Real-time scattering of interacting quasiparticles in quantum spin chains

TL;DR

This paper introduces a tensor network-based method to generate and analyze localized quasiparticle excitations in quantum spin chains, enabling the study of scattering, phase shifts, and impurity interactions in real-time.

Contribution

It presents a novel approach to create and manipulate localized quasiparticles in strongly-correlated spin chains using tensor networks and variational principles.

Findings

Wavepackets propagate almost dispersionless

Phase shifts can be extracted from scattering data

Reflection and transmission coefficients are characterized

Abstract

We develop a method based on tensor networks to create localized single particle excitations on top of strongly-correlated quantum spin chains. In analogy to the problem of creating localized Wannier modes, this is achieved by optimizing the gauge freedom of momentum excitations on top of matrix product states. The corresponding wavepackets propagate almost dispersionless. The time-dependent variational principle is used to scatter two such wavepackets, and we extract the phase shift from the collision data. We also study reflection and transmission coefficients of a wavepacket impinging on an impurity.

Equations

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$$|{\Phi }_{\mathrm{rs}}(m)⟩$$

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\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@moveto}44.81279pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}0.80.00.00.839.72658pt-1.00044pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}{s}_{n-1}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@moveto}70.42009pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}0.80.00.00.867.20056pt-1.24pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}{s}_n\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@moveto}96.0274pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}0.80.00.00.889.94563pt-0.81378pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}{s}_{n+1}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@moveto}121.6347pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}0.80.00.00.8118.6347pt-2.0pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\dots \text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture},$$

$$B_n={\int }_0^{2\pi }f(k)B_k{e}^{ik(n-m)}dk.$$

$$B_k\to B_k+e^{ik}A{X}_k-X_{k}A$$

$$B_n=\sum _{j=1}^{N}f(k_j)B_{k_j}{e}^{i{k}_{j}n},k_j=j\frac{2\pi }{N}$$

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$$\left|B_n\right|<ϵ,\forall n\in [-L,-\Lambda ]\cup [+\Lambda ,+L]$$

$$f(k_j)\to f(k_j)e^{\frac{(k_0-k_j{)}^2}{2v^2}}.$$

$$\begin{gathered} |\Psi (A;V_1,...,V_n)⟩ \\ 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$$H=\sum _{⟨ij⟩}{S}_i^z{S}_j^z+\lambda \sum _i{S}_i^x.$$

$$H=\sum _i{S}_i^x{S}_{i+1}^x+S_i^y{S}_{i+1}^y+\Delta S_i^z{S}_{i+1}^z,$$

$$\bar{x}=\frac{1}{N}\sum _{i=0}^{L/2}i{e}_i^2,N=\sum _{i=0}^{L/2}{e}_i^2,$$

$$H=\sum _i{\vec{S}}_i\cdot {\vec{S}}_{i+1},$$

$$H_{\mathrm{imp}}={\vec{S}}_{-1}\cdot \vec{T}+\vec{T}\cdot {\vec{S}}_{+1}+h{T}^z,$$

Full text

The field of strongly-correlated quantum many-body physics is one of the most interesting and diverse fields in contemporary physics, both theoretically and experimentally. Here the notion of a quasiparticle is commonly understood as carrying the low-energy degrees of freedom on top of a strongly-correlated ground state. With traditional experimental probes, these quasiparticles are targeted in the momentum-energy plane—think of inelastic neutron-scattering experiments on low-dimensional quantum magnets as an outstanding example Stone et al. (2006); Coldea et al. (2010); Mourigal et al. (2013). Recently, however, progress in the design and probing of strongly-correlated quantum phases has made it possible to directly observe the dynamics of quasiparticles in real-space and real-time Jurcevic et al. (2014); Chiu et al. (2019); Koepsell et al. (2019); Vijayan et al. (2020).

On the theoretical side, understanding low-energy dynamics in terms of effective quasiparticles originates from Fermi liquid theory, where quasiparticles are close to their free counterparts but the interaction, treated perturbatively, leads to dressing and finite lifetimes. The notion of quasiparticles has also been made precise for integrable spin chains by the recent development of generalized hydrodynamics Bertini et al. (2016); Castro-Alvaredo et al. (2016); Bulchandani et al. (2017), which fully describes the large-scale dynamics of a system only in terms of the semi-classical motion of stable quasiparticles. This idea can also in principle work for non-integrable systems, where local quasiparticles are usually thought of as almost stable excitations with long lifetimes that can be treated semi-classically Damle and Sachdev (1998); Moca et al. (2017). However, there is much less information available about the dynamics of local excitations in generic interacting systems and there is no generic procedure to extract them from microscopics.

Recently, a variational framework was developed for representing quasiparticles in gapped one-dimensional quantum spin systems in the thermodynamic limit, which does not rely on integrability. This framework starts from a variational ansatz Haegeman et al. (2012, 2013a) for describing elementary excitations on top of strongly-correlated ground states parametrized by the class of matrix product states (MPS) Verstraete et al. (2009); Schollwöck (2011); Orús (2013). Inspired by the single-mode approximationFeynman (1954); Arovas et al. (1988), the ansatz effectively boils down to a plane wave of a local perturbation on top of a strongly-correlated ground state. This approach is very reminiscent of a quasiparticle running through the system, but differs from the traditional quasiparticle concept since these states are exact eigenstates (up to variational errors) with infinite lifetimes with respect to the fully-interacting hamiltonian Haegeman et al. (2013b), much like electrons and protons in the standard model. In a natural next step, two-particle wave functions were constructed and a generic definition of the two-particle $S$-matrix was proposed Vanderstraeten et al. (2014, 2015).

In this Letter, a method is presented that yields insight in the propagation and mutual interaction of these quasiparticle excitations in real space and time. Whereas in some past works this was done for integrable systems on top of product states Vlijm et al. (2015); Ganahl et al. (2012), our method is used to study two-particle interactions in both integrable and non-integrable systems on the correlated ground state, and is therefore truly capturing the low-energy degrees of freedom for strongly-interacting systems. We also study the scattering of a particle off an impurity and how this impedes the transport of energy.

Methodology. As a first step, we approximate the ground state in the thermodynamic limit as a uniform matrix product state Haegeman et al. (2011); Vanderstraeten et al. (2019a) (MPS), which we can represent as

[TABLE]

where every node corresponds to a tensor and every connected node is a contraction over the corresponding index. This class of states has a number of desirable properties: the ground state of generic gapped quantum spin chains can be faithfully represented as an MPS Hastings (2006); Verstraete and Cirac (2006), which allows for efficient calculation of all observables Verstraete et al. (2009); Schollwöck (2011); Orús (2013). Approximating the ground state of a given model hamiltonian within this manifold can be done using variational optimization methods Zauner-Stauber et al. (2018a); Vanderstraeten et al. (2019a).

Next, the quasiparticle excitations on this ground state can be described using the MPS quasiparticle ansatz Haegeman et al. (2012),

[TABLE]

Because the tensor $B$ acts on the virtual level of the MPS, it can capture the deformation of the ground state over an extended region, and therefore represents a dressed object on top of a correlated ground state. For a given momentum $k$, the problem of minimizing the energy can be reduced to an eigenvalue problem for the tensor $B$, which can be efficiently solved using iterative eigensolvers. The resulting solution depends on $k$ and yields the dispersion relation $\epsilon_{k}$ as eigenvalue, while the corresponding eigenvector is henceforth denoted as $B_{k}$. It was shown Haegeman et al. (2012); Zauner-Stauber et al. (2018b) that this approach yields quasi-exact results for the dispersion relation of generic spin chains. Additionally, these variational quasiparticles are experimentally relevant, as in gapped systems they carry the major fraction of the spectral weight of local operators, and they can be used to compute spectral functions to high precision Bera et al. (2017).

Using these momentum-resolved quasiparticle states, we can now localize a wave packet in real space via a procedure reminiscent to the construction of Wannier orbitals Marzari et al. (2012). A real-space localized wave packet, centered around site $m$, is given by

[TABLE]

where the real-space tensor $B_{n}$ is

[TABLE]

The determination of $B_{n}$ is ambiguous in three ways. First, since the tensors $B_{k}$ are found as a solution of an eigenvalue problem, they are only defined up to a phase, just as in the original Wannier orbital problem, and the sampling function $f(k)$ should compensate for this. But there are additional $k$-dependent gauge redundancies in the tensor $B_{k}$, since the transformation

[TABLE]

with $X$ a matrix that multiplies the MPS tensor $A$ on the virtual level, gives rise to the same wave function $\ket{\Phi_{k}(B_{k})}$. A third difficulty concerns the fact that, in a practical computer simulation, we only have access to a finite set of momenta, such that the real-space tensor

[TABLE]

is periodic with $n\to n+N$ and thus contains an infinite number of translated copies of the wave packet.

We first resolve the gauge freedom by tuning $X_{k}$ for the tensor $B_{k}$, so as to make its effect well localized and symmetric. A natural choice is to minimize the matrix norm of

[TABLE]

which ensures that the perturbation of the ground state by the tensor is as local as possiblesup ; here, we have used the ‘center-gauged’ MPS tensors Vanderstraeten et al. (2019a) $\{A_{L},A_{R},A_{C}\}$ in order to orthogonalize the state locally (tracing on the left and right in the above tensor yields zero). Next, we find the sampling function $f(k)$ such that $B_{n}$ is confined to the smallest possible region in space. For an odd number of sampling points $N=2L+1$, we find the minimal length $\Lambda<L$ for which there still exists an $f(k)$ such that

[TABLE]

for some chosen small parameter $\epsilon$ sup . Because the corresponding $B_{n}$ is then localised within the region $[-\Lambda,\Lambda]$, we can safely truncate the shifted copies by convoluting with a filter function, and as such obtain the definition of our wave packet. Finally, we then prepare wave packets centered around a given momentum $k_{0}$ by multiplying the sampling function $f$ from the previous construction with an additional gaussian distribution, i.e.

[TABLE]

This completely solves the problem of constructing a narrow wave packet with well defined momentum.

We can simulate the collision of two such wave packets centered around different momenta at different locations in real space by representing this state as a finite string of tensors $V_{i}$ surrounded by the uniform MPS tensors,

[TABLE]

and simulate the real-time evolution on this finite window using the time-dependent variational principle (TDVP) for site-dependent MPS Haegeman et al. (2016). The time evolution can be fully captured by changing $V_{i}$, as long as the wave packets propagate within the window Milsted et al. (2013); Zauner et al. (2012).

The Ising chain. We illustrate our method by simulating the scattering process between two wave packets in an infinite Ising spin chain defined by the hamiltonian

[TABLE]

In the fully polarized case ($\lambda\to\infty$) the excitations are localized spin flips, and at finite $\lambda$ these become dressed by the quantum fluctuations in the ground state. We work in the symmetric phase ($\lambda>0.5$), but we could similarly study the domain-wall excitations in the symmetry-broken phase with only mild adaptations (i.e. using the two different ground states left and right from the local excitation tensors).

We first illustrate the construction of the wave packet. In Fig. 1 we have plotted the norm of the tensor $B_{n}$ that was optimized as described before, showing the localized wave packet and its shifted copies. In the same figure, we plot the excess energy density (i.e. after subtracting the ground state energy density) of the wave function, showing that we have indeed found well-localized packets of energy – i.e. particles – on top of the correlated ground state. This form of $B_{n}$ is then truncated with a filter function to keep a single wave packet.

In Fig. 2 we display the time evolution of two wave packets that were prepared around momenta $k_{0}=\pm 1.5$ with a spreading of $v=0.5$. From this figure we can see that the two wave packets nicely propagate through the chain – there is a small dispersion effect, because of the finite width $v$ of the gaussian – until they collide and interact. After the collision we find two outgoing wave packets, showing that we have indeed identified stable quasiparticles on top of the correlated ground state. We observe no position shift in the wave packets’ trajectory compared to the freely propagating case, which reflects the fact that particles in the Ising chain behave as free fermions Sachdev (2011). To confirm this quantitatively, we have subtracted away the single particle trajectories finding an energy density difference of at most $5\times 10^{-4}$, in agreement with the expected TDVP error of order $dt^{3}$.

Spinon-spinon scattering. Let us now consider the XXZ hamiltonian

[TABLE]

which is integrable Bethe (1931); Gaudin and Caux (2009). For $\Delta>1$, the elementary excitations are massive particles with fractional spin $s=1/2$ Faddeev and Takhtajan (1981); Mourigal et al. (2013). The quasiparticle ansatz readily generalizes to this caseHaegeman et al. (2012), and we find a spinon dispersion relation that agrees with the exact result to high precision. Similarly as before, we create two wave packets with parameters $v=0.3$ and $k_{0}=\pm 0.7$, and find two well-defined outgoing modes (not shown), a consequence of the fact that particle scattering is purely elastic in integrable systems. Fig. 3 depicts the location of the particles as a function of time, where the approximate position of the wave packet was found by

[TABLE]

where $e_{i}$ is the excess energy density at site $i$. This can then be compared to the freely propagating case and, in contrast to the Ising case, we find that the collided particles have undergone a displacement. As one can see in the inset of Fig. 3, we find a displacement that is constant in time with a magnitude $d\approx 0.476$. Using the exact (dressed) scattering phase of excitations on top of the ground state, provided by the integrability framework, the predicted displacement at momenta $k=\pm 0.7$ (corrected by calculating the weighted average of the phase shift with two gaussianssup ) yields a value of $d=0.4721$, in good agreement with our numerical value. We remark that this constitutes the first direct numerical measurement of the dressed spinon scattering phase shift.

Magnon-magnon scattering. Next we study scattering in the non-integrable spin-1 Heisenberg model,

[TABLE]

for which it is well known Haldane (1983); White and Huse (1993) that the elementary excitation is a gapped magnon with spin $s=1$.

The magnon-magnon scattering properties are qualitatively described by the $S$-matrix of the non-linear $\sigma$-model, which was confirmed quantitatively in Refs. Lou et al., 2000; Vanderstraeten et al., 2014. Because this model is not integrable, two-particle scattering is not purely elastic. An initial state with two particles can also scatter to three- or four-particle state with the same total energy.

A two-particle state can have total spin $s=0$, $s=1$ or $s=2$, and we can prepare two-wave-packet states within these three sectors separately. Fig. 4 plots the difference between the approximate position (according Eq. (14)) of the colliding and freely propagating wave packets. Even though the model is non-integrable, we observe a well-defined phase shift. We find that the $s=0$ sector is leading as compared the freely propagating particles, whereas the wave packets are lagging in the $s=1$ and $s=2$ sectors. This is in agreement with previously known results for the signs of the scattering lengths in the different sectors Lou et al. (2000); Vanderstraeten et al. (2014).

Impurity scattering. We can also describe a wave packet impinging on an impurity in an infinite spin chain using the same techniques. Consider again the spin-1 Heisenberg but with a single spin-2 site at $x=0$ coupling to its neighbors as

[TABLE]

where $T^{x,y,z}$ are the spin-2 operators acting at the impurity site and $h$ is a small magnetic field polarizing the impurity spin. We find the ground state of this configuration by optimizing the variational energy within a finite window, embedded in a uniform MPS that corresponds to the ground state of the spin-1 chain.

We have prepared a wave packet, polarized along the $z$, direction with $k_{0}=2.65$ (where the group velocity is negative) and let it impinge on the impurity from the right side. It is interesting to note that, while there may exist local excitations on this impurity spin, these are orthogonal to our initial state if the wave packet is prepared sufficiently far, and will thus remain orthogonal throughout this unitary evolution. Therefore, the wave packet can only reflect on or tunnel through the impurity. In our simulations (see inset of Fig. 5), we confirm that no excess in energy stays around the impurity and we find only two propagating wave packets, corresponding to a reflected and transmitted part. By comparing the total energy density left and right from the impurity, we can calculate the transmission and reflection coefficients to be $0.4206$ and $0.5794$ respectively (see Fig. 5).

Outlook. We have constructed localized wave packets of quasiparticles on top of correlated ground states in generic spin chains, and simulated collision processes in real space and time. We emphasize that our real-space localized wave packets are exceptionally stable through time, against a strongly-correlated background state—this should be compared to the time evolution of a local quench in a system, for which quasiparticles of different energies and momenta lead to an inextricable tangle. We have no knowledge of other approaches that can create these states for generic gapped (non-integrable) systems. Our results for the scattering displacements are in excellent agreement with known analytical results in integrable spin chains. The absence of integrability does not change the fundamental picture as demonstrated by magnon-magnon scattering in the spin-1 Heisenberg model. We have also simulated the scattering of particles off an impurity in the spin chain.

Our results open up the possibility of simulating the low-energy degrees of freedom of generic spin chains and one-dimensional electrons directly in real space and time. In particular, it would be interesting to look at bound-state formation and particle confinement in spin chains, or hole dynamics in electronic chains and ladders. In the context of integrability, we can study the effect of violations of the Yang-Baxter equation and three-particle processes on the real-time dynamics. These results could be used to add extra dissipative terms to the generalized-hydrodynamic equations responsible for decay processes in perturbed integrable systems Groha and Essler (2017) and to compute transport coefficients, related to the dressed scattering shiftsDe Nardis et al. (2018); Gopalakrishnan et al. (2018), in non-integrable models. Moreover as the dressed scattering shift plays the role of an important input parameter of the non-linear Luttinger liquid field theory Imambekov and Glazman (2009), our method allows to extract such data in non-integrable systems.

Finally, we believe that our approach can be extended to capture stable localized modes on a representative finite-temperature state; here, we expect to observe a finite lifetime due to multi-particle scatterings with thermally excited particles. Also, starting from the quasiparticle ansatz for two-dimensional systems based on the formalism of projected entangled-pair states Vanderstraeten et al. (2019b), we can be apply our approach to scatter quasiparticles in spin and electron systems in two dimensions.

Acknowledgements. We would like to thank Boye Buyens, Karel Van Acoleyen and Gertian Roose for inspiring discussions. This work has received funding from the Research Foundation Flanders (G087918N), FWF (BeyondC) and the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 715861 (ERQUAF) and 647905 (QUTE)).