Say a ring $R$ is an isolated hypersurface singularity if $R = k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}/(W)$, where $k$ is a field and $W \in k[x_1, \ldots, x_n]$ is such that the ideal $(\partial_1 W, \ldots, \partial_n W)$ is $(x_1, \ldots, x_n)$-primary. For finitely generated $R$-modules $M$ and $N$ define the function $\theta$ to be $\theta(M, N) = \lambda( \operatorname{Tor}^R_{2i} (M,N)) - \lambda( \operatorname{Tor}^R_{2i-1}(M,N))$ for any $i \gg 0$. Here $\lambda$ denotes the length of a module. The definition makes sense because all modules over $R$ have eventually 2-periodic resolutions (since $R$ is a hypersurface) and the $\operatorname{Tor}$'s have finite length for $i \gg 0$ (since $R$ is an "isolated singularity"). Hochster made this definition in his 1981 paper "The dimension of an intersection in an ambient hypersurface."

I'm looking for examples (or preferably a family) of isolated hypersurface singularity rings with $n \geq 4$ and modules $M$ over such rings with $\theta(M, - )$ non-zero. See Hailong's answer below for an equivalent formulation of this in terms of certain Chow groups when $k = \mathbb{C}$ and $W$ is homogeneous. I would prefer if $W$ were not homogeneous but am interested in all cases.

Of course any insight as to when $\theta$ is non-zero would be great but in general this is hard. For instance in the paper above Hochster showed that the direct summand conjecture is true if $\theta$ is non-zero for an explicit family of modules and rings. It is conjectured that $\theta$ is zero when $n$ is odd (this is known when $W$ is homogeneous, see http://arxiv.org/abs/0910.1289v1), and it is known that when $n=4$ the function $\theta$ is nonzero if and only if the class group of $R$ is nonzero.

EDIT: There is a physical interpretation of the above in the spirit of this post: Matrix factorizations and physics. My knowledge of physics is limited so I apologize in advance for any mistakes.

D-branes in a B-twisted topological Landau-Ginzburg models with potential $W$ are given by matrix factorizations of $W$. We only care about values of $\theta$ on maximal Cohen-Macaulay (MCM) $R$-modules, and all such modules are given by matrix factorizations of $W$. Thus MCM modules over $R$ can be thought of as D-branes. Now physicists talk about the BRST-cohomology of two branes $M,N$ (which I don't understand) but it seems that it is given by $\operatorname{Ext}_R^2(M \oplus \Omega M, N \oplus \Omega N)$ (or equivalently the stable homomorphisms between these modules) where $\Omega( - )$ denotes the first syzygy; see for instance http://arxiv.org/abs/0802.1624. It is not hard to see, viewing the modules as matrix factorizations, that for two MCM modules $M$ and $N$ we have $\theta(M, N) = \lambda( \operatorname{Ext}^1_R( M^*, N) ) - \lambda( \operatorname{Ext}^2_R(M^*, N) )$, where $M^*$ is the MCM module given by $\operatorname{Hom}_R(M, R)$. Thus to find an example of modules with nonzero $\theta$ is equivalent to finding branes whose "even" and "odd" BRST cohomology have different dimensions over $k$.