. 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.

DOI: 10.1103/physrevb.111.045112

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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})

{\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.

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})

\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.

For a parameterized family of invertible states (short-range-entangled states) in

(1+1)

dimensions, we discuss a generalization of the Berry phase. Using translationally invariant, infinite matrix product states (MPSs), we introduce a gerbe structure, a higher generalization of complex line bundles, as an underlying mathematical structure describing topological properties of a parameterized family of MPSs. Furthermore, we introduce a generalization of a quantum mechanical inner product, which we call the ``triple inner product,'' defined for three matrix product states. The triple inner product proves to extract a topological invariant, the Dixmier-Douady class over the parameter space.

X

. This is called a higher pump. It is conjectured that a

(1+1)

-dimensional bosonic invertible state parametrized by

X

is classified by

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

. In this paper, we construct two higher pumping models parametrized by $X=\mathbb{R}{P}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ and $X=\mathrm{L}(3,1)\ifmmode\times\else\texttimes\fi{}{S}^{1}$ that corresponds to the torsion part of

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

. As a consequence of the nontriviality as a family, we find that a quantum mechanical system with a nontrivial discrete Berry phase is pumped to the boundary of the

(1+1)

-dimensional system. We also study higher pump phenomena by using matrix product states, and construct a higher pump invariant which takes value in a torsion part of

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

. This is a higher analog of the ordinary discrete Berry phase that takes value in the torsion part of

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

. In order to define the higher pump invariant, we utilize the smooth Deligne cohomology and its integration theory. We confirm that the higher pump invariant of the model has a nontrivial value.

d

-dimensional lattice systems, for which we construct the closed

(d+2)

-form higher Berry curvature, which is a generalization of the well-known 2-form Berry curvature. Such

(d+2)

-form higher Berry curvature characterizes a flow of

(d+1)

-form higher Berry curvature in the system. Our construction is equally suitable for constructing other higher pumps, such as the (higher) Thouless pump in the presence of a global onsite

\mathrm{U}(1)

symmetry, which corresponds to a closed

d

-form. The cohomology classes of such higher differential forms are topological invariants and are expected to be quantized for short-range entangled states. We illustrate our construction with exactly solvable lattice models that are in nontrivial higher Berry classes in

d=2

D

spatial dimensions which can detect gapless or degenerate points in the phase diagram of a system. Field theory predicts that in spatial dimension

D

the analog of the Berry curvature is a closed

(D+2)

-form on the parameter space (the Wess-Zumino-Witten form). We construct such closed forms for arbitrary families of gapped interacting lattice systems in all dimensions. We show that whenever the integral of the Wess-Zumino-Witten form over a

(D+2)

-dimensional surface in the parameter space is nonzero, there must be gapless edge modes for at least one value of the parameters. These edge modes arise even when the bulk system is in a trivial phase for all values of the parameters and are protected by the nontrivial topology of the phase diagram.