundary conditions give these two partition functions? Intuitions and derivations?

2. How are the two boundary conditions related to spin structures? (Ramond and Neveu-Schwarz spin structures)

3. What are the ground state sectors (bosonic vs fermionic) for the above boundary conditions?

This post imported from StackExchange Physics at 2020-12-12 20:06 (UTC), posted by SE-user annie marie heart

+ 2 like - 0 dislike

I have some insight on the first question. For the ordinary partition function: $$\text{Tr} \ e^{-\beta H} = \sum_n \langle n | e^{-\beta H} | n \rangle$$ Then one inserts the fermionic path integral: $$\int dc \ dc^{*} e^{-c c^{*}}\sum_n \langle n| c \rangle \langle c | e^{-\beta H} | n \rangle = \int dc \ dc^{*} e^{-c c^{*}} \sum_n \langle -c | e^{-\beta H} | n \rangle \langle n| c \rangle = \int dc \ dc^{*} e^{-c c^{*}} \langle -c | e^{-\beta H} | c \rangle$$ Where one has used : $$\langle n | c \rangle \langle c | n \rangle = -\langle c | n \rangle \langle n | c \rangle$$.

Then one inserts resolution of identity $$N$$ times (here $$c_N = -c_0$$): $$\int \prod dc_n dc_n^{*} e^{-\sum_n c_n c_n^{*}} \langle c_N | e^{-\Delta \tau H} | c_{N-1} \rangle \ldots \langle c_1 | e^{-\Delta \tau H} | c_{0} \rangle$$ Where $$\tau = \beta / N$$. In the continuum limit one gets the path integral after: $$\Delta \tau \sum_n^{N} \ldots \rightarrow \int_0^{\beta} d \tau \ldots \qquad \prod_n dc_n dc_n^{*} \rightarrow \mathcal{D} c \mathcal{D} c^{*} \qquad \frac{c_n - c_{n-1}}{\Delta \tau} \rightarrow \partial_\tau$$ $$Z = \int \mathcal{D} c \mathcal{D} c^{*} e^{-\int_0^{\beta} d \tau \ L (c, c^{*})}$$ With the boundary condition $$c(\beta) = - c(0)$$.

For the Witten index: $$\text{Tr} \ (-1)^{F} e^{-\beta H}$$ One uses the anticommutation of $$(-1)^{F}$$ with fermions: $$\{(-1)^{F}, c\} = 0$$ Proceeding as above, but and evaluating $$(-1)^{F}$$ on a particular state, one gets now: $$\langle c | e^{-\beta H} | n \rangle \langle n| c \rangle$$ And after the insertions of identity the path integral with $$c(\beta) = c(0)$$

This post imported from StackExchange Physics at 2020-12-12 20:06 (UTC), posted by SE-user spiridon_the_sun_rotator
answered Oct 29, 2020 by (70 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.