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In various topological systems, it is common to encounter the concept of smooth deformation, which introduces changes in spectra of topological systems without allowing topological phase transitions. As a result, two homotopic phases are connected by continuous deformations.

My main concern is how to compute this deformation for a specific Hamiltonian, say $H$, with open or periodic boundary conditions. For the SPT phases specifically, this deformation should preserve symmetry constraints, so in principle, my question should boil down to finding an operator $U$ which satisfies $H' = U^{-1} H U$, where $H'$ is the Hamiltonian describing the deformed system. What are the different possibilities to write $U$? Would the exponential of a generator of the symmetry be an option? Answers with an example Hamiltonian are very appreciated.

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One simple way to deform an SPT Hamiltonian without breaking the symmetry is to add terms which respect the symmetry. For example, the 1D ZXZ cluster state is an SPT phase with Hamiltonian $$H_0=-\sum_i Z_iX_{i+1}Z_{i+2}.$$ This state has $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry represented by $\prod_{i\text{ even}}X_i$ and $\prod_{i\text{ odd}}X_i$. Any string of $X$ operators clearly respect this symmetry, so you can explicitly add a symmetric, local perturbation $H'=X_1X_2X_3X_4$ such that $H=H_0+tH'$, where $t$ is a sufficiently small parameter. This deformation is an example of a homotopy connecting $H_0$ and $H_0+tH'$, parameterized by $t$.

An equivalent definition of SPT order in 1D on the level of states is as follows: two 1D gapped states are in the same SPT phase if they can be mapped into one another by a constant depth local unitary circuit that preserves the symmetry, meaning that the local unitaries which comprise the circuit each commute with the unitaries representing the symmetry. (See here and here.) This is another way of parameterizing the symmetry-preserving homotopy you describe. It has the nice benefit that it avoids the possibility of phase transitions by restricting to local unitaries and constant depth. (A general symmetry-preserving homotopy does not guarantee this, as it may close the gap.) It can also be applied to Hamiltonians instead of states, and this would yield a unitary like the one you're looking for. For example, the local unitary gate $U=Z_1Z_2Z_3Z_4$ is symmetric with respect to our $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry. Under this gate, we have $$\begin{align}U^\dagger HU&=Z_1X_2Z_3+Z_2X_3Z_4+Z_3X_4Z_5-\sum_{i\neq1,2,3}Z_iX_{i+1}Z_{i+2}\\ &=H_0+2(Z_1X_2Z_3+Z_2X_3Z_4+Z_3X_4Z_5)\end{align}.$$ This is not a super interesting peturbation, but we only used a single layer and a single gate, after all, so hopefully it at least demonstrates the principle.

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  • $\begingroup$ Thank you for your response and for sharing references. This answer does not address all of my questions, but it gives me ways to address the rest independently. I thus accept it. $\endgroup$
    – Shasa
    Commented Mar 11, 2022 at 14:00

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