Introduction

Exploration of non-Hermitian physics has flourished in the last few years1,2,3. It covers a wide range of classical wave systems such as lossy acoustic cavities4,5,6,7,8 or photonic crystals8,9,10,11,12,13,14,15, open quantum systems connected to the environment16,17,18, and quasiparticles with finite lifetimes in condensed matter19,20. Complex features, including non-real eigenvalues21, non-orthogonal eigenvectors22,23, and exceptional points24,25 are characteristics of non-Hermitian systems. They give rise to a variety of intrinsic phenomena without parallels in the Hermitian limit. A prime example is the non-Hermitian skin effect (NHSE)26,27,28,29, the anomalous localization of extensive bulk states at the open boundaries. Numerous platforms have witnessed the effective observation of NHSE. Its potential applications, including unidirectional amplifiers30, optical funnels31, high-efficiency energy harvesting32, and enhanced sensors33,34,35, also have been explored.

In NHSE, extensive boundary-localized bulk states lead to the failure of traditional Bloch band theory and the breakdown of conventional bulk-boundary correspondence36. These motivate the novel concept of the generalized Brillouin zone, formed by introducing an imaginary part to the Bloch wave vector to account for the exponentially localized skin states37. In dynamics, NHSE manifests as the nonreciprocal propagation of the wave packet, which suppresses the entanglement propagation and thermalization38. At the heart of the NHSE lies the point-gap topology of the energy spectrum under the periodic boundary condition (PBC). In one dimension (1D), the eigenvalues under PBC can form a closed curve on the complex energy plane, deviating from a specific reference point, i.e., point gap. Then, a spectral winding number can be defined with respect to the reference point. The necessary and sufficient condition for the presence of NHSE is the non-zero spectral winding number or nontrivial point-gap topology39. When there are higher dimensions, displaying more possible symmetries and geometries, NHSE will have richer connotations and be the subject of debate40,41,42. It has been suggested that a global criterion connects the spectral area to the occurrence of NHSE43. In this case, corresponding eigenenergies under PBC form a closed curve with nontrivial point-gap topology, as the Bloch wave vector, perpendicular to the open boundary which hosts skin states, goes through the Brillouin zone.

In this paper, we report the NHSE outside the framework of the point-gap topology in 2D quasicrystals. We consider a 2D anisotropic square lattice with nonreciprocal hoppings in one direction, say the y dimension, and an incommensurate magnetic field penetrating the lattice. Under PBCs, the incommensurate magnetic field can induce Anderson localization of states in the y dimension, eliminating effects of the nonreciprocity on spectra. The spectrum is real, with zero winding numbers. However, the system possesses huge degeneracy. On the other side, under open boundary conditions (OBCs), the open boundary in the x dimension causes degeneracy breaking. Degenerated y-dimension-localized states superpose to form extended states, which turn into skin states in the presence of nonreciprocal hoppings. The degeneracy-breaking-induced NHSE manifests in dynamics as an anomalous nonreciprocal propagation of the wave packet.

Results and Discussion

Nonreciprocal Hofstadter model

Our exploration of the anomalous NHSE commences with a single particle within a nonreciprocal square lattice, subjected to a vertical uniform magnetic field. It is described by the nonreciprocal Hofstadter model, which is written as

$$H= {\sum }_{n,m} \left({t}_{x}{c}_{n+1,m}^{{{\dagger}} }{c}_{n,m}+{t}_{x}{c}_{n,m}^{{{\dagger}} }{c}_{n+1,m}\right.\\ \left.+{t}_{y}{e}^{i2\pi \beta n+g}{c}_{n,m+1}^{{{\dagger}} }{c}_{n,m}+{t}_{y}{e}^{-i2\pi \beta n-g}{c}_{n,m}^{{{\dagger}} }{c}_{n,m+1}\right).$$
(1)

\({c}_{n,m}^{{{\dagger}} }\) (cn,m) is the creation (annihilation) operator at the lattice site (n, m). tx represents the nearest-neighbor hopping amplitude in the x direction. ty is the geometric average of the nonreciprocal hopping amplitudes in the y direction, and g characterizes the strength of nonreciprocity. The phase factor β is defined by the magnetic flux through a lattice cell (lattice constant 1). In the presence of a homogeneous magnetic field, the Hofstadter model has a magnetic translation group and an enlarged magnetic unit cell44. Above, the Landau gauge A = (0, βn) has been adopted. We focus on the situation that β is an irrational number like the golden ratio \((\sqrt{5}-1)/2\), which makes the model quasiperiodic. The quasiperiodicity is the source of all phenomena and makes the model distinguishable from others45,46. In practice, the irrational \(\beta =(\sqrt{5}-1)/2\) is approximated by rational numbers β = Fl/Fl+1 with Fl the lth Fibonacci number47. Correspondingly, the total number of lattice sites is L × L with the linear size L = Fl+1. Without loss of generality, we will use L = 144, the 12th one. Traditionally, the Hofstadter model is distinguished by its butterfly-shaped energy spectrum with multi-fractal properties48. It also serves as an essential platform for investigating quantum Hall effects and Chern topological insulators49,50.

Mirror-time symmetry breaking

The properties of system strongly depend on boundary conditions, and we first focus on the case under all PBCs (PBCs in both x and y dimensions). The Hamiltonian Eq.(1) possesses combined mirror-time (MT) symmetry as28

$$({{{\mathcal{MT}}}})H{({{{\mathcal{MT}}}})}^{-1}=H.$$
(2)

Operators of mirror reflection (\({{{\mathcal{M}}}}\)) about the y-axis and time reversal (\({{{\mathcal{T}}}}\)) are defined by \({{{\mathcal{M}}}}{c}_{n,m}{{{{\mathcal{M}}}}}^{-1}={c}_{-n,m}\) and \({{{\mathcal{T}}}}i{{{{\mathcal{T}}}}}^{-1}=-i\) respectively. As a result, the energy spectrum can be either entirely real or composed of complex conjugate pairs. A real-to-complex spectral phase transition occurs along with the spontaneous breaking of the MT symmetry. To detect the MT symmetry breaking, in Fig. 1a we present the largest value of imaginary parts of all eigenenergies E, as a function of ty. From now on, tx = 1 is set as the energy unit. The MT symmetry breaking happens at ty = e−∣g∣ (dash line). When ty < e−∣g∣ the system is in the MT symmetry unbroken phase with a real spectrum, while the system is in the MT symmetry broken phase and the spectrum is complex when ty > e−∣g∣. To explore more features of the spectrum, in Fig. 1b we show typical spectra on the complex energy plane before and after the symmetry breaking. When ty < e−∣g∣, the spectrum is real without point-gap topology, and contains bands, each having subbands due to the fractal caused by quasiperiodicity51. Moreover, energy levels are highly degenerated with 2L degree of degeneracy (See Supplementary Note 1). When ty > e−∣g∣, the spectrum is complex with loops, which implies nontrivial spectral topologies. Energy levels are with L degree of degeneracy, and the lack of factor 2 is because of the broken MT symmetry.

Fig. 1: Mirror-time (MT) symmetry breaking.
figure 1

a The largest value of imaginary parts of all eigenenergies (Max∣Im(E)∣), as a function of ty. b Spectra on the complex energy plane before (ty = 0.6) and after (ty = 1.4) the MT symmetry breaking. Systems are under periodic boundary conditions (PBCs) and with nonreciprocity g = 0.1.

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The Hamiltonian Eq.(1) has translational symmetry in the y dimension and we can perform dimension reduction by conducting Fourier transformation in this dimension. It results in

$${H}^{x}({k}_{y})={\sum }_{n=1}^{L}[{t}_{x}{c}_{n+1}^{{{\dagger}} }{c}_{n}+h.c.+2{t}_{y}\cos (2\pi \frac{{F}_{l}}{L}n-ig+\frac{{k}_{y}}{L}2\pi ){c}_{n}^{{{\dagger}} }{c}_{n}]$$
(3)

living in the x dimension, a 1D non-Hermitian Aubry-André-Harper (AAH) model52,53. In recent years, non-Hermitian AAH models have been extensively studied54,55,56, focusing on Anderson localization and topology. Driven by the quasiperiodic potential (the last term in Eq.(3)), which acts as a quasirandom disorder, the model undergoes triple phase transitions at ty = e−∣g∣56. Due to the breaking of parity-time symmetry, which is similar to the MT symmetry, as ty increases the spectrum turns from real to complex with loops, i.e., the phase transition of parity-time symmetry breaking. The emergence of loops also leads to a spectral topological phase transition. Furthermore, the spectrum is independent of the momentum ky, which results in the L degree of degeneracy described above. The Hamiltonian remains the same, when we simultaneously add an integer k to ky and shift the lattice index n to n + n0 so that \({{{\rm{mod}}}}({F}_{l}{n}_{0},L)=L-k\). As for the eigenstates, they undergo an extended-localized phase transition. All states are extended when ty < e−∣g∣ while Anderson localized when ty > e−∣g∣. Localization lengths of states are energy- and ky-independent, but localization centers are ky-dependent and distributed all over the lattice. On the other hand, by performing the gauge transformation cn,m → ei2πmnβcn,m, we can rewrite the Hamiltonian Eq.(1) in another gauge A = ( − βm, 0) (See Supplementary Note 2). The resulting Hamiltonian has translational symmetry in the x dimension. After performing dimension reduction, it leads to

$$\begin{array}{rcl}{H}^{y}({k}_{x})= & & {\sum }_{m=1}^{L}\left[{t}_{y}{e}^{g}{c}_{m+1}^{{{\dagger}} }{c}_{m}+{t}_{y}{e}^{-g}{c}_{m}^{{{\dagger}} }{c}_{m+1}\right.\\ & & \left.+2{t}_{x}\cos (2\pi \frac{{F}_{l}}{L}m-\frac{{k}_{x}}{L}2\pi ){c}_{m}^{{{\dagger}} }{c}_{m}\right]\end{array}$$
(4)

living in the y dimension, another non-Hermitian AAH model57. This Hamiltonian is independent of kx. It has the same spectrum as Hx. Nevertheless, states have opposite localization properties: all states are Anderson localized when ty < e−∣g∣, while extended when ty > e−∣g∣.

Backing to the nonreciprocal Hofstadter model, we introduce spectral winding numbers43,56

$${\nu }_{p}={lim}_{L\to \infty }\frac{1}{2\pi i}\int_{0}^{2\pi /L}{{{\rm{d}}}}{k}_{p}\frac{\partial }{\partial {k}_{p}}{{{\rm{ln}}}}[\det (H({k}_{x},{k}_{y}))],$$
(5)

along x (p = x) and y (p = y) dimensions. They characterize how the spectral trajectory encircles the origin of the complex energy plane when the momentum kp crosses the Brillouin zone. H(kx, ky) is the Bloch Hamiltonian defined over the enlarged magnetic unit cell. It is equivalent to Hx(ky) with twisted boundary terms \([{t}_{x}{e}^{iL{k}_{x}}{c}_{1}^{{{\dagger}} }{c}_{L}+h.c.]\). Analytical calculation shows that νx = 0, indicating that the spectrum is topologically trivial in the x dimension (See Supplementary Note 3). But in the y dimension

$${\nu }_{y}={{{\rm{sign}}}}(g)\theta ({t}_{y}-{e}^{-| g| }),$$
(6)

with θ the step function. There is a topological phase transition, characterizing the presence of loops in the spectrum. On the other hand, known from the dimension reduction described above, the nonreciprocal Hofstadter model supports the directional localization of states58. Crucially, this is a global property of the entire band. When ty < e−∣g∣ all eigenstates are Anderson localized in the y dimension while extended in the x dimension, whereas they are extended in the y dimension while Anderson localized in the x dimension when ty > e−∣g∣ [see See Supplementary Note 5 for simulations]. The transition at ty = e−∣g∣ thus marks a collective switching of the localization direction for all eigenstates.

NHSE under all OBCs

Since we focus on NHSE without point-gap topology, we concentrate on the situation ty < e−∣g∣, where the spectrum under all PBCs is real and topologically trivial. Under all OBCs (OBCs in both x and y dimensions), we utilize the imaginary gauge transformation59,60

$${c}_{n,m}={e}\,^{gm}{a}_{n,m},{c}_{n,m}^{{{\dagger}} }={e}^{-gm}{\widetilde{a}}_{n,m}.$$
(7)

a(\(\widetilde{a}\)) is the annihilation (creation) operator, but \(\widetilde{a}\) is not the Hermitian conjugate of a. We transform the Hamiltonian Eq.(1) into the same form but without the nonreciprocal hopping (g = 0). Thus, the model has the same real spectrum as the Hermitian Hofstadter model. The above degeneracy of 2L degrees is broken! (See Supplementary Note 1)

As for eigenstates, we present the spatial distribution of a typical bulk one in Fig. 2a. All bulk states are exponentially localized at the large (small) y edge when g > ( < )0, exceptionally showing NHSE. To quantitatively study NHSE, we adopt the exponential wave function \({\phi }_{n,m}=f(n){e}^{{{{\rm{sign}}}}(g){\gamma }_{n}m}\), where γn are Lyapunov exponents (LEs or inverse of localization lengths) in the y dimension. Fitting eigenstates with the above wave function, we extract mean LEs γ = ∑nγn/L and mean squared errors of the LEs, summing over the lattice index n (in the x dimension). In Fig. 2b we present mean LEs, as a function of eigenenergy, of all eigenstates for a system with ty < e−∣g∣. Compared to Fig. 1b, states emerge in gaps, whose mean LEs and mean squared errors are larger than that of majority states (See also Supplementary Note 4). These are edge states caused by the Chern topological nature of the nonreciprocal Hofstadter model61. On the other hand, the majority states, which are bulk states, have approximately the same mean LE. To obtain the LE of these bulk states, we extract the median LE \(\overline{\gamma }\) over all eigenstates. In Fig. 2c we show its relation with the nonreciprocity g. The LE of bulk states \(\overline{\gamma }=| g|\), independent of other parameters. Furthermore, we count the number of eigenstates whose mean LEs lie in the interval \(\overline{\gamma }\times [1-\varepsilon ,1+\varepsilon ]\) with a small ε = 0.1, considering it the number of bulk states. We present the number versus the lattice size L2 in Fig. 2d. It is fitted by a straight line with the slope 98%, which can be seen as the proportion of bulk states. The linear relation further indicates that the NHSE in the nonreciprocal Hofstadter model is of the first order, where an extensive number \({{{\mathcal{O}}}}({L}^{d})\) of skin modes emerge in a d dimensional system62,63,64.

Fig. 2: Non-Hermitian skin effect under open boundary conditions (OBCs).
figure 2

a Semi-log plot of the spatial distribution of the bulk eigenstate with energy E = 0.2. b Mean Lyapunov exponents (LEs) in the y dimension of all eigenstates, as a function of eigenenergy. The red dashed line indicates the value of nonreciprocity g. c Median LE of all eigenstates (LE of bulk eigenstates) versus the nonreciprocity g. d The number of bulk skin states versus the lattice size L2. Systems are under all OBCs and with g = 0.1 and ty = 0.6 if not specified.

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Open-boundary-induced degeneracy breaking

To explain the mechanism behind, we start from the case under all PBCs and open the boundaries in the y and x dimensions successively. Numerical simulations under various boundary conditions are presented in the Supplementary Note 5. From above, we know that when ty < e−∣g∣ the system supports directional localization of states under all PBCs. Eigenstates are extended in the x dimension and Anderson localized in the y dimension. Like in nonreciprocal AAH models57, the localization of states in the y dimension suppresses the effects of nonreciprocity and leads to a real spectrum. Furthermore, for AAH models in the Anderson localized phase, bulk eigenstates are exponentially localized in the bulk, which makes them independent of boundary conditions57,65. Correspondingly, when ty < e−∣g∣ and under all PBCs, eigenstates are exponentially localized in the y dimension while extended in the x dimension. Then, the localization and energies of bulk states remain the same when we turn the PBC into OBC in the y dimension (y-PBC into y-OBC). Except for the emergence of edge states in gaps, which is not the focus here, the degeneracy of bulk states is approximately the same, and the reality of bulk spectrum still holds. Things are different when we change x-PBC to x-OBC.

Under y-OBC, the above-introduced imaginary gauge transformation works regardless of the boundary condition in the x dimension. Note that it does not work under y-PBC. The nonreciprocal and the Hermitian Hofstadter models share the same spectrum, when under y-OBC and the same boundary condition in the x dimension. A correspondence can be made through the transformation, between the right eigenstates of two models. Given that φn,m is an eigenstate of the Hermitian Hofstadter model under y-OBC and x-PBC(OBC), ϕn,m = e gmφn,m is a right eigenstate of the nonreciprocal model under y-OBC and x-PBC(OBC). It clearly shows how nonreciprocal hopping affects states in different phases. For states of the Hermitian Hofstadter model, which are extended in the y dimension, corresponding wave functions ϕn,m are exponentially localized at the large (small) y edge when g > ( < )0. This works for the case under all OBCs, and the resulting states are skin states with LEs γ = ∣g∣. For Anderson localized bulk eigenstates of the Hermitian Hofstadter model, wave functions ϕn,m are also Anderson localized in the y dimensional bulk but with different right and left side LEs (the case under y-OBC and x-PBC).

Through the imaginary gauge transformation, we obtain that when ty < e−∣g∣ and under y-OBC and x-PBC, the spectrum of the Hermitian Hofstadter model has huge degeneracy. Moreover, bulk states of the Hermitian Hofstadter model are Anderson localized in the y dimensional bulk while extended in the x dimension. We choose that eigenstates have translational symmetry along the x direction, and the momentum kx is a good quantum number. Supposing that an eigenstate \(\left|\varphi \right\rangle\) of energy E has a momentum kx in the x dimension and is exponentially localized around m0 in the y dimensional bulk, we can obtain new states that have the same energy E and construct the corresponding degenerate subspace. Easily seen in the dimension-reduced Hamiltonian H y(kx), when we add an integer k to the momentum kx, the Hamiltonian remains the same, by shifting the lattice index m to m + m1 so that \({{{\rm{mod}}}}({F}_{l}{m}_{1},L)=k\) and emerging phases in the cosine function cancel. New states in the degenerate subspace are obtained by performing the same operation on the state \(\left|\varphi \right\rangle\), as long as they are localized in the y dimensional bulk. Furthermore, the state with momentum  − kx and being localized around  − m0 is also in the degenerate subspace, ensured by the MT symmetry. All these states contribute the approximate 2L degree of degeneracy.

Now, we further change the x-PBC into x-OBC. This is the pivotal step that induces the NHSE. To meet the need of x-OBC, the state with energy E and momentum kx has to superpose with the state which has the same energy E but opposite momentum  − kx to form a standing wave in the x dimension. However, these two states are spatially separated in the y dimension, which makes the superposition impossible [see Fig. 3a]. To bridge these two states to satisfy the x-OBC, all bulk states in the degenerate subspace, being Anderson localized at different positions in the y dimension and having different momenta kx, must superpose together. This superposition across the degenerate subspace breaks the degeneracy and hybridizes the y-localized states into states that are extended in both dimensions [see Fig. 3b]. However, the essences of the “extended" in x and y dimensions are different, which will be shown later. Finally, through the imaginary gauge transformation again, we know that under all OBCs, all bulk eigenstates of the nonreciprocal Hofstadter model are skin states with LEs γ = ∣g∣, which explains the median LE. For reference, we summarize in Table 1 the properties of the spectrum and bulk eigenstates under different boundary conditions. Further details are provided in Supplementary Notes 1 and 4–6.

Fig. 3: Open-boundary-induced degeneracy breaking.
figure 3

For the Hermitian Hofstadter model, degenerated bulk eigenstates under x-PBC, which are localized in the y dimensional bulk and extended in the x dimension, have to superpose together to meet the need of x-OBC (schematically shown in (a)). b Spatial distribution of the bulk eigenstate with energy E = 1.45, for the system under all OBCs and with g = 0 and ty = 0.6.

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Table 1 Summary of the spectral and bulk eigenstate properties for the nonreciprocal Hofstadter model with ty < e−∣g∣ under different boundary conditions
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Anomalous wave packet dynamics

Not only the spectrum and eigenstates, but the NHSE also significantly affects several dynamic properties of the system even in bulk. The appearance of a directional (chiral) bulk flow is a typical phenomenon17,66. We simulate the time evolution of a wave packet, which is initially in the Gaussian form \(\left|{\phi }_{0}\right\rangle ={{{\mathcal{N}}}}{\sum }_{n,m}\exp [-{(n-{n}_{0})}^{2}/25-{(m-{m}_{0})}^{2}/25]{c}_{n,m}^{{{\dagger}} }\left|0\right\rangle\). Here, \({{{\mathcal{N}}}}\) is the normalization factor, and n0 = 70, m0 = 30 are coordinates of the center of the initial wave packet. Let us indicate by YCM the center of mass in the y dimension of the wave packet. It is given by the normalized expectation value of the position operator

$${Y}_{{{{\rm{CM}}}}}(t)=\left\langle \phi (t)\right|{\sum }_{n,m}m{c}_{n,m}^{{{\dagger}} }{c}_{n,m}\left|\phi (t)\right\rangle .$$
(8)

In Fig. 4 we present the time evolution of YCM for the system under all OBCs and with ty < e−∣g∣, while the one for ty > e−∣g∣ is shown in Supplementary Note 7. In the directional bulk flow caused by normal NHSE, the center of mass increases linearly with time in the Lieb-Robinson velocity67. In contrast, here, YCM is constant in the beginning, followed by a rapid increase to the maximum. Finally, the wave packet accumulates at the large y edge, showing NHSE.

Fig. 4: Anomalous wave packet dynamics.
figure 4

Time evolution of the center of mass YCM, in the dynamics of an initial Gaussian wave packet on the lattice under all OBCs and with nonreciprocity g = 0.1 and ty = 0.6. Besides, spatial distributions of the wave packet at specific times are also presented. The dynamics show anisotropic bulk transport and directional edge flow.

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To investigate the physics behind the anomalous directional flow, we further present in Fig. 4 spatial distributions of normalized states \(\left|\phi (t)\right\rangle ={{{\mathcal{N}}}}(t){e}^{-iHt}\left|{\phi }_{0}\right\rangle\) at specific times. Initially, the wave packet only expands in the x dimension, and the center of mass in the y dimension keeps constant. Bulk eigenstates under all OBCs are skin states that are localized in the y dimensional edge, but they are superpositions of states that are Anderson localized in the y dimensional bulk, while they are extended in the x dimension. The system has an anisotropic conductivity with vanishing bulk transport in the y dimension68, because of the directional Anderson localized nature of states. When the wave packet expands to the edges, chiral edge states start to play roles, due to the non-orthogonality of states in non-Hermitian systems. These edge states result from the Chern topological nature of the nonreciprocal Hofstadter model, and they are ballistic along edges. NHSE of the chiral edge states forces the wave packet to move along the edge to the top-right corner, and correspondingly, the center of mass in the y dimension increases dramatically. After expanding in the large y edge, which does not cause apparent changes in YCM, the wave packet finally evolves into a skin state.

Conclusion and discussion

We have reported NHSE without point-gap topology in the 2D nonreciprocal Hofstadter model. It results from the OBC-induced degeneracy breaking and manifests an anomalous directional flow of wave packet. Techniques to generate (artificial) magnetic fields have been well established in various candidate non-Hermitian systems, such as ultra-cold atoms69, photonic and acoustic structures70,71, and electric circuits72, where the NHSE has been observed17,31,73,74. Thus, the physics shown above is highly accessible in experiments. For example, in electric circuits, the single-particle problem can be simulated by Kirchhoff’s current law \({I}_{a}(\omega )={\sum }_{b=1}^{L}{J}_{ab}(\omega ){V}_{b}(\omega )\), where the Laplacian J of the circuit acts as the effective Hamiltonian, and Ia and Va are the current and voltage at node a. Periodic arrays of capacitors and inductors are known to simulate the physics in crystal lattices75. Nonreciprocal hopping amplitudes are realized by negative impedance converters with current inversion (INICs)76. Artificial magnetic fields are generated by spatially varying electric elements72. Further studies on (higher-order) NHSE and topology in highly degenerated systems, such as Moiré materials, systems with flat bands, and (artificial) magnetic lattices, will be interesting.

Methods

The eigenenergies and eigenstates are obtained by performing exact diagonalization of the non-Hermitian, non-interacting Hamiltonian, \(H\left|{\phi }_{i}\right\rangle ={E}_{i}\left|{\phi }_{i}\right\rangle\), where \(\left|{\phi }_{i}\right\rangle\) denotes the i-th right eigenstate. The spatial distribution (wavefunction) of an eigenstate is given by \({\phi }_{n,m}^{i}=\left\langle n,m| {\phi }_{i}\right\rangle\), with \(\left\langle n,m\right|\) representing the Wannier state localized at lattice site (n, m). For the time evolution of a wave packet, \(\left|\phi (t)\right\rangle ={e}^{-iHt}\left|{\phi }_{0}\right\rangle\), we numerically compute the action of the matrix exponential e−iHt on the initial state vector \(\left|{\phi }_{0}\right\rangle\). The resulting state is then normalized such that \(\left\langle \phi (t)| \phi (t)\right\rangle =1\).