Peratic Phase Transition by Bulk-to-Surface Response

The study of dynamical phase transitions has been attracting considerable research efforts in the last decade. One theme of present interest is to search for exotic scenarios beyond the framework of equilibrium phase transitions. Here, we establish a duality between many-body dynamics and static Hamiltonian ground states for both classical and quantum systems. We construct frustration free Hamiltonians whose ground state phase transitions have rigorous duality to chaotic transitions in dynamical systems. By this duality, we show the corresponding ground state phase transitions are characterized by bulk-to-surface response, which are then dubbed"peratic"meaning defined by response to the boundary. For the classical system, we show how the time-like dimension emerges in the static ground states. For the quantum system, the ground state is a superposition of geometrical lines on a two dimensional array, which encode the dynamical Floquet evolution history of one dimensional disordered spin chains. Our prediction of peratic phase transition has direct consequences in quantum simulation platforms such as Rydberg atoms and superconducting qubits, as well as anisotropic spin glass materials. The discovery would shed light on the unification of dynamical phase transitions with equilibrium systems.


I. INTRODUCTION
Characterization of different phases of matter is fundamental to our understanding of nature. With rapid developments in controlling many-body states away from equilibrium in condensed matter and quantum information experiments, the study of non-equilibrium phase transitions has attracted much present research interest. Non-equilibrium many-body physics previously unreachable can now be probed in experimental systems such as light-driven electronic matters [1,2] and highly controllable quantum simulation [3][4][5][6][7] or quantum computing platforms [8][9][10][11]. Quantum many-body localization (MBL) happens for a disorder system in its dynamical properties at infinite temperature [12,13]. Anomalous protected edge modes at zero Chern number which are unexpected for equilibrium systems, could appear in periodically driven Floquet quantum dynamics [14][15][16]. Spontaneous translation symmetry breaking is generalized to the temporal domain, giving rise to crystallization in time [7,8,[17][18][19][20]. There is mounting evidence suggesting non-equilibrium phases may contain exotic scenarios beyond equilibrium setups.
Here, we construct frustration free Hamiltonians whose ground states have rigorous duality to dynamical systems. Through this theoretical construction, we find a novel phase transition mechanism-bulk-to-surface (BTS) response defines a peratic phase transition in Hamiltonian ground states. This mechanism is established by building exact duality to order-to-chaos and MBL transitions for classical and quantum systems, respectively. The Hamiltonian ground states have two distinctive phases as determined by whether the bulk is rigid against manipulations at the surface. The phase transi- * xiaopeng li@fudan.edu.cn The BTS response is averaged over random surface terms (h surf ), with p the probability of each surface term h surf,i taking a positive value. The inset of (a) shows its first derivative for I = 100, 300, 500 with p = 0.5. This derivative develops a peak near the transition point, which sharpens up as we increase the system size. This implies a divergent second derivative in the thermodynamic limit. (b), the dependence of local BTS response (χ j ) on the j index. This quantity has an exponential decay at small σ when χ BTS vanishes. The decay becomes power law at the critical point (σ c ). Here we choose J = 5I and M = 2. The results are calculated by averaging over 10 4 samples, and the statistical data error is smaller than the symbol size.
tion is characterized by a BTS response being zero or finite, with O bulk an observable in the bulk, Var( O bulk ) the variance of the bulk observable as we manipulate the surface. The BTS response quantifies the stability of the bulk against surface manipulations, which vanishes if the bulk is stable and acquires a finite value otherwise. The phase transition in the classical ground state has an exact duality to the order-to-chaos transition in classical nonlinear dynamical systems [21]. The quantum ground state phase transition con-structed by forming superposition of fluctuating line configurations has a rigorous duality with the quantum ergodic [22][23][24] to MBL transition [12,13] in quantum many-body dynamics. Unlike the standard phase transitions described either by spontaneous symmetry breaking [25,26] or quantum state topology [27][28][29], there is no change in symmetry or topology across the peratic phase transition. Our theory implies that the study of unconventional phases in non-equilibrium systems would rather inspire discovery of more exotic equilibrium phases and transitions than reaching completely beyond.

II. EMERGENT CHAOTIC DYNAMICS IN A HAMILTONIAN GROUND STATE
We first consider a discrete classical system which contains binary degrees of freedom z i j = ± on a two dimensional (2d) grid, with the indices i ∈ [0, I − 1], and j ∈ [0, J − 1]. Our theory starts from constructing a Hamiltonian ground state that supports the phase transition scenario described by the BTS response in Eq. (1). The Hamiltonian contains a bulk and a surface term, H bulk , and H surf , where w [i j] represents Ising couplings, u j is a local field in the bulk, and h [i] surf is a local field on the surface introduced to study the BTS response. The Ising couplings are randomly drawn from a Gaussian distribution with zero mean and variance σ 2 . The bulk local field u j takes random binary values ±1 with equal probability. We adopt an open boundary condition along the j-axis and periodic boundary along the other axis (Fig. 1). The system has a layer structure along the j-axis-we have inter-layer couplings only.
Minimizing each term of the Hamiltonian in Eq. (2) leads to a set of equations, which can all be satisfied. The Hamiltonian is thus frustration free. The binary degrees of freedom in the Hamiltonian ground state is given by z g i j . We then observe that the j-dependence of the binary variables in the static ground state is time-like, for the j-th layer is completely determined by the ( j − 1)-th layer. A time-like dimension thus arises in the static ground state.
By treating the j-axis as a time evolution direction, Eq. (3) maps onto classical nonlinear dynamics having an order-tochaos dynamical phase transition [21]. In the ordered phase (σ < σ c ), the dynamical state is pinned by the local field u j -the difference between two initial states as measured by Hamming distance is quickly washed away in the dynamical evolution. In the chaotic phase (σ > σ c ), the difference in the initial states would either remain or gain amplification by the nonlinear dynamics. Consequently, the ground state of the Hamiltonian in Eq. (2) also has two phases. For σ < σ c , the system is in a rigid phase where the bulk is stable and immune to surface manipulations with different h surf . For σ > σ c , we have a volatile phase with the bulk sensitive to surface manipulations. Quantitatively, the BTS response for this specific system is defined as with the variance Var z i j = 1− z i j 2 , and the average z i j obtained by randomly sampling h surf . This BTS response quantifies the degree of bulk fluctuations induced by perturbations at the boundary, and reflects the chaotic structures of the dual dynamics. We also introduce a space resolved local BTS response χ j = I −1 i Var z i j | h surf to diagnose the criticality. Their behavior across the phase transition is shown in Fig. 1. The BTS response vanishes in the rigid phase, and becomes finite in the volatile phase, defining our peratic phase transition. The BTS response is consistent with the bulk entropy of the system (see Appendix A). By increasing the system size, we find the BTS response has a divergent second derivative at the critical point. The local BTS response shows an exponential decay in the rigid phase, with a decay length that tends to diverge approaching the critical point. At the critical point, the local BTS response exhibits a power law scaling, a signature of nontrivial criticality for the peratic phase transition. We argue the critical behavior is described by a universal finite size scaling function with I, J ∝ L, t = (σ − σ c )/σ c , and determine the anomalous dimension η = 0.172 (2), and ν-exponent ν = 2.73(8) (see Appendix B). This phase transition scenario does not rely on the symmetry or the topology of the system, in sharp contrast to the standard phase transitions. Whether it can be described by the replica type of spontaneous symmetry breaking [30][31][32] as established in disorder spin systems [33][34][35] is worth future investigation. We emphasize that the chaotic structures or the BTS response reported here for the highly anisotropic disorder spin system may shed light on understanding anisotropic spin glasses [33,36,37], for which the exact models are lacking to our knowledge. One fascinating property of the volatile phase is its robustness-the BTS response is stable irrespective of different choices of surface manipulations. In our numerical calculation, each surface term h surf,i takes a positive value with a probability p, and a negative value with 1 − p. The BTS response in the volatile phase as constructed above does not vary with the p-value ( Fig. 1(b)), i.e., having robustness against different surface manipulations. This nontrivial property can be attributed to the presence of a dynamical fixed point of the Hamming distance evolution in the dual chaotic dynamics [38]. This makes the volatile phase sharply distinctive from a trivial case with the bulk trivially determined by the surface, for example with z i j = z i0 , where the BTS response would strongly depend on p.

III. EXPERIMENTAL CANDIDATES
Although the frustration free Hamiltonian in Eq.
(2) has a nice property of being polynomially solvable, its experimental realization is challenging. For experimental realization, we further consider a frustrated Hamiltonian with two-body Ising couplings only, which could describe a broad range of spin systems from Rydberg atomic systems [39] and superconducting qubits [40] to anisotropic spin glasses [33]. We study its ground state phase transition by numerically minimizing the energy. The results are shown in Fig. 2 (a). We observe that the BTS responses for the frustrated and frustration-free models are quite similar to each other with a tiny difference barely noticeable. The peratic phase transition still preserves in the frustrated model. The two-body Hamiltonian also allows a natural way to generalize the phase transition to quantum systems, by simply promoting the Ising variables z i j to Pauli operatorsẑ i j , and adding a transverse field coupling ∆H = i j h Txi j (x the Pauli-x operator) to generate quantum fluctuations. As shown in Fig. 2 (b), the peratic phase transition still persists in presence of quantum fluctuations. We find quantum effects further stabilize the rigid phase and shift the transition point towards the volatile phase, which is somewhat counterintuitive. This can be attributed to that the transverse field couples the degenerate ground states and develops a tendency for gap opening, rendering the bulk more rigid.
As a concrete experimental candidate, we consider a system of Rydberg p-wave dressed atoms, which has been used to construct quantum spin ice Hamiltonians [41] and programmable quantum annealing [42]. Two atomic hyperfine states |↑ = |5 2 S 1/2 , F = 2, m F = 0 and |↓ = |5 2 S 1/2 , F = 1, m F = 0 of 87 Rb atoms are selected to form a spin-1/2 lattice with I rows and J columns. Their couplings are controllable by performing a microwave induced transition, which are described by a transverse field H T = r h Txr , with the field strength h T determined by the Rabbi frequency of the microwave. Taking a Rydberg p-wave dressing scheme where the |↑ state is selectively dressed with a Rydberg pwave state |n 2 P 3/2 , m = 3/2 (the quantization axis along the i-direction in Fig. 3 (a)), we introduce interactions between neighboring layers as labeled by j [41]. One key feature of this system is it has inter-layer interactions but no intra-layer interactions, and thus the description of this system closely resembles Eq. (7). Arranging the atoms periodically in a 2D array as shown in Fig. 3, the resultant interaction between the two qubits at r and r is given by with the coupling strength V and the Rydberg interaction range r c determined by the Rabbi frequency and the detuning of the one-photon transition of the Rydberg dressing scheme [41]. The on-site longitudinal field is tunable by manipulating the detuning of the microwave with respect to the hyperfine splitting of 6.8 GHz. The longitudinal fields are described by H L = r h rẑr .  The quantum ground state phase transition as constructed is dual to the quantum ergodic to many-body-localized dynamical phase transition. At small σ, the bulk is robust against surface manipulations giving a vanishing BTS, which is dual to the dynamical quantum ergodic phase. Above a certain threshold at σ > σ c , the bulk is stringently tied with surface, corresponding to the dynamical MBL phase. The results are calculated by averaging over 100 samples, with the standard deviations illustrated by the shaded error bands. The inset of (c) shows the system size dependence of BTS response (with p fixed at 0.5), which indicates the phase transition becomes sharper at larger system size.
As shown in Fig. 3 (b), the BTS response shows distinctive behaviors at small and large Rydberg interaction range r c . At small r c , the system is in a rigid phase having a vanishing χ BTS with bulk spin polarization robust against different choices of surface polarizations. When r c is larger than a certain threshold (roughly 0.8 in units of lattice constant), the bulk becomes volatile, fluctuating with different surface polarizations, as characterized by a finite χ BTS . The Rydberg system thus supports a peratic phase transition characterized by the BTS response. Since two spin states in our proposed setup has a hyperfine splitting at the order of GHz, the peratic phase transition can be detected by microwave spectroscopy, which is a standard technique in cold atom experiments. The local addressability can be reached by creating spatial-resolved Stark shifts using focused laser fields [43].

IV. QUANTUM PERATIC PHASE TRANSITION
We now provide a rigorous quantum model supporting the quantum peratic phase transition. This is achieved by constructing a quantum frustration free Hamiltonian, whose ground state phase transition has an exact duality with the dynamical MBL transition of the Floquet quantum dynamics of a one-dimensional disordered spin chain [12,13].
We consider a 2d qudit lattice model with a four dimensional local Hilbert space. The four levels are labeled as ν z with ν (= 0, 1) and z (= ±1) (Fig. 4 (a)). The lattice contains I rows and J rungs, with two types of rungs labeled by A and B. Different qudits are labeled according to their posi-tion on the lattice by (i, j), with i ( j) the row (rung) index. The large 4 I J -dimensional Hilbert space of the qudit system is constrained to a low-energy subspace by introducing local projectors. The forbidden configurations are illustrated in Fig. 4 (b), with the corresponding Hamiltonian realization given in Appendix C. With the forbidden configurations by the horizontal rules, there is at most only one '·' in one row. All sites on the left (right) of the '·' have to be '⊗' (' '). By the vertex rules, the '·' sites on each rung have to form a continuous line, which can either go straight on the same rung or bend over to its nearby rungs. By the vertical rules on the A (B) rungs, the continuous line has to reach to the bottom (top) on the A (B) rungs. With all these constraints, the allowed states in the low-energy subspace correspond to the two types of continuous lines-Type A and Type B, as illustrated in Fig. 4 (a). We construct a 2d local qudit Hamiltonian (see Appendix C), whose projection on the low-energy subspace takes a form of Feynman-Kitaev clock Hamiltonian [44,45], Here, the γ-states are |γ l ( z ) = z ψ l ( z)|l; z , and the sequential index, l, is introduced for a compact representation of (i, j)-l i j = jI + i for j ∈ A, and l i j = ( j + 1)I − i − 1 for j ∈ B (see one explicit example in Appendix C, Fig. 7). The wave functions ψ l ( z) are defined through a sequential unitary transformation starting from ψ 0 ( z) = δ z z . The update of ψ l is designed to follow the Floquet quantum dynamics of a one-dimensional spin-1/2 system [46]. From l = 0 to l = I −1, the update of ψ l corresponds to a unitary gate e −ix ixi+1 /10 (i from 0 to I − 1), and these unitary gates are then applied in a backward order from l = I to 2I − 1. Then in the same order, we apply the unitary gates e −iŷ iŷi+1 /10 from l = 2I to l = 4I − 1, and e −i(ẑ iẑi+1 /10+δ iẑi ) from l = 4I to 6I − 1. This unitary update process is then repeated forward for J/6 periods. The amplitude δ i is a random number drawn from a Gaussian distribution with zero mean and variance σ 2 .
The constructed 2d local qudit Hamiltonian is semi-positive definite and frustration free [44], whose ground state is , having a 2 I -fold degeneracy as labeled by z . The ground state is an equal-amplitude quantum superposition of those geometrical line configurations in Fig. 4 (a).
We introduce a bulk observable to diagnose the peratic phase transition, given by χ BTS = 1 with . . . averaging over different z . Treating unitary update from l to l + 1 as quantum time evolution, the dynamics of ψ l ( z) has two distinctive phases-quantum ergodic and MBL. For the MBL dynamics, the wave function holds the memory of the initial configuration of z , and the resultant BTS response χ BTS of the dual 2d quantum ground state is finite, and strongly depends on z , namely the p-value in sampling h [i] surf . On the contrary for the quantum ergodic dynamics, the local observables would thermalize as l proceeds and are then independent of z , rendering a vanishing BTS response for the 2d ground state. The numerical results are provided in Fig. 4 (c), which agree well with the theoretical analysis. The BTS response thus defines a peratic quantum phase transition in the frustration free quantum ground states. Since the existence of the MBL phase has been proven for one-dimensional systems by Imbrie [47], our constructed exact duality establishes a rigorous scenario for the quantum peratic phase transition.
With the exact construction presented above and the numerical results for the transverse field Ising model, we expect the quantum peratic phase transition to be generic, not relying on the duality with the MBL to ergodic phase transition. The unconventional phase transition defined by bulk-to-surface response could arise in a broad range of quantum simulation platforms as well as anisotropic spin glass materials.

V. CONCLUSION AND OUTLOOK
We propose a peratic phase transition defined by a bulk-tosurface response in both classical and quantum ground states. By constructing frustration free models, we establish rigorous duality from the peratic phase transition to order-to-chaos transition in the classical system, and to MBL-to-ergodic transition in the quantum setting. With numerical results, we show the peratic phase transition also preserves in anisotropic spin glass models with two-body Ising couplings only. We predict the system of Rydberg p-wave dressed atoms supports the peratic phase transition. Our theory implies dynamical phase transitions would inspire exotic scenarios in equilibrium systems rather than reaching beyond. Our approach also provides an alternative way for characterizing dynamical phase transitions from the perspective of equilibrium phase transitions, which could unify the description of non-equilibrium phases within the equilibrium framework, for the constructed duality from dynamical phases to static Hamiltonian ground states is quite generic (see Appendix D).

ACKNOWLEDGMENTS
We would like to thank Wei Wang for suggesting the ancient Greek "péras" for naming the phase transition, and thank Dong-Ling Deng and Meng Cheng for helpful discussion. In this section, we show that the bulk-to-surface (BTS) response reflects the entropy of bulk fluctuations in the system. To probe the bulk phase transition directly in our model, we define a half-system entropy S = − σ P(σ) log 2 P(σ), with σ indexing the configuration of one half of the system and P(σ) the corresponding probability. Its behavior is shown in Fig. 5. We find that the half-system entropy is vanishing and sub-extensive on the two sides of the peratic phase transition.
Here, sub-extensive means that the entropy increases linearly with the system size I instead of I × J. Both of the BTS response and the half-system entropy characterize the fluctuations of the bulk system, and serve properly as an order parameter for the peratic phase transition. But for our frustrationfree model, it is so much more convenient to compute the BTS response than the entropy. Computing the BTS response takes polynomial time, whereas computing the entropy scales exponentially. The BTS response is thus more convenient to diagnose peratic phase transition.

Appendix B: Phenomenological theory and scaling analysis
To analyze the peratic phase transition, we propose a phenomenological theory and perform scaling analysis. Since the BTS response that characterizes the phase transition also probes the degree of fluctuations in the bulk system, resembling the entropy, the phenomenological free energy near the phase transition takes a form of where we have t = (σ − σ c )/σ c , r > 0 (r < 0) for t > 0 (t < 0), and κ > 0. This free energy does not have Ising symmetry. Nonetheless, minimizing the free energy under the physical constraint χ BTS 0 produces a phase transition, i.e., the peratic phase transition established in the main text. The scaling analysis is performed by postulating the phase transition is described by a certain renormalization group fixed point [30]. Taking a renormalization group transformation, t → tζ 1/ν , L → L/ζ (the system size I, J ∝ L), with ζ a scaling factor, and ν a critical exponent, we have  I=500  I=480  I=460  I=440  I=420  I=400  I=300  I=200  at the neighborhood of the fixed point. Here, η is the anomalous dimension. The ν and η exponents are to be fixed with our numerical results. The scaling form implies that with A(. . .) a universal function. It then follows directly that a thermodynamic limit system right at the peratic phase transition point has a power law local BTS response, The scaling of the total BTS response obeys χ BTS (tζ 1/ν , L/ζ) = ζ η χ BTS (t, L), which implies χ BTS (t, L) = L −η G(Lt ν ), with G(. . .) a universal function. The η-exponent is determined by fitting our numerical results to χ j ∝ j −η at the critical point ( Fig. 6 (a)), from which we get η = 0.172 (2). We first fit all numerical results near the critical point for χ j to a power law, and locate the critical point by minimizing the root mean square (RMS) error of the fitting. The value of the η-exponent and its error are obtained by fitting the data right at the critical point.
The ν-exponent is extracted by performing a data collapse taking the scaling form in Eq. (B6) (Fig. 6 (b)), and we get ν = 2.73 (8). The ν exponent is calculated using the analysis in Ref. [48], with the error estimated by bootstrap.