Dynamical quantum phase transitions in the one-dimensional extended Fermi-Hubbard model

In spite of being the object of intense research for several decades, interacting many-body quantum systems continue posing some of the most exciting and challenging problems in modern physics. Key examples manifest during their unitary dynamics resulting from nonequilibrium setups such as sudden quenches, where fundamental phenomenology including thermalization [1, 2, 3, 4, 5] and transport of conserved quantities [6, 7, 8] emerges. The understanding of these problems has received an enormous boost largely due to the development of powerful numerical methods [9, 10, 11], and due to the implementation of quantum simulators which allow for exquisite unprecedented control of the degrees of freedom of many-body systems [12, 13, 14].

A fascinating effect identified within this combined effort, known as dynamical quantum phase transitions (DQPT), results from taking general ideas of quantum criticality to the nonequilibrium scenario [15, 16]. This concept relies on the return (or Loschmidt) amplitude of the evolved state $|\psi_{0}(t)\rangle$ of a quantum system to its initial state $|\psi_{0}\rangle$,

during the dynamics arising from a sudden quench. Here $|\psi_{0}\rangle$ is the ground state of a Hamiltonian $H_{0}$, and $H$ is the post-quench Hamiltonian. The Loschmidt amplitudes have been argued to be similar to canonical partition functions in equilibrium [15]. Namely, for a system of $L\gg 1$ sites, the associated Loschmidt echo $\mathcal{L}(t)$ has a dependence with $L$ of the form

with $\lambda(t)$ a real-valued (intensive) rate function which is obtained in the thermodynamic limit as

A vanishing value of the return amplitude, and thus of the Loschmidt echo, results in nonanalyticities of the rate function $\lambda(t)$. This is similar to how the complex zeros of partition functions, known as Fisher or Lee-Yang zeros, lead a nonanalytic behavior of the free energy density (i.e. the associated rate function) at the critical point of a phase transition. Extending this idea to the nonequilibrium realm, a DQPT at a critical time $t^{*}$ (instead of a critical control parameter) is said to occur when during the dynamics, such zeros are crossed [15].

A vast amount of recent theoretical research has been devoted to study this beautiful insight in systems of different nature, namely spin [15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], fermionic [34, 35, 36, 37, 38, 39], bosonic [40, 41, 42, 43, 44, 45], and hybrid models [46]. This effort also includes the analysis of the impact of ingredients such as disorder [47, 48, 49] and topological order [50, 51, 52]. Crucially, experimental demonstrations of DQPTs have been achieved in several quantum simulation platforms [53, 54, 55, 56, 57]. Many of these studies have shown evidence that DQPTs emerge when there is a quench across an equilibrium quantum phase transition, i.e. when $|\psi_{0}\rangle$ and the ground state of $H$ correspond to different phases. However, several exceptions have been reported [16, 19, 20, 45, 58], so no one-to-one correspondence between DQPTs and equilibrium quantum phase transitions can be established. In addition, it is common to look for manifestations of DQPTs beyond nonanalyticities of $\lambda(t)$. In several scenarios, such as particular quenches in the XXZ [18, 51], O$(N)$ [59] and symmetry-broken Ising [53] models, these nonanalyticities coincide with vanishing values of order parameters; in other cases they are shifted from zeros or minima of order parameters by a constant factor, as found in Ising [15, 26] and Bose-Hubbard models [41] respectively. However, in many others such an association remains elusive [15, 16, 17, 18].

To help unravel the connections between DQPTs, equilibrium quantum criticality and the time evolution of observables, it is valuable to consider simple models with different phases that are accessible experimentally. In this regard, systems of strongly interacting spin-$1/2$ fermions constitute very attractive candidates to be the object of such analysis. In spite of this, DQPTs on these systems have been discussed only a few times  [34]. In the present work we perform such a study in the one-dimensional extended Fermi-Hubbard (EFH) model at half filling, which incorporates on-site and nearest-neighbor density-density interaction. This model not only generalizes the seminal and widely-studied Fermi-Hubbard model, whose dynamics has been intensively explored experimentally with cold atoms in optical lattices [6, 60, 61, 62, 63]; it is also known to possess a rich ground-state phase diagram [64, 65, 66, 67], and has been recently proposed to explain properties of one-dimensional quantum materials [68, 69]. We analyze DQPTs emerging at early times from quantum quenches in one or both of the interaction terms, with special focus on the Fermi-Hubbard limit and charge density wave (CDW) states. For the former case, we also discuss how symmetries of the model are manifested in the time evolution of the rate function. Furthermore, we establish two main general scenarios where a clear connection between DQPTs and the dynamics of observables exists. Finally we show that DQPTs can be induced solely by a time periodic modulation of an on-site potential, which provides a timely strategy for observing such phenomena given the state-of-the-art advances in Floquet engineering with cold fermions in optical lattices [70, 71, 72, 73].

The manuscript is organized as follows. In section 2 we describe the EFH model and the method used to study its dynamics. In section 3 we show the existence of DQPTs in the Fermi-Hubbard limit ($V=0$) for noninteracting, weakly and strongly interacting initial states. The transitions resulting from CDW states, namely a ground state of the extended model for finite $V>0$ and a product degenerate CDW, are discussed in section 4. The emergence of DQPTs from Floquet modulation is presented in section 5. Finally, section 6 contains our main conclusions.

We study the DQPTs of the one-dimensional EFH model at half filling, zero magnetization and with open boundary conditions. Its Hamiltonian is given by

where $\hat{c}_{j,\sigma}^{\dagger}$ ($\hat{c}_{j,\sigma}$) creates (annihilates) a fermion with spin $\sigma=\uparrow,\downarrow$ on site $j$, $\hat{n}_{j\sigma}=\hat{c}_{j,\sigma}^{\dagger}\hat{c}_{j,\sigma}$ is the number operator for site $j$ and spin $\sigma$, $\hat{n}_{j}=\hat{n}_{j\uparrow}+\hat{n}_{j\downarrow}$ is the total number operator at site $j$, $J$ is the hopping (taken as $J=1$ to set the energy scale), $U$ the on-site coupling, and $V$ the nearest-neighbor interaction. Thus this model extends the standard (integrable) Fermi-Hubbard Hamiltonian, which corresponds to $V=0$. Importantly, we allow $U$ and $V$ to represent both repulsive ($U,V>0$) or attractive ($U,V<0$) interactions.

The ground state phase diagram of the model at half filling and zero magnetization, i.e. with

is well known [64, 65, 66, 67], and features several (quasi-long range ordered) phases, as sketched in figure 1. When the system has on-site attraction and nearest-neighbor repulsion, or when the nearest-neighbor repulsion dominates over on-site repulsion, a charge density wave (CDW) is favored. On the other hand, if the on-site repulsion is dominant, a spin density wave (SDW) is formed. For weak and intermediate repulsive interactions, a bond-order wave (BOW) emerges between the CDW and SDW states, characterized by an alternating expectation value of the hopping term of the Hamiltonian [65, 74, 75, 76, 77, 78, 79, 80]. For strong nearest-neighbor attraction, the fermions cluster together forming a phase separation (PS). Finally, for weak and intermediate nearest-neighbor attraction, superconducting states are established, either of singlet (SS, with on-site attraction) of triplet (TS) pairing [65, 81].

The dynamical properties of the one-dimensional EFH model are known to a much lesser degree, since most research has focused on the $V=0$ limit. The latter includes analysis on the expansion of initially-confined particles [82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 62, 94], doublon dynamics [95], melting of product ordered states [96, 90, 97, 93], and quantum quenches from ground [90, 98, 99, 100, 101] and thermal states [102, 103]. On the other hand, for finite nearest-neighbor interactions, the study on dynamical properties has been much more limited, focusing on the time evolution of doublons [104], time-resolved single-particle spectrum [105], and spectral functions for detecting nonequilibrium superconductivity [106]. Thus the nonequilibrium properties of the EFH model still constitute a largely uncharted territory.

In the present work we discuss the time evolution induced by different sudden quenches in the EFH model, focusing on early times where the existence of DQPTs is unveiled. We consider three main scenarios. First we discuss transitions solely within the Fermi-Hubbard limit. Then we explore quenches from CDW states, corresponding to the ground state of the EFH with finite $V$ or a product state. Finally we study DQPTs emerging from a periodic modulation of the Hamiltonian. To reach systems of hundreds of sites, our simulations are based on a matrix product state description. Namely, we calculate the initial state $|\psi_{0}\rangle$ as the ground state of a Hamiltonian $H_{0}$ (with interactions $U_{0}$ and $V_{0}$) with the density matrix renormalization group [9], and use its time-dependent extension [107, 108] to obtain the evolved state $|\psi_{0}(t)\rangle$ under Hamiltonian $H$ (with interactions $U$ and $V$). We show results for chains of up to $L=160$ for correlated initial states, and of $L=256$ for the product CDW (finite-size effects are discussed in B). We also incorporate the conservation of number of fermions with spin up and down, to work directly in the symmetry sector of half filling and zero magnetization (5). The ground states have been calculated with maximal truncation error per sweep down to $10^{-10}$. In addition, during the dynamics we take time steps down to $\delta t=0.001$ and maximal truncation errors per step down to $10^{-10}$ (up to a maximal truncation parameter $\chi=3000$). Our codes are based on the open-source Tensor Network Theory library [109, 110].

We initiate by discussing DQPTs in the Fermi-Hubbard limit $V_{0}=V=0$. In this case, in addition to the rate function $\lambda(t)$ of  (3), we calculate the average double occupation in the lattice, given by

We also consider two distinct cases: when the initial state $|\psi_{0}\rangle$ is noninteracting ($U_{0}=0$) and when it has finite on-site interactions ($U_{0}\neq 0$).

Now we discuss DQPTs from different initial CDW states, namely the ground states of the EFH model for finite (section 4.1) and infinite (section 4.2) nearest-neighbor attraction $V$. In addition to $d(t)$, here we calculate the mean imbalance between the population of odd and even sites, given by

In contrast to cases discussed in section 3, here the imbalance shows a sizeable nontrivial dynamical behavior.

Finally, we discuss an alternative and far less explored method to induce DQPTs. This corresponds to driving the system by an external periodic excitation [54, 112, 113, 114, 115, 116], where the average dynamics can be described by an effective (Floquet) Hamiltonian. To perform this study, we first calculate the ground state of the EFH model at half filling for particular values of $U_{0}$ and $V_{0}$. Then we perform a real time evolution with $U=U_{0}$, $V=V_{0}$, and a time-periodic electric field of the form

with frequency $\omega$. Moving to a rotating frame, the time-periodic electric field is transformed into a time-dependent hopping parameter given by

Thus the hopping is multiplied by a Peierls phase. In this way, the emerging DQPTs are a direct consequence of the driving protocol, which is controlled by the frequency $\omega$ and the amplitude $A$.

To discuss these Floquet DQPTs, we focus on a high-frequency driving ($\omega=15$ and $A=3$), which is known to correspond to an effective Hamiltonian in which the hopping is renormalized by the Bessel function $\mathcal{J}_{0}(A)$ [117, 118, 119]; this is equivalent to having new density-density interactions $U/\mathcal{J}_{0}(A)$ and $V/\mathcal{J}_{0}(A)$. The results are shown in figure 8 for two initial states in different phases, namely an insulating SDW state for the Fermi-Hubbard Hamiltonian and a CDW for the extended model; we see in both cases that the evolution of the double occupation and the rate function is very well captured, in average, by that of the effective Hamiltonian.

For a SDW initial state with $U_{0}=2$ and $V_{0}=0$ the effective Floquet Hamiltonian has $U=-2/\mathcal{J}_{0}(3)\approx-7.69$ and $V=0$, which corresponds to a metallic ground state; considering the results of section 3.2, a DQPT is anticipated, and indeed observed in figure 8(a). A similar result is obtained for a CDW case with $U_{0}=-2$ and $V_{0}=0.5$, since the effective Hamiltonian corresponds to $U=2/\mathcal{J}_{0}(3)\approx 7.69$ and $V=0.5/\mathcal{J}_{0}(3)\approx-1.92$, which has a ground state deep in the SDW regime. In the light of the results of section 4.1, DQPTs are expected and observed, as depicted in figure 8(a).

Our results thus show that the form and time of occurrence of DQPTs can be controlled with a periodic external field, by the amplitude of the driving. In addition to the existing capabilities for implementing Fermi-Hubbard models in cold-atom architectures [6, 60, 61, 62, 63], recent advances in simulating periodic driving of the form (23) by mechanical shaking [70, 71, 72, 73] make our proposal a very-promising scheme for observing DQPTs in experiments.

The rich ground state phase diagram of systems of interacting fermions makes them an ideal testbed for observing DQPTs and establishing connections to particular characteristics of observables. With this in mind, we performed an analysis of DQPTs in the one-dimensional EFH model at half filling, for several interaction regimes.

We focused our discussion on initial noninteracting, metallic and CDW ordered states. In these regimes, DQPTs clearly emerge on two main general scenarios. The first one corresponds to initial states with zero or weak on-site interaction $U_{0}$ and strong final interactions, with zero (figures 2 and 3) or finite nearest-neighbor coupling $V$ (figure 5). The second one corresponds to initial ground states of strong interactions, namely large finite coupling $U_{0}$ (figure 4) and infinite attraction $V_{0}$ (product CDW, figures 6 and 7), and much weaker final interactions. Thus, DQPTs are favored when the initial and final Hamiltonians involved in the quench correspond to opposed interacting regimes.

A clear coincidence of DQPTs and particular properties of observables emerges in a much more reduced parameter space. Two main regimes were identified. The first one is that of zero or weak initial interactions and very strong final interactions (figures 2 and 3). Here the evolution of the rate function and the double occupation display the same timescale, given by $U$, and nonanalyticities of the former almost exactly coincide with minima of the latter. Introducing weak nearest-neighbor interactions degrades this correspondence (figure 5), but it is still observed as long as $U$ is largely dominant. The second regime corresponds to an initial product CDW (effectively equivalent to a Néel state) and a final noninteracting Hamiltonian (figure 6), where the vanishing of the charge imbalance agrees (up to a constant shift) with the occurrence of DQPTs. This connection is broken when interactions in the final Hamiltonian are introduced. These findings would be crucial for an experimental demonstration of DQPTs and their coincidence with particular features of observables with interacting fermions.

Finally, we demonstrated the emergence of DQPTs solely from a time-periodic on-site potential. Performing a high-frequency modulation on initial insulating SDW and CDW states, DQPTs were induced and captured by a time evolution given by an effective Floquet Hamiltonian with renormalized interactions.

Our results pave the way for experimentally observing DQPTs in state-of-the-art quantum simulators. On the one hand, protocols for obtaining the Loschmidt echo in ultracold atom setups have been proposed [120, 121, 122]. Also, Fermi-Hubbard systems [63] and product CDWs [61] have been implemented in fermionic optical lattices. Furthermore, existing capabilities for performing time-periodic modulation [70, 71, 72, 73] makes the observation of Floquet DQPTs in such platforms very promising. On the theoretical side, our work motivates the exploration of DQPTs from different phases, e.g. superconducting or BOW states, and in richer models such as Bose-Fermi mixtures, which are experimentally available [123, 124, 125] and efficiently simulable [126, 127]. Resonant driving could be also exploited, since its effective Hamiltonian, being quite different to the original [128], seems appealing to induce alternative Floquet DQPTs.