Finite-temperature dynamical correlations for the dimerized spin-$\frac{1}{2}$ chain

We use the density matrix renormalization group method to compute the frequency and momentum-resolved spin-spin correlation functions of a dimerized spin-1/2 chain under a magnetic field at finite temperature. The spectral features strongly depend on the regime of the magnetic field. For increasing magnetic fields, the transitions from a gapped spin liquid phase to a Tomonaga-Luttinger liquid, and then to a totally polarized phase, can be identified in the spectra. Compared to the zero-temperature case, the finite-temperature excitations give rise to additional spectral features that we compute numerically and identify analytically as transitions from thermally excited states. We compute quantitatively the broadening of the dispersion of a single spin-flip excitation due to the temperature and find a strong asymmetric broadening. We discuss the consequences of these findings for neutron experiments on dimerized one-dimensional quantum chains.

Figure 1

Dimerized chain: to each spin-1/2 corresponds a black square, the coupling (nearest-neighbor only) is alternated with strong bonds $J_s=J+\delta J$ and weak bonds $J_w=J-\delta J$. Each spin has two indexes: the first indicating the strong bond in which it is located, the second indicating the position in the strong bond (1 for left, 2 for right). We define the lattice spacing $a$ as the distance between two unit cells, i.e., between next-nearest-neighbor sites.

Figure 2

Sketch of the excitation spectrum for the dimerized chain under a magnetic field $h$. In the strong dimerization limit, i.e., $J_w\ll J_s$, the physics of the model can be essentially restricted to that of two spins-1/2 on a strong bond. At $h=0$ there is a gap between the singlet $|s\rangle =\frac{1}{\sqrt{2}}\left(|\uparrow \downarrow \rangle -|\downarrow \uparrow \rangle \right)$, and the three triplets $|t^{+}\rangle =|\uparrow \uparrow \rangle , |t^0\rangle =\frac{1}{\sqrt{2}}\left(|\uparrow \downarrow \rangle +|\uparrow \downarrow \rangle \right)$, and $|t^{-}\rangle =|\downarrow \downarrow \rangle$. The field removes the degeneracy in energy of the triplets bringing down the energy of $|t^{+}\rangle$ (upper part) and closing the gap. Due to the weak coupling between strong bonds, a triplet can be delocalized, which results in a dispersion in energy (dotted lines). When the field is high enough such that the lowest triplon excitation crosses the singlet excitation ($h>h_{c1}$), the triplon band begins to fill leading to a massless phase, the Tomonaga-Luttinger liquid, up to the point $h_{c2}$ where the band is totally filled and the system again forms a gap. After Fig. 2 in Ref. [11].

Figure 3

Case $T=0.339J_w, h=0, \langle S^z{S}^z\rangle$ correlations. A cosine shaped band at finite energy shows the gapped nature of the system in this case. The blue curve represents the cosine, which one gets by approximating the spectrum of the system with the one of a single triplet in a sea of singlets, hopping from one strong bond to the neighboring one with a tight-binding Hamiltonian and a corresponding amplitude $J_w/2$. Green dotted lines indicate the regions where cuts have been performed to study temperature effects on the cosine band represented in Fig. 5.

Figure 4

Case $T=1.356J_w, h=0$. $\langle S^z{S}^z\rangle$ correlations. The cosine band at finite energy is still there but with a reduction in intensity and a stronger broadening especially at $q$ close to 0 and $2\pi /a$. New structures appear at low energy (localized below the purple dashed curve, see text and Appendix pp3). Green dotted lines indicate the regions where cuts have been performed to study temperature effects on the cosine band represented in Fig. 5.

Figure 5

Position of intensity maxima at $q=0,\frac{\pi }{2},\pi$ as a function of $k_{B}T/J_w$ for $\langle S^z{S}^z\rangle$ correlations in the case $h=0$ (left plot). On the right, cuts at fixed momentum as a function of energy, as obtained, i.e., from Fig. 3 or Fig. 4. All intensities have been renormalized in order to have 1 as a maximum. A different color corresponds to a different temperature with black $\leftrightarrow 0.082J_w$, purple $\leftrightarrow 0.204J_w$, light blue $\leftrightarrow 0.339J_w$, blue $\leftrightarrow 0.543J_w$, red $\leftrightarrow 1.356J_w$, green $\leftrightarrow 2.712J_w$, yellow $\leftrightarrow 3.39J_w$, orange $\leftrightarrow 6.78J_w$. Curves from $T=0.082J_w$ to $T=0.543J_w$ are all on top of each other (see blue curve).

Figure 6

Case $T=0.339J_w, h=2.868J_w$. $\langle S^{-}{S}^{+}\rangle$ (left), $\langle S^{+}{S}^{-}\rangle$ (center), and $\langle S^z{S}^z\rangle$ (right) correlations. The gap of the system is almost closed by the magnetic field. Cosine bands displace depending on the correlation considered and temperature effects (broadening, lower-energy structures) can be already seen. Green dotted lines indicate the regions where cuts have been performed to study temperature effects on the cosine band.

Figure 7

Case $T=1.356J_w, h=2.868J_w$. $\langle S^{-}{S}^{+}\rangle$ (left), $\langle S^{+}{S}^{-}\rangle$ (center), and $\langle S^z{S}^z\rangle$ (right) correlations. Temperature effects are enhanced, low-energy features look more pronounced. For $\langle S^{-}{S}^{+}\rangle$ correlations the cosine band is substantially lost especially close to $q=0$ and $q=2\pi /a$. Green dotted lines indicate the regions where cuts have been performed to study temperature effects on the cosine band.

Figure 8

Position of intensity maxima at $q=0,\frac{\pi }{2},\pi$ as a function of $k_{B}T/J_w$ for $\langle S^{-}{S}^{+}\rangle$ correlations in the case $h=2.868J_w$ (left plot). The squeezing of the cosine band is clearly seen and a nontrivial behavior is observed at low T. On the right, cuts at fixed momentum as a function of energy, as obtained, i.e., from Fig. 6 or Fig. 7. All intensities have been renormalized in order to have 1 as a maximum. A different color corresponds to a different temperature as in Fig. 5.

Figure 9

Position of intensity maxima at $q=0,\frac{\pi }{2},\pi$ as a function of $k_{B}T/J_w$ for $\langle S^{+}{S}^{-}\rangle$ correlations in the case $h=2.868J_w$ (left plot). The squeezing of the cosine band is clearly seen and a nontrivial behavior is observed at low T. On the right, cuts at fixed momentum as a function of energy, as obtained, i.e., from Fig. 6 or Fig. 7. All intensities have been renormalized in order to have 1 as a maximum. A different color corresponds to a different temperature as in Fig. 5.

Figure 10

Position of intensity maxima at $q=0,\frac{\pi }{2},\pi$ as a function of $k_{B}T/J_w$ for $\langle S^z{S}^z\rangle$ correlations in the case $h=2.868J_w$ (left plot). The squeezing of the cosine band is clearly seen and a nontrivial behavior is observed at low T. On the right, cuts at fixed momentum as a function of energy, as obtained, i.e., from Fig. 6 or Fig. 7. All intensities have been renormalized in order to have 1 as a maximum. A different color corresponds to a different temperature as in Fig. 5.

Figure 11

Case $T=0.082J_w, h=3.716J_w$. $\langle S^{-}{S}^{+}\rangle$ (left), $\langle S^{+}{S}^{-}\rangle$ (center) and $\langle S^z{S}^z\rangle$ (right) correlations. Green arrows indicate the positions of the zero-energy points according to the low-energy approximated description (mapping to an homogeneous spin chain). The agreement between the numerical results and the theoretical expectations is excellent.

Figure 12

Case $T=0.339J_w, h=3.716J_w$. $\langle S^{-}{S}^{+}\rangle$ (left), $\langle S^{+}{S}^{-}\rangle$ (center), and $\langle S^z{S}^z\rangle$ (right) correlations. The temperature starts to play a big role with respect to the previous picture, deforming the structures and redistributing the intensities.

Figure 13

Case $T=0.339J_w, h=4.674J_w$. $\langle S^{-}{S}^{+}\rangle$ (left), $\langle S^{+}{S}^{-}\rangle$ (center), and $\langle S^z{S}^z\rangle$ (right) correlations. At relatively low temperature $\langle S^{-}{S}^{+}\rangle$ correlations are zero because all spins are up. In the central plot two cosines both with amplitude $J_w/2$ are superposed to the structures we associate to the $|s\rangle$ (lower cosine) and to the $|t^0\rangle$ (upper cosine) excitations, according to the single strong dimer treatment.

Figure 14

Case $T=1.356J_w, h=4.674J_w$. $\langle S^{-}{S}^{+}\rangle$ (left), $\langle S^{+}{S}^{-}\rangle$ (center), and $\langle S^z{S}^z\rangle$ (right) correlations. Some signal is now present in $\langle S^{-}{S}^{+}\rangle$ correlations at low energy and $q\sim 0$ or $2\pi /a$ because thermal fluctuations can make some spin flips in the system. The other two correlations get broadened and there is a different redistribution of relative intensity.

Figure 15

Convergence checks for $h=0$ (gapped regime) at $T=0.339J_w$. $\langle S^{-}{S}^{+}\rangle (d=0,t)$ correlations (left) and the corresponding differences to the chosen value (right) plotted as a function of time, for three different values of retained states (top) and minimal truncation (bottom).

Figure 16

Convergence checks for $h=0$ (gapped regime) at $T=0.339J_w$. $\langle S^{-}{S}^{+}\rangle (d=5,t)$ correlations plotted as a function of time (left), for three different values of retained states (top) and minimal truncation (bottom). On the right, plot of the absolute difference between the selected value for that specific parameter.

Figure 17

Case $h=3.016J_w$ (gapless regime) at $T=0.339J_w$. $\langle S^{-}{S}^{+}\rangle (d=0,t)$ correlations (left) and the corresponding differences to the chosen value (right) plotted as a function of time, for three different values of retained states (top) and minimal truncation (bottom).

Figure 18

Case $h=3.016J_w$ (gapless regime) at $T=0.339J_w$. $\langle S^{-}{S}^{+}\rangle (d=5,t)$ correlations (left) and the corresponding differences to the chosen value (right) plotted as a function of time (left), for three different values of retained states (top) and minimal truncation (bottom).