PHYSICAL REVIEW LETTERS 126, 040602 (2021)

Entanglement View of Dynamical Quantum Transitions

Stefano De Nicola , Alexios A. Michailidis , and Maksym Serbyn IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria (Received 27 August 2020; accepted 23 December 2020; published 29 January 2021)

The analogy between an equilibrium partition function and the return probability in many-body unitary dynamics has led to the concept of dynamical quantum (DQPT). DQPTs are defined by nonanalyticities in the return amplitude and are present in many models. In some cases, DQPTs can be related to equilibrium concepts, such as order parameters, yet their universal description is an open question. In this Letter, we provide first steps toward a classification of DQPTs by using a matrix product state description of unitary dynamics in the thermodynamic limit. This allows us to distinguish the two limiting cases of “precession” and “entanglement” DQPTs, which are illustrated using an analytical description in the quantum Ising model. While precession DQPTs are characterized by a large entanglement gap and are semiclassical in their nature, entanglement DQPTs occur near avoided crossings in the entanglement spectrum and can be distinguished by a complex pattern of nonlocal correlations. We demonstrate the existence of precession and entanglement DQPTs beyond Ising models, discuss observables that can distinguish them, and relate their interplay to complex DQPT phenomenology.

DOI: 10.1103/PhysRevLett.126.040602

Introduction.—The rapid development of different quan- early times [3]. However, fðtÞ corresponds to the free tum simulation platforms [1,2] fuels the exploration of new at complex temperature, and a precise relation nonequilibrium phenomena that can be probed in isolated between the behavior of fðtÞ and the equilibrium phase interacting quantum systems. Because of experimental diagram was not established [28]. Phenomenologically, limitations, phenomena observable at short times in quan- quenches from a state jψ 0i that realizes a different phase tum quenches are of particular interest. Dynamical quan- compared to the of H often give rise to DQPTs tum phase transitions (DQPTs) have recently emerged as an [3–5,10,18]; however, there are exceptions to this rule interesting phenomenon within this regime [3,4]. Since the [6,7,14,15,27,28]. early work of Heyl et al. [3], who introduced the notion of In order to connect DQPTs to equilibrium concepts, such DQPT considering the quantum Ising model, DQPTs have as order parameters, the behavior of local observables was attracted great interest [5–28]. Moreover, they were exper- explored [22,26,35]. A direct correspondence was estab- imentally observed in trapped ion quantum simulators [29], lished for systems with broken symmetries, involving a superconducting qubits [30], and other platforms [31–34]. generalized notion of DQPTs [11,17,20,36]. Recently, local In the framework of DQPTs, one considers quantum string observables capable of revealing DQPTs were quenches from an initial state jψ 0i and monitors the introduced [37,38]. However, the general relation between normalized logarithm of the return probability in the DQPTs and local expectation values remains elusive. process of unitary evolution under a Hamiltonian H, Connections to the entanglement were also explored: DQPTs may correspond to regions of rapid growth [29] or peaks [21] in the entanglement entropy, 1 −iHt 2 fðtÞ¼− lim log jhψ 0je jψ 0ij ; ð1Þ L→∞ L and, for certain quenches in integrable models, they occur at crossings in the entanglement spectrum [9,10,39]. Nonetheless, the underlying mechanism and the conditions where we restrict to one-dimensional cases and denote under which DQPTs may be related to entanglement system size as L. This quantity is identified as the non- signatures are not clearly understood. Thus, in spite of equilibrium analog of the free-energy density, with DQPTs many advancements, the rich phenomenology of DQPTs corresponding to nonanalyticities in the behavior of fðtÞ at and their relation to other physical quantities still call for a more general understanding [28]. In this Letter, we utilize the matrix product state Published by the American Physical Society under the terms of (MPS) [40] language for DQPTs that was applied in the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to numerical studies [5,10,11,14,18,26]. We show that, in the author(s) and the published article’s title, journal citation, the low-entanglement regime, it is possible to distinguish and DOI. between precession and entanglement DQPTs, which

0031-9007=21=126(4)=040602(6) 040602-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 126, 040602 (2021)

(a) (d) (e) |ψ(t) = Λ(t) Γ(t) Λ(t) Γ(t)

Λ(0) Γ(0) v T f = = Λ(t) Γ(t) Λ(t) Γ(t)

(b) (c)

- - - -

FIG. 1. (a) Representation of the fidelity density transfer matrix Tf obtained from the iMPS canonical form when the initial state is a product state. The evolution of the two leading eigenvalues of Tf in the complex plane, fidelity density, entanglement spectrum, and the overlaps are illustrated for (b),(d) a pDQPT and (c),(e) an eDQPT. Red circles in (b) and (c) correspond to times where DQPTs occur. (d), (e)ffiffiffi Comparison of fidelity density, entanglement spectrum, and overlaps. Solid lines show iTEBD data obtained with χ ≤ 8 (truncating p −9 λi < 10 ), and dashed lines correspond to the analytical Ansätze; quench parameters are listed in the main text. correspond to different , as highlighted by analytical conjugate at t ¼ 0.Thus,Tfð0Þ coincides with the conven- MPS Ansätze. We illustrate the existence of entanglement tional transfer matrix and has je1j¼1 and all other jeij < 1, and precession DQPTs in different models, discuss ways to as follows from normalization of jψ 0i. At later times, distinguish them experimentally, and suggest how their the eigenvalues of TfðtÞ perform complicated evolution interplay may lead to the rich phenomenology reported in in the complex plane. As illustrated in Figs. 1(b) the literature. and 1(c), singularities in fðtÞ emerge from the initially — MPS description of DQPTs. DQPTs are typically subleading eigenvalue e2 surpassing in magnitude the largest studied at short times for quenches from area-law entangled one. initial states. In this regime, the time-evolved state jψðtÞi ¼ Two limiting cases of DQPTs.—To distinguish between −iHt e jψ 0i has area-law entanglement due to Lieb-Robinson different physical mechanisms that drive the crossing bounds [41,42] and admits a MPS description [40].For between transfer matrix eigenvalues, we use the canonical translation-invariant initial states (possibly with a finite- form of the MPS tensor and focus on the case when the size unit cell), infinite MPS (iMPS) [43] provide an initial state jψ 0i¼⊗i jvii is a product state. The contrac- efficient representation of jψðtÞi. In Fig. 1(a) we show tion of the time-evolved MPS with the product state does an iMPS in the canonical form [43,44], where the standard not affect bond indices [see Fig. 1(a)], resulting in σ building block of MPS, the tensor AijðtÞ, is decomposed σ σ  pffiffiffi   as AijðtÞ¼ΛiiðtÞΓijðtÞ. Here σ ¼ ↑; ↓ is the physical λ 0 o o Tf t 1 ffiffiffi 11 12 ; and i; j ¼ 1; …; χ are bond indices. The diagonal matrix ð Þ¼ p pffiffiffi 0 λ2 o21 o22 ΛiiðtÞ¼ λi contains the ordered, λi > λi−1, singular X o vσ Γσ ; values of the Schmidt decomposition across a bond. ij ¼ ð Þ ij ð3Þ σ λi determine the entanglement spectrum,P so that the bipartite entanglement entropy is S ¼ − i λi log λi. The σ where we retained the leading 2 × 2 part of the MPS virtual tensor Γ ðtÞ carries a physical index andP together with Λ 2 σ σ space, corresponding to the two largest singular values. For satisfies a set of canonical conditions, ijσ Λ Γ Γ ¼ P ij jk il o 2 σ σ initial product states, the elements of the overlap matrix ijσ ΛijΓkjΓli ¼ δkl [44]. σ are obtained via contraction of the tensor Γij with the Using the iMPS representation, the fidelity density is σ single-site spinor wave function v . expressed directly in the thermodynamic limit via the Tf t e Equations (2) and (3) single out the contribution of the spectrum of the fidelity transfer matrix, ð Þ, f ig,as[7,45] entanglement spectrum, encoded in the diagonal of the matrix Λ, to the transfer matrix TfðtÞ and DQPTs. When f t −2 e : ð Þ¼ log max ðfj ijgÞ ð2Þ the entanglement spectrum features a large gap, λ1 ≫ λ2, the switch in magnitude between eigenvalues of Tf is The transfer matrix TfðtÞ is defined in Fig. 1(a) as a necessarily driven by the evolution of the overlap matrix. contraction of the time-evolved MPS tensor with its This is a precession DQPT (pDQPT) that is of semiclassical

040602-2 PHYSICAL REVIEW LETTERS 126, 040602 (2021) nature, as we explain below. In the opposite limit, when the excellent agreement between our analytical results and two leading singular values λ1 and λ2 exhibit an avoided iTEBD predictions for the singular values. The Γ matrix in crossing, the system features entanglement of order ln 2. the canonical form reads DQPTs happening near such points are dubbed entangle-   ment DQPTs (eDQPTs). We illustrate these two limits of ↓ ↑ −it h σx h σz −iJb t σy j iji ¯ DQPTs in the quantum Ising model using analytical MPS ΓðtÞ¼e ð x þ z Þe ð Þ Λ; ð5Þ Ansätze. ij↑i −ij↓i Precession DQPTs in the Ising model.—In order to illustrate pDQPTs, we study the dynamics under the where Λ¯ ¼ diagðsignfcos½JaðtÞg; signfsin½JaðtÞgÞ and transverse- and longitudinal-field Ising model 2 2 2 2 2 bðtÞ¼hx½hx cosð6htÞþ3ðh þ3hzÞcosð2htÞ−4ðh þ2hzÞ= X 4 H Jσzσz h σx h σz : 12h [46]. The matrix of overlaps o is obtained by contracting ¼ ½ i iþ1 þ x i þ z i ð4Þ i all entries of ΓðtÞ with the h↓j state on the left. The behavior of o o 11 and od obtained from (5) agrees with numerically exact The initial state jψ 0i¼⊗i j↓ii is the ground state of the iTEBD results, Fig. 1(d).Sinceλ1 ≫ λ2 within the range of Hamiltonian (4) in the ferromagnetic phase, J → −∞, considered times, the precession of spins in ΓðtÞ induced by hz > 0. The evolution is performed with J ¼ 0.1, exponentials of Pauli matrices plays the main role in driving hx ¼ 1, hz ¼ 0.15, so that single-spin terms are dominant. the pDQPT. The top panel of Fig. 1(d) shows the fidelity density The dominant component of the MPS corresponds to the calculated using infinite time-evolving block decimation top diagonal entry in Eq. (5), and it coincides with the (iTEBD) [43]. It exhibits a cusp at t ≈ 1.5, signaling a initial state j↓i at t ¼ 0. The off-diagonal entries in Eq. (5) DQPT. By bringing the MPS to the canonical form we givepffiffiffiffiffiffiffiffiffiffiffi subleading contributions suppressed by powers of extract the entanglement and overlap contributions. The λ2=λ1, as follows from Eq. (3). However, as both the middle plot shows the evolution of the two leading singular dominant component j↓i and its correction j↑i precess, see values, which remain very well separated at the time when Eq. (5) and [46], the overlap of the dominant contribution o the DQPT occurs. At the same time, j 11j exhibits a decreases, while the subleading state rotates closer to the minimum near the DQPT, while the off-diagonal compo- j↓i state. A pDQPT occurs when the formerly subleading o o ≡ o nent j 12j¼j 21j j odj shows a clear maximum. Thus, contribution becomes important enough to flip the magni- the overlap matrix is predominantly responsible for the f tude of thep eigenvaluesffiffiffiffiffiffiffiffiffiffiffi of T , which happens when switch of the transfer matrix eigenvalues, providing a o =o ∼ λ =λ ≪ 1 j 11 odj 2 1 . The DQPT is then closely asso- prototypical example of pDQPT. o Analytical pDQPT Ansatz.—The precession nature of ciated with the minimum of 11, with corrections given by pDQPTs can be illustrated by analytically constructing a off-diagonal terms; see Fig. 1(d). Note that, although free precession dominates the dynamics for the present quench, suitable χ ¼ 2 MPS Ansatz. In the limit when J ≪ hx, hz we a minimal χ ¼ 2 is required to capture DQPTs due to splitP the Hamiltonian (4) into an interactingP part V ¼ J σzσz H h σx Eq. (2), reflecting the quantum nature of such phenomena. i i iþ1 and a free-precessing part 0 ¼ i½ x i þ z Entanglement DQPTs in the Ising model.—We consider hzσi that contains only single-spin terms. Then, we move to a quench from the initial state jψ 0i¼⊗i j →ii, correspond- the rotating frame with respect toR H0, rewriting the time t ing to the free paramagnet ground state of (4) for hx → −∞. −iH t −i V˜ ðt0Þdt0 evolution as jψðtÞi ¼ e 0 Te 0 jψ 0i.Theinterac- The dynamics is governed by the Ising Hamiltonian with V˜ t eiH0tVe−iH0t J 1 h 0 1 h 0 15 PtionP term in the rotating frame reads ð Þ¼ ¼ ¼ , x ¼ . , and z ¼ . . Figure 1(e) shows that a s t s t σασβ α; β ∈ x; y; z DQPT happens near an avoided crossing in the entangle- i α;β αð Þ βð Þ i iþ1, where f g, the time- 2 2 ment spectrum. The overlaps o11 and o also display the dependent coefficients are sxðtÞ¼2hxhz sin ðhtÞ=h , j j j odj 2 2 2 evolution characteristic of an avoided crossing. This syðtÞ¼hx sinð2htÞ=h,andszðtÞ¼½hx cosð2htÞþhz=h , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi provides an example of eDQPT. h h2 h2 and ¼ x þ z is the magnitude of the applied field. Analytical eDQPT Ansatz.—The smallness of all but the Finally, we exploit the slow initial buildup of entanglement first two singular values for this quench allows us to analyti- z σx σz V˜ t along the axis to replace the and operators in ð Þ by cally construct a χ ¼ 2 MPS Ansatz describing eDQPTs, their expectation values under free precession, −sx and −sz, ˜ which agrees well with numerically exact iTEBD. To this end, respectively. This allows us to approximately write VðtÞ as a we approximate the time-evolution operator by a second-order matrix product operator (MPO) of χ ¼ 2 (see Supplemental Trotter decomposition, splitting the Hamiltonian into a single- ↓ Material [46]). Acting by this MPO on the initial j i-product spin term H0 and a two-spin term V. The decomposition reads state gives a MPS Ansatz for ψ t . Bringing this Ansatz to −iHt −iH0t=2 −iVt −iH0t=2 −iVt j ð Þi pffiffiffi e ≈ e e e ,wheree admits an exact canonical form [46]ffiffiffi, we obtain the singular values as λ1 ¼ MPO representation with χ ¼ 2 (see Supplemental Material p 2 j cos½JaðtÞj and λ2 ¼jsin½JaðtÞj, where aðtÞ¼hx½4ht− [46]). Applying the resulting MPO to the initial state, we obtain sinð4htÞ=8h3. The middle panel of Fig. 1(d) reveals an the analytical MPS Ansatz

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 −iJt iJt  e c↑ðtÞj↑ðtÞi e c↑ðtÞj↑ðtÞi (a) (b) A t ; ð Þ¼ iJt −iJt ð6Þ e c↓ðtÞj↓ðtÞi e c↓ðtÞj↓ðtÞi

x z (c) (d) where j↑ðtÞi ¼ exp½−itðhxσ þ hzσ Þ=2Þj↑i and c↑ðtÞ¼ h↑j→ ðtÞi, and likewise for ↓. Casting the Ansatz (6) in canonical form yields the tensor Γ, which generally has a complicated expression but can be − (e) (f) simplified in certainp limitsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi[46], and the entanglement spectrum λ1;2 ¼½4 fðtÞþ13=8, whose avoided cross- ings are expected to drive the DQPT. Here fðtÞ¼ 4cos½4θðtÞ−cos½8θðtÞþ8sin4½2θðtÞcosð4JtÞ is expressed in terms of a time-dependent angle 2 cos2 θðtÞ¼1 þ½1− 2 cosðhtÞhxhz=h . The special cases when either hx or hz vanishes correspond to classical [12,22] or inte- FIG. 2. The qualitative behavior of the fidelity density is very similar for the pDQPTs in (a) and the eDQPTs in (b) that occur in grable [3,9,10,39,49] Ising models, discussed in the the XXZ spin chain. In contrast, x magnetization and MI have Supplemental Material [46]. qualitatively different behavior for (c),(e) pDQPTs and (d),(f) h ;h ≠ 0 In the generic case with x z , the top and middle eDQPTs. Simulations are performed using iTEBD with χ ¼ 200. panels of Fig. 1(e) show that the Ansatz (6) accurately captures the dynamics of the rate function, singular values, While local expectation values evolve smoothly and and overlaps. The avoided crossing of the singular values cannot indicate the precise location of DQPTs, they provide leads to a much faster growth of entanglement compared to an additional test for the underlying physical mechanisms. pDQPTs and drives the switch of the transfer matrix Namely, near pDQPTs the dominant component of the state eigenvalues: near the DQPT the quantum state undergoes is maximally far away from the initial state, thus the local a rearrangement whereby the initially off-diagonal compo- magnetization along the orientation of the initial state has nent, which for λ2 ≪ λ1 provides a correction to the leading opposite sign compared to its value at t ¼ 0, see Fig. 2(c). top-diagonal component, becomes the dominant contribu- Near eDQPTs, which are characterized by larger entangle- tion. Thus, eDQPTs manifest a change in the leading ment, local expectation values are expected to be small; this component of the quantum state and can be revealed by the is indeed confirmed by Fig. 2(d), where the magnetization structure of nonlocal correlations, as we discuss below. along the x direction assumes its minimal magnitude near DQPTs in the XXZ model.—To demonstrate the exist- an eDQPT. ence of pDQPTs and eDQPTs beyond the Ising chain, we The mutual information (MI) can be used to reveal the consider quenches from the fixed initial product state nontrivial entanglement pattern near eDQPTs. The MI ψ ⊗ → XXZ j 0i¼ i j ii. The dynamicsP is governed by the between two regions A and B is defined as IA;B ¼ H J σασα h σα model with a field ¼ i;α ½ α i iþ1 þ α i , where SðAÞþSðBÞ − SðA ∪ BÞ, where Sð·Þ is the von Neumann Jx ¼ Jy and we set hy ¼ 0. Figure 2(a) shows dynamics entropy of a given region. Regions A and B are chosen to for Jx ¼ Jy ¼ 0.9, Jz ¼ 1, hx ¼ 0.1, and hz ¼ 1, which contain one or two spins. Because of translational invari- displays pDQPTs, as it can be seen from the behavior of the ance, only relative distances between regions are important. entanglement spectrum in the inset. In Fig. 2(b),we MI provides a basis-independent upper bound on connected consider the same initial state evolved with Jx ¼ Jy ¼ correlation functions, which could reveal qualitatively 0.3, Jz ¼ 1, hx ¼ 0.3, and hz ¼ 0.1. In this case, cusps in similar behavior provided an appropriate basis is chosen. fðtÞ are close to avoided crossings in the entanglement Figure 2(e) shows that the MI for all choices of regions A and spectrum (see inset), suggesting eDQPTs. The behavior of B undergoes slow monotonic growth in the case of a pDQPT. the overlaps shown in the Supplemental Material [46] In contrast, eDQPTs correspond to complex oscillatory confirms these expectations. dynamics of the MI; this is demonstrated in Fig. 2(f), where Experimental signatures.—pDQPTs and eDQPTs have DQPTs correspond to broad maxima in the MI I1;2;3 between very different physical mechanisms, yet the fidelity density spins f1; 2g and f3g. behaves qualitatively similarly, cf. Figs. 1(d) and 1(e) or In the Supplemental Material [46], we show a similar Figs. 2(a) and 2(b). An immediate distinction between pattern for the DQPTs of Figs. 1(d) and 1(e) in the Ising different DQPTs is provided by the bipartite entanglement model. The quick growth and nonmonotonic behavior of entropy S: pDQPTs are of semiclassical nature and occur in the MI for eDQPTs signal a change in the dominant low-entanglement regions, whereas eDQPTs are triggered component of the wave function. The MI can be probed by avoided crossings in λi at early times, reflected in rapid by the connected correlation functions between the two entanglement growth. subsystems, which are typically accessible in experiments.

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Discussion.—We introduced the notions of precession [3] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical and entanglement DQPTs as two limiting cases, which Quantum Phase Transitions in the Transverse-Field Ising have different underlying mechanisms and are associated to Model, Phys. Rev. Lett. 110, 135704 (2013). different physics. pDQPTs can be understood analytically [4] M. Heyl, Dynamical quantum phase transitions: A review, Rep. Prog. Phys. 81, 054001 (2018). by relying on the large entanglement gap λ1 ≫ λ2 and the [5] C. Karrasch and D. Schuricht, Dynamical phase transitions dynamics being driven by single-spin terms in the after quenches in nonintegrable models, Phys. Rev. B 87, Hamiltonian. In contrast, eDQPTs happen near avoided λ ∼ λ ≫ λ 195104 (2013). level crossings in the entanglement spectrum 1 2 3 [6] S. Vajna and B. Dóra, Disentangling dynamical phase and can also be analytically described by ignoring λi transitions from equilibrium phase transitions, Phys. 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Dóra, Topological classification of dynami- interesting to develop analytical Ansätze to characterize cal phase transitions, Phys. Rev. B 91, 155127 (2015). [14] S. Sharma, S. Suzuki, and A. Dutta, Quenches and dynami- situations with more than one dominant mechanism, as well cal phase transitions in a nonintegrable quantum Ising as long-range interacting models [16,19,20,29] and other model, Phys. Rev. B 92, 104306 (2015). cases that violate typical phenomenology [4,19,50]. [15] M. Schmitt and S. Kehrein, Dynamical quantum phase The connection between DQPTs and the spectrum of the transitions in the Kitaev honeycomb model, Phys. Rev. B fidelity transfer matrix, which is generically non-Hermitian, 92, 075114 (2015). calls for exploring the relation between DQPTs and the [16] J. C. Halimeh and V. Zauner-Stauber, Dynamical phase theory of non-Hermitian matrices [51,52], which may allow diagram of quantum spin chains with long-range inter- a classification of DQPTs. 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