We consider families of invertible many-body quantum states in $d$ spatial dimensions that are parameterized over some parameter space $X$ . The space of such families is expected to have topologically distinct sectors classified by the cohomology group $\mathrm{H}^{d+2}(X;\mathbb{Z})$ . These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work, we introduced a generalized inner product for three one-dimensional many-body quantum states, (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in $\mathrm{H}^{3}(X;\mathbb{Z})$ , can be extracted. In this paper, we introduce an inner product of four two-dimensional invertible quantum many-body states. We use it to measure the topological nontriviality of parameterized families of 2d invertible states. In particular, we define a topological invariant of such families that takes values in $\mathrm{H}^{4}(X;\mathbb{Z})$ . Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of non-trivial parameterized families of 2d invertible states parameterized over $\mathbb{R}P^4$ and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.We consider families of invertible many-body quantum states in $d$ spatial dimensions that are parameterized over some parameter space $X$. The space of such families is expected to have topologically distinct sectors classified by the cohomology group $\mathrm{H}^{d+2}(X;\mathbb{Z})$. These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work, we introduced a generalized inner product for three one-dimensional many-body quantum states, (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in $\mathrm{H}^{3}(X;\mathbb{Z})$, can be extracted. In this paper, we introduce an inner product of four two-dimensional invertible quantum many-body states. We use it to measure the topological nontriviality of parameterized families of 2d invertible states. In particular, we define a topological invariant of such families that takes values in $\mathrm{H}^{4}(X;\mathbb{Z})$. Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of non-trivial parameterized families of 2d invertible states parameterized over $\mathbb{R}P^4$ and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.

arXiv:2405.05325May 8, 20240 citations

Higher Berry phase from projected entangled pair states in ( 2 + 1 ) dimensions

Shuhei Ohyama, Shinsei Ryu

We consider families of invertible many-body quantum states in $d$ spatial dimensions that are parameterized over some parameter space $X$ . The space of such families is expected to have topologically distinct sectors classified by the cohomology group ${\mathrm{H}}^{d+2}(X;\mathbb{Z})$ . These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work [Phys. Rev. B 109, 115152 (2024)], we introduced a generalized inner product for three one-dimensional many-body quantum states (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in ${\mathrm{H}}^{3}(X;\mathbb{Z})$ , can be extracted. In this paper, we introduce an inner product of four two-dimensional invertible quantum many-body states. We use it to measure the topological nontriviality of parameterized families of 2d invertible states. In particular, we define a topological invariant of such families that takes values in ${\mathrm{H}}^{4}(X;\mathbb{Z})$ . Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of nontrivial parameterized families of 2d invertible states parameterized over $\mathbb{R}{P}^{4}$ and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.We consider families of invertible many-body quantum states in $d$ spatial dimensions that are parameterized over some parameter space $X$. The space of such families is expected to have topologically distinct sectors classified by the cohomology group ${\mathrm{H}}^{d+2}(X;\mathbb{Z})$. These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work [Phys. Rev. B 109, 115152 (2024)], we introduced a generalized inner product for three one-dimensional many-body quantum states (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in ${\mathrm{H}}^{3}(X;\mathbb{Z})$, can be extracted. In this paper, we introduce an inner product of four two-dimensional invertible quantum many-body states. We use it to measure the topological nontriviality of parameterized families of 2d invertible states. In particular, we define a topological invariant of such families that takes values in ${\mathrm{H}}^{4}(X;\mathbb{Z})$. Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of nontrivial parameterized families of 2d invertible states parameterized over $\mathbb{R}{P}^{4}$ and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.

Physical review. B./Physical review. BJan 6, 202510 citations

We consider families of invertible many-body quantum states in

d

spatial dimensions that are parameterized over some parameter space

X

. The space of such families is expected to have topologically distinct sectors classified by the cohomology group

\mathrm{H}^{d+2}(X;\mathbb{Z})

. These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work, we introduced a generalized inner product for three one-dimensional many-body quantum states, (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in

\mathrm{H}^{3}(X;\mathbb{Z})

, can be extracted. In this paper, we introduce an inner product of four two-dimensional invertible quantum many-body states. We use it to measure the topological nontriviality of parameterized families of 2d invertible states. In particular, we define a topological invariant of such families that takes values in

\mathrm{H}^{4}(X;\mathbb{Z})

. Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of non-trivial parameterized families of 2d invertible states parameterized over

\mathbb{R}P^4

and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.

We consider families of invertible many-body quantum states in

d

spatial dimensions that are parameterized over some parameter space

X

. The space of such families is expected to have topologically distinct sectors classified by the cohomology group

{\mathrm{H}}^{d+2}(X;\mathbb{Z})

. These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work [Phys. Rev. B 109, 115152 (2024)], we introduced a generalized inner product for three one-dimensional many-body quantum states (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in

{\mathrm{H}}^{3}(X;\mathbb{Z})

{\mathrm{H}}^{4}(X;\mathbb{Z})

. Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of nontrivial parameterized families of 2d invertible states parameterized over

\mathbb{R}{P}^{4}

and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.