# Quantum order-by-disorder induced phase transition in Rydberg ladders with staggered detuning

### Submission summary

 As Contributors: Arnab Sen · Krishnendu Sengupta Preprint link: scipost_202205_00008v1 Date submitted: 2022-05-12 10:06 Submitted by: Sengupta, Krishnendu Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

{\bf $^{87}{\rm Rb}$ atoms are known to have long-lived Rydberg excited states with controllable excitation amplitude (detuning) and strong repulsive van der Waals interaction $V_{{\bf r} {\bf r'}}$ between excited atoms at sites ${\bf r}$ and ${\bf r'}$. Here we study such atoms in a two-leg ladder geometry in the presence of both staggered and uniform detuning with amplitudes $\Delta$ and $\lambda$ respectively. We show that when $V_{{\bf r r'}} \gg(\ll) \Delta, \lambda$ for $|{\bf r}-{\bf r'}|=1(>1)$, these ladders host a plateau for a wide range of $\lambda/\Delta$ where the ground states are selected by a quantum order-by-disorder mechanism from a macroscopically degenerate manifold of Fock states with fixed Rydberg excitation density $1/4$. Our study further unravels the presence of an emergent Ising transition stabilized via the order-by-disorder mechanism inside the plateau. We identify the competing terms responsible for the transition and estimate a critical detuning $\lambda_c/\Delta=1/3$ which agrees well with exact-diagonalization based numerical studies. We also study the fate of this transition for a realistic interaction potential $V_{{\bf r} {\bf r'}} = V_0 /|{\bf r}-{\bf r'}|^6$, demonstrate that it survives for a wide range of $V_0$, and provide analytic estimate of $\lambda_c$ as a function of $V_0$. This allows for the possibility of a direct verification of this transition in standard experiments which we discuss.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission scipost_202205_00008v2 on 9 August 2022

Submission scipost_202205_00008v1 on 12 May 2022

## Reports on this Submission

### Strengths

1. The system proposed realizes examples of quantum phase transitions arising from an order-by-disorder mechanism.
2. The nature of the phase transition is examined both numerically and analytically.

### Weaknesses

1. The study does not fully elucidate the reason why the effective model Eq. 17 well explains the critical detuning obtained from ED.

### Report

The paper studies the ground-state phase diagram and quantum phase transitions in a class of Rydberg ladders with uniform and staggered detunings. When the uniform detuning $\lambda$ is negatively large, the ground state does not have any Rydberg excitation. On the other hand, when $\lambda$ is positively large, the ground state is two-fold degenerate and maximizes the number of possible Rydberg excitations. In between, the ground state of the classical part is exponentially degenerate. However, quantum fluctuations lift this degeneracy and lead to a ground state with a definite order, which is called an order-by-disorder mechanism. To investigate the nature of the system, the authors first study the phases of the effective Hamiltonian given by Eq. 6. Then they demonstrate that a simple variational calculation based on the above physical picture well explains the ED results and argue that the phase transition on the $n=1/4$ plateau belongs to the 2D Ising universality class. This is further justified by examining the effective Hamiltonian Eq. 17. In Sec. 4, the effect of the long-range van der Waals interaction is discused numerically and analytically. The authors conclude that the further neighbor interactions do not obliterate the Ising transition, although they shift the transition to a higher value of $\lambda/\Delta$. The models on $n$-leg ladders are discussed in Appendices.

The manuscript is overall well written and nicely illustrates how this seemingly less obvious phase transition occurs in the system. In addition, the subject is certainly timely considering the recent research interest in Rydberg-atom systems. Thus I think the manuscript is well suited for publication in SciPost Physics. Nevertheless, I would like to suggest that the authors address the following point before publication:

The authors emphasize that a simple estimate based on a variational excited state for the effective Hamiltonian for $H'$ yields the approximate transition point, which is remarkably close to the one obtained from ED. However, this may not be a big mystery because the value of the transition point $\lambda_c = \Delta/3$ is exact for $H'$. This can be seen by mapping $H'$ in Eq. 17 to a transverse-field Ising chain, which is achieved by defining an effective spin-1/2 degree of freedom for each even rung. (Since the spin configurations on odd rungs are totally frozen, we can ignore them in Eq. 17.) With the identification $|\uparrow_{2j,1}, \downarrow_{2j,2}\rangle$ <-> $|\Uparrow_{2j}\rangle$ and $|\downarrow_{2j,1}, \uparrow_{2j,2}\rangle$ <-> $|\Downarrow_{2j}\rangle$, one can rewrite Eq. 17 as a Hamiltonian of the transverse field Ising chain, where the first and second terms correspond to the Ising-interaction and transverse-field terms, respectively, for the effective spins $|\Uparrow\rangle$ and $|\Downarrow\rangle$. Then it follows from the Kramers-Wannier duality that the transition point is exactly at $\alpha_0 = 2$, leading to $\lambda_c = \Delta/3$. The same story applies to the effective Hamiltonian of the 3-leg ladder model (Eq. 34). One can deduce the exact value of the transition point $\lambda_c = \Delta/2$ from the known result for the 3-state quantum Potts chain. It would be beneficial to the reader if the authors could include these discussions in the revision.

### Requested changes

1. In Eq. 1, the first sum is written as $\sum_{j=1,2L}$. However, it is less ambiguous to write the sum as $\sum_{j=1}^{2L}$. The same applies in numerous places in the draft like Eq. 5, 6, ...

2. The argument around Eq. 5 is confusing, as $H_0$ pops up suddenly. I guess $H_0$ there means $H_a$ in Eq. 4.

3. In Fig.2, the variational and ED results are shown for comparison. But what is the system size for the ED results? (In the main text, it is just said that $N \le 40$.)

4. I think Eq. 18 should be replaced with the superposition of all Fock states with one Rydberg excitation. The current one is a superposition of just two states. (There may be a better way to express this excited state using an operator acting on the ground state $|\psi_G\rangle$ defined below Eq. 18.) A similar problem happens in Eq. 36 in Appendix B.

5. The $\ell_0$-state quantum Potts model exhibits a first-order phase transition for $\ell_0 > 4$. Thus it is quite natural to expect that the same applies to the current $\ell_0$-leg ladder models. I wonder if the authors can comment on this. (I suggest that the authors at least remark that the nature of the phase transition of the Potts model depends on the value of $\ell_0$.)

• validity: high
• significance: good
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent

### Strengths

1. Rydberg lattice are readily realisable in the state of the art experiments.
2. An interesting phenomenon of order by disorder transition is discussed.

### Weaknesses

The draft does not explicitly discuss how to realise experimentally the staggered detuning in a Rydberg chain.

### Report

The authors studied the ground state configurations in a Rydberg ladder in the presence of both staggered (Delta) and uniform (lambda) detunings. In the case of the nearest neighbour blockade, the system exhibits macroscopic degenerate states with a Rydberg filling of n=1/4 for a finite range of lambda/Delta ratio. Within the n=1/4 plateau, above a critical value of uniform detuning, the system chooses one of the classical configurations in which Rydberg atoms are excited only in one of the chains, breaking the Z2 symmetry via quantum order by disorder mechanism. In Sec. 3, the physics is analyzed using effective Hamiltonians. In contrast, in Sec. 4, the original Hamiltonian for a Rydberg ladder is used and shows the existence of the critical point in the n=1/4 plateau. The quantum order by disorder transition emerges from the staggered detuning and the Rabi coupling. Finally, the study is extended beyond the two-leg ladder scheme. In particular, the three-leg ladder possesses a quantum critical point with a three-states Pott universality class in 2D. This work is exciting; at first, it was less obvious, but later it became more apparent in the draft. I recommend the paper to publish in the scipost physics. More specific comments are provided below.

### Requested changes

1. On multiple occasions in the draft, calling “amplitude” of realizing the Rydberg excitation at any site as “detuning” is very misleading, especially from an atom-light interaction point of view. Detuning has units of frequency (equivalently energy) and well-defined interpretation even without any Rydberg excitations.

2. One of the essential ingredients of the model is the staggered detuning; it would be beneficial if the authors could provide a scheme to implement such a pattern in detuning along the ladder. In any case, it requires local manipulations using other optical fields and may not be as straightforward as the authors think. Along these lines, the papers cited in the introduction, Refs [50-59] do not provide how to implement that either.

3. Note that quantum order by disorder transition is also discussed using a Rydberg + square lattice setup in https://journals.aps.org/prx/pdf/10.1103/PhysRevX.4.041037 but in a completely different interaction scheme.

4. In Eq. 1, the Hamiltonian for the Rydberg ladder, the Rabi term or the transverse magnetic field term (\sigma_x) has an additional negative sign, in contrast to the Hamiltonians describing the original Rydberg setups. It doesn’t look like the sign matters for the Physics discussed in the manuscript, as in the effective Hamiltonian in Eq. 17, it is w^2 in the first term, but it is better to clarify it.

5. When the exact 1/r^6 interaction is considered, the transition point shifts to a larger value of \lambda, and a further increase in lambda, makes a jump in the Rydberg density to n=1/2. Then, mentioning lambda>lambda_c, the system chooses a single classical configuration in the presence of weak quantum fluctuations arising from the Rabi coupling. Is not better to say, lambda_c < lambda < lambda_u, where lambda_u defines the upper value of n=1/4 plateau, the system chooses a single configuration?

6. If we do an ED with the model (1) and truncate to nearest neighbour interaction, do we get the critical lambda=0.33 Delta?

7. It is also good to comment on what happens if the Rabi coupling becomes sufficiently large. The ground state will have contributions from the degenerate manifold of the first excited state in Fig. 5d.

• validity: top
• significance: top
• originality: top
• clarity: high
• formatting: excellent
• grammar: excellent