Scattering particles in quantum spin chains

A variational approach for constructing an effective particle description of the low-energy physics of one-dimensional quantum spin chains is presented. Based on the matrix product state formalism, we compute the one- and two-particle excitations as eigenstates of the full microscopic Hamiltonian. We interpret the excitations as particles on a strongly correlated background with nontrivial dispersion relations, spectral weights, and two-particle $S$ matrices. Based on this information, we show how to describe a finite density of excitations as an interacting gas of bosons, using its approximate integrability at low densities. We apply our framework to the Heisenberg antiferromagnetic ladder: we compute the elementary excitation spectrum and the magnon-magnon $S$ matrix, study the formation of bound states, and determine both static and dynamic properties of the magnetized ladder.

Figure 1

A typical momentum-energy excitation spectrum of a one-dimensional lattice system. We have depicted three elementary (one-particle) excitations (full lines) and the many-particle continuum (grey). Both ${\alpha }_2$ and ${\alpha }_3$ are stable in the whole Brillouin zone; the latter remains stable even inside the continuum, possibly because it cannot decay in a two-particle state through symmetry constraints. Particle ${\alpha }_1$ becomes unstable upon entering the continuum, so that it ceases to be a one-particle excitation (cannot be created by a local operator).

Figure 2

Graphical representation of an MPS ground state. The circles represent the ($D\times d\times D$)-dimensional tensor $A$: every outgoing leg corresponds to a tensor index. Whenever two legs are connected, this corresponds to a contraction of the two indices. In the MPS, all virtual indices are contracted, while the physical indices correspond to the physical degrees of freedom in the MPS wave function (1). The matrix product structure contains the (quantum) correlations of this ground state.

Figure 3

Graphical representation of the one-particle excitation ansatz. The ground-state tensor $A$ is changed at site $n$ into a new tensor $B$ (square) and a momentum superposition is taken. The matrix product structure allows that the tensor $B$ can change the ground state over a finite distance.

Figure 4

Graphical representation of the basis states (7). The ground state is changed at two sites at a distance of $n$ sites and a momentum superposition is taken (with the total momentum $K$).

Figure 5

(a) The Fermi sea of pseudo momenta, filled up to the Fermi level $q$. Physical excitations can be pictured as particle-hole excitations close to the Fermi level. (b) The physical excitation spectrum, the grey area represents a continuum of states. Because of the fact that physical excitations always come in pairs, the spectrum has its minima at momentum 0 and $2k_F$. The slope of the dispersion relation at these momenta is the Fermi velocity $u$.

Figure 6

The ladder geometry with $J_{\parallel }$ and $J_{\perp }$ the couplings along the leg, respectively, rung. We will always put $J_{\parallel }=1$ and define the coupling ratio $\gamma =J_{\perp }/J_{\parallel }$.

Figure 7

The rescaled gap $\frac{\mathrm{\Delta }}{\sqrt{1+{\gamma }^2}}$ as a function of the interchain coupling $\gamma$. The blue dashed lines are the first-order correction from the strong-coupling limit ($\gamma \to \infty$) and results from bosonization in the weak-coupling limit ($\gamma \to 0$) [79, 82].

Figure 8

The one-particle spectrum consists of a triplet (magnon), which is stable over the whole Brillouin zone (lowest lying blue curve), a singlet (bound state), which is stable for momenta between ${\kappa }_{\text{BS1}}\approx 0.39\pi$ and $\pi$ (second blue curve), and a triplet (bound state), which is stable for momenta between ${\kappa }_{\text{BS2}}\approx 0.46\pi$ and $\pi$ (third blue curve). Note that the determination of ${\kappa }_{\text{BS1}}$ and ${\kappa }_{\text{BS2}}$ is not very precise because the one-particle ansatz is not accurate near the transition. The red region is the two-magnon continuum and the green region is the three-magnon continuum; the other continua (e.g., triplet-singlet continuum) are not shown.

Figure 9

The (${log}_{10}$ of the modulus of the) variance of the one-particle excitations; dots and crosses are positive and negative variances, respectively (see Appendix pp1-s4 for the meaning of a negative variance). The magnon (green) is clearly a well-defined particle excitation in the whole Brillouin zone. The singlet (red) and triplet (blue) get larger variances as they come closer to the two-particle band, until they actually dive in and are no longer stable. Calculations were done at $\gamma =2$ with bond dimension $D=30$; the ground-state variance density is $2.27\times {10}^{-8}$ at that bond dimension.

Figure 10

The $S$ matrix as a function of relative momentum ${\kappa }_1-{\kappa }_2$ at total momentum $K=0$. Plotted are the phases of the $S$ matrix in the $S=0$ (red), $S=1$ (blue), and $S=2$ (green) sector. Calculations were done at $\gamma =2$ and with bond dimension $D=32$.

Figure 11

The scattering phase in the $S=2$ sector for eight equally spaced values of the total momentum between $K=0$ (upper line) and $K=\pi /3$ (lower line). Around $K=0$, there is a region where the $S$ matrix is independent of total momentum, which points to some Galilean invariance around the minimum of the dispersion relation. Calculations were done at $\gamma =2$ and with bond dimension $D=32$.

Figure 12

The scattering lengths $a_0$ (red), $a_1$ (blue), and $a_2$ (green) as a function of the total momentum $K$. In the $S=2$ sector, nothing spectacular happens, although it does change sign. In the other sectors we see a divergence at the momentum where a bound state forms. The plotted range does not show all data points around the divergences, the full lines are a guide to the eye and give an indication on where the other points are situated. Calculations were done at $\gamma =2$ and bond dimension $D=32$.

Figure 13

The one-particle spectral weights; these appear in the spectral functions $S_{0/\pi }(\kappa ,\omega )$ as the prefactor of the $2\pi \delta (\omega -\mathrm{\Delta }(\kappa \left)\right)$ function [where $\mathrm{\Delta }\left(\kappa \right)$ is the dispersion relation of the particle]. We have plotted the magnon weights with respect to the odd operator (green) and the weight of the triplet bound state with respect to the even operator (blue). All the other one-particle spectral weights are identically zero. These results are in accordance with Ref. [81]. Note that the one-particle description of the bound state gets worse when coming closer to the continuum, so that the calculation of its spectral weight loses accuracy in this region. It is nevertheless clear that the spectral weight goes to zero as the bound state loses stability.

Figure 14

The two-particle contribution to the spectral function $S_0(\kappa ,\omega )$ for equally spaced values of the momentum between $\kappa =0$ and $\pi /2$. The $\kappa =0$ curve is not shown as it is equal to zero everywhere. Calculations were done at $\gamma =2$ with bond dimension $D=32$.

Figure 15

The maximum of the two-particle contribution to the spectral function $S_0(\kappa ,\omega )$ for different momentum slices. The full line is a guide to the eye. In the inset, we show a close-up of the small momentum region, the full line is a quadratic fit. Calculations were done at $\gamma =2$ with bond dimension $D=32$.

Figure 16

The integrated spectral function $\int d\omega /2\pi S(\kappa ,\omega )$ as a function of the momentum $\kappa$ (red dots) compared with the momentum space correlation function $s_0\left(\kappa \right)$ (blue line). In the inset, we plot the (${log}_{10}$ of the) difference between the two; values below ${10}^{-2}$ are not shown. Calculations were done at $\gamma =2$ with bond dimension $D=32$.

Figure 17

The magnetization of the ladder ($\gamma =2$) as a function of the applied magnetic field $h$. The dots are calculated with a direct MPS optimization (using an adapted version of Ref. [32]), the red line is the free-fermion result [Eq. (41)], the green one is with the scattering length correction [Eq. (42)], and the blue line is a full approximate Bethe ansatz calculation.

Figure 18

The LL parameter as a function of the magnetization for $\gamma =5$ (blue), $\gamma =2$ (red), $\gamma =1$ (green) and $\gamma =1/2$ (magenta).

Figure 19

The scattering length for different values of the interchain coupling $\gamma$.

Figure 20

The magnetization as a function of the magnetic field $h$ for three values of the temperature: $T=0.01\mathrm{\Delta }$ (blue), $0.045\mathrm{\Delta }$ (green), and $0.08\mathrm{\Delta }$ (red).

Figure 21

Graphical representation of the bound state ansatz. The $B$ tensor of the one-particle ansatz in Fig. 3 is spread over more than one site.