Question B1. The model seems strongly related to quantum dimer models in the
frustrated magnets community. It will be nice if some information/
reference elaborates on the connection with the quantum dimer model
can be provided in the supplemental materials.
Answer: For the limit of |\mu| \gg J_x, J_y, \mu>0, charges are pinned to the B-sublattice and the dynamics is reduced to the spins on the links. Indeed, this limit corresponds to a model of tightly packed hard-core dimers, a quantum dimer model on a square lattice.
Traditionally, as in Refs.~[32,33,40] the dimer-model Hamiltonian contains ring-exchange terms, which - on the square lattice - have been shown to introduce several phase transitions: Phases found and discussed e.g. in Ref.[40] are Neel, columnar, or a plaquette-ordered (also dubbed RVB-solid) phase. At the phase transition-point between columnar and plaquette-ordered phase, lies the so called Rokhsar-Kivelson point, discussed in the paper. Here we do not have such strong ring-exchange terms (only in terms from fourth order perturbation) and the physics for the strong-coupling dimer limit is indeed different from the phase diagram of Refs.~[32,33,40].
We have added this discussion to the supmat.
Question B2: 2. At the limit of small $\mu$ (in equation (1)), the system should
have a relatively small gap. At this limit, I will expect the fermion
hopping term dominates. Especially, At the limit of $\mu=0$, I would
expect the system is in a gapless phase. If the model is indeed a
gapless phase at $\mu=0$, the DMRG approach might need a high bond
dimension to capture the wave function correctly. Is it possible to
calculate current data with different bond dimensions and conclude the
disordered phase is robust as we increase the bond dimension? Or, is
it possible to have an analysis/argument of the energy gap and
established the robustness of the disordered phase?
Answer: Actually a small mu does not correspond necessarily to a small gap, as becomes clear also for the case of the 1D-toy model, which is in a 1D Haldane-phase with a finite energy gap of order Jx~Jy. Also for the 2-leg ladder case the gap is ...?. (SG: I will check if I can generate some results for the gap just in 2-leg and 3-leg ladders. I think it is _not_ small, if you look at entanglement scaling...)
In the supplement material, we added a graph (the current Figure 6) showing
a clear exponential convergence of the ground-state energy as a function of the
bond dimension, for a fixed chain length.
The curves suggest that the ground-state phase diagram obtained with MPS in
manifolds with bond-dimensions up to $\chi$=400, relatively small with regard to
the local Hilbert space dimension, is qualitatively correct.
Question B3: In the supplement, the author mentioned the classical RK state is
obtained by Metropolis sampling of classical QLM. Is it possible to
provide further details about the setup of the numerical approach? Or
provide some related reference on the wave function. Naively, I will
expect the wave function can be solved exactly at the RK point. Why
does the comparison need to be done numerically with a classical QLM
instead of comparing it with analytic results? 3-a. If I understand it
correctly, the Monte Carlo will sample states in different sectors. In
DMRG, if the algorithm is designed to keep the symmetry, the algorithm
will minimize the energy within that sector. If the state is
initialized in a random way, that means the wave function could be the
lowest energy state withing a random sector. Will that change the
result significantly? If the sector structure in the $U(1)$ QLM is not
relevant, why is it so? If it is relevant, how it changes the current
results?
Answer: The Metropolis sampling creates a infinite temperature (random) superposition of all classical configurations reachable by single particle-tunnelings. Actually, the sampling does not mix different symmetry sectors and we make sure that we initialize the algorithm in a state of the gauge vacuum sector compatible with DMRG-simulations and it does not leave this sector. We added a discussion of this algorithm to the supmat.
Our main problem with analytical results is that in Fig. 2c of the main text as well and Fig. 5 of the supmat we compare properties of the RK-like state on a finite ladder and cylinder geometry.
Even thought analytical calculations seem possible, due the strong influence of the boundary conditions the numerical sampling of the state seems a practically easiest way to access the reduced density
matrix. In the future work we will try to extend our numerical DMRG method to more 2D-settings on larger cylinders, and here a comparison with analytical results will be indeed an interesting objective.
As mentioned above, also in our Metropolis sampling we do not mix symmetry sectors.
Initialization of the system in a random gauge symmetry sector can be interpreted a random localized charges to the problem. This can lead to interesting localization and dynamical effects as discussed in Ref.[34]. In this work we limit our discussion to the interesting physics of the gauge vacuum sector, which is a good case to start - maybe also for experiments, which typically also would probably fix the gauge sector. Furthermore, as the referee pointed out above, there are some interesting connections to the quantum dimer models (a special case of two localized defects is actually shown in the paper while discussion the string tension).
The physics of different gauge sectors will be an interesting problem, but we believe that this goes by far beyond the scope of the present work.
Question B4: As the author mentioned, the effective Hamiltonian (2) indicates the
Haldane like phase.
The Haldane phase is a symmetry protected topological order.
In the Hamiltonian (2), which symmetry protects the numerical observed
Haldane-like phase?
It will be nice if the author can provide explicit symmetry analysis and tell
the readers which symmetry protects the Haldane-like phase in this model.
Answer: The quasi 1D toy model, resembles in some sense closely a traditional spin-1 chain. Indeed, as already suggested by the string- and parity-order we were discussing in the manuscript and the supmat, the protecting symmetries are similar; in this case the SPT-phase is protected by a Z2xZ2 symmetry, pi-rotations with the (generalized) Sz and Sx operators. We have added a discussion of the symmetries and a calculation of the generalized topological "order-parameter" for our trial state to the supmat.
Question B5: In the paragraph of large mass limit, line 6.
The expression of $S^z_k(r)$ contains a parameter $v$, which is not defined.
Is it a typo, or is it defined elsewhere?
Answer: It was a typo, thanks for the correction.
Question B6: Not related to the criteria of publishing.
One question that I think is interesting for future study is: In the Hamiltonian
(1), the simulation is focusing on the half-filling case.
It will be interesting to know the result if we move away from the half-filling
limit.
Answer: We thank Referee B for the stimulating comment, hinting towards possible
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For the limit of $|\mu| \gg J_x, J_y$, $\mu>0$, charges are pinned to the B-sublattice and the dynamics is reduced to the spins on the links. Indeed, this limit corresponds to a model of tightly packed hard-core dimers, a quantum dimer model on a square lattice.
Traditionally, as in Refs.~\cite{Shastry1981,Anderson1987, Moessner2002, Ralko2005} the dimer-model Hamiltonian contains ring-exchange terms, which - on the square lattice - have been shown to introduce several phase transitions: Phases found and discussed e.g. in Ref.~\cite{Tschirsich2019} are Neel, columnar, or a plaquette-ordered (also dubbed RVB-solid) phase. At the phase transition-point between columnar and plaquette-ordered phase, lies the so called Rokhsar-Kivelson point, discussed in the paper. Here we do not have such strong ring-exchange terms (only in terms from fourth order perturbation) and the physics for the strong-coupling dimer limit is indeed different from the phase diagram of Refs.~\cite{Ralko2005}.
@@ -106,12 +114,12 @@ A simple example is given by the following matrices, with the lowest non-trivial
\sin\phi& 0
\end{array}\right) \nonumber\\
\Gamma_\beta&= \Gamma_A^T \nonumber\\
\Gamma_{\tilde{0}_\pm}&= \left(
\Gamma_{\tilde{0}_\pm}&= \sqrt{2}\left(
\begin{array}{cc}
\cos\phi& 0 \nonumber\\
0 & 0
\end{array}\right) \\
\Gamma_{0_\pm}&= \left(
\Gamma_{0_\pm}&= \sqrt{2}\left(
\begin{array}{cc}
0 & 0 \\
0 &\cos\phi
...
...
@@ -121,6 +129,7 @@ A simple example is given by the following matrices, with the lowest non-trivial
and $\Lambda=\{\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\}$ which resembles an AKLT-model like state~\cite{aklt}. This ansatz has minimal energy for $\phi_{min}=\arctan\left(\frac{2J_x-J_y}{2J_x + J_y}\right)$. With this $E_{min}=-\frac{(2J_x + J_y)^2}{4J_x}$, which gives a reasonably good approximation of the actual ground-state energy in this regime obtained by a DMRG simulation, as shown in Fig.~\ref{fig:S_1d_m0}. Also the entanglement entropy of $\log(2)$ describes already well the actual values obtained by DMRG simulations with a higher bond-dimension~(see Fig.~\ref{fig:S_1d_m0}).
The string order is given by $O_{SO}=\lim_{|x-x'|\to\infty}\la S^z_x \e^{\ii\pi\sum_{x<k<{x'}} S^z_k} S^z_{x'}\ra=(-1)^{i-j}\frac{\cos^4\phi}{4}$, while the parity order $O_{PO}=\lim_{|x-x'|\to\infty}\la\e^{\ii\pi\sum_{x<k<{x'}} S^z_k}\ra$ is exponentially suppressed.
As the Haldane-phase in this toy model, the SPT-phase is protected by a $Z_2\times Z_2$ symmetry, with two $\pi$-rotations in the pseudo-spin space $\e^{\pi S_z}$ and $\e^{\pi S_x}$. Here, $S_z$ is defined as above and a corresponding $\e^{\pi S_x}$ maps the states $\alpha\to\beta$ and $0_\pm\to\tilde{0}_\pm$. For the trial-state of Eq.~\eqref{eq:lgt1dMPS} we may now define, following the discussion of Pollmann and Turner in Ref.~\cite{pollmann2012symmetry}, a generalized order parameter $\mathcal{O}_{Z_2\times Z_2}=\tr\left(U_x U_z U_x^\dagger U_z^\dagger\right)/2$. The 2-dimensional representations of the symmetry group corresponding to the MPS ansatz $U_x$ and $U_z$ are derived from a generalized transfer matrix and are given by by the actual Pauli-matrices $\sigma_z$ and $\sigma_x$. With this we find $\mathcal{O}_{Z_2\times Z_2}=-1$ indicating the topologically non-trivial character of the phase.
%%
\begin{figure*}[tb]
...
...
@@ -226,7 +235,7 @@ In Figs.~\ref{fig:S_entanglement}~(b) and (c) and ~\ref{fig:S_stringorder}~(b) a
In Fig.~\ref{fig:S_class_plt} we provide further comparisons to the classical RK states, obtained by a simple Metropolis sampling for the classical QLM analogue at infinite temperature, with the DMRG results of the intermediate phase in the QLM. For three- and four-leg ladders and four-leg cylinders the local spin and charge configurations compare very accurately.
In Fig.~\ref{fig:S_class_plt} we provide further comparisons to the classical RK states, obtained by a simple Metropolis sampling for the classical QLM analogue at infinite temperature, with the DMRG results of the intermediate phase in the QLM. The Metropolis sampling does not mix different symmetry sectors and we make sure that we initialize the algorithm in a state of the gauge vacuum sector compatible with DMRG-simulations. For three- and four-leg ladders and four-leg cylinders the local spin and charge configurations compare very accurately.