Diagnosing Potts criticality and two-stage melting in one-dimensional hard-core boson models

We investigate a model of hard-core bosons with infinitely repulsive nearest- and next-nearest-neighbor interactions in one dimension, introduced by Fendley, Sengupta, and Sachdev [Phys. Rev. B 69, 075106 (2004)]. Using a combination of exact diagonalization, tensor network, and quantum Monte Carlo simulations, we show how an intermediate incommensurate phase separates a crystalline and a disordered phase. We base our analysis on a variety of diagnostics, including entanglement measures, fidelity susceptibility, correlation functions, and spectral properties. According to theoretical expectations, the disordered-to-incommensurate-phase transition point is compatible with Berezinskii-Kosterlitz-Thouless universal behavior. The second transition is instead nonrelativistic, with dynamical critical exponent $z>1$. For the sake of comparison, we illustrate how some of the techniques applied here work at the Potts critical point present in the phase diagram of the model for finite next-nearest-neighbor repulsion. This latter application also allows us to quantitatively estimate which system sizes are needed to match the conformal field theory spectra with experiments performing level spectroscopy.

Figure 1

Phase diagram of the model Hamiltonian Eq. (1). Ordered phases are colored in light blue. Red (blue) lines indicate second (first) order phase transitions. On the green dotted lines the model is integrable. The integrable line in the lower half plane is on top of the transition line when the transition is of the first order. The two lines separate at the tricritical point ${\mathcal{M}}_2$, where the transition becomes continuous. After this point, the second-order phase transition belongs to the Ising universality class. The integrable line in the upper half plane crosses the second-order transition line exactly at the ${\mathcal{M}}_3$ critical point, belonging to the Potts model universality class. Below this point, on the transition line, a gapless phase (lower yellow region, not in scale) opens, enclosed within a Japaridze-Nersesyan-Pokrovsky-Talapov (JNPT) and Berezinskii-Kosterlitz-Thouless (BKT) transition. Above this point, the opening of a gapless phase (upper yellow region, not in scale) is under debate. Purple dashed lines are studied in this work.

Figure 2

(a) Power-law fit of the peak position $U^{*}\left(L\right)$ of the first derivative of the concurrence Eq. (5), for $L$ from 24 to 36 sites. The scaling exponent extracted from this range of system sizes is $\gamma =4.0\pm 0.1$, but it is not stable including smaller sizes. The critical position we get from the fit is $U_c=-3.03\pm 0.04$. (b) Power-law fit of the peak position $U^{*}\left(L\right)$ in the fidelity susceptibility Eq. (6), for $L$ from 30 to 42 sites. The result is stable when smaller system sizes are included. Taking into account small variations with respect to the range of lengths employed in the fit, the scaling exponent and the critical point position we get are $\gamma =2.4\pm 0.1$ and $U_c=-3.03\pm 0.01$. (c) Scaling of the maximum of ${\chi }_F$ according to Eq. (7) for $L$ from 30 to 42 sites. The correlation length critical exponent slightly increases when smaller system sizes are included in the fit. By taking into account variations with respect to the range of lengths fitted we get $\nu =0.84\pm 0.01$, in good agreement with the exact value $\nu =5/6=0.8333...$.

Figure 3

(a) Finite-size scaling of the boson density at the transition point and close to it. Exactly at the transition point, the density does not scale. (b) Boson density as a function of $U$ for $V=V_c$ and different chain lengths. The lines sharply cross at the transition point (dashed red line) for any system size, since finite-size corrections vanish exactly at the critical point.

Figure 4

(a) Finite-size scaling of the entanglement entropy of a half partition, for $L$ from 12 to 36 sites. The slope is the one expected from CFT already for system sizes $L\le 24$, indicating negligible finite-size corrections to the CFT predictions Eq. (8). (b) Effective central charge as defined in Eq. (9). The peak is sharpening as the system size is increased and the peak position is moving towards the expected value $U_c=-3.0299...$. Curves for different lengths cross almost exactly at this value of $U$, indicating the presence of a single critical point in which the effective central charge is nondecreasing.

Figure 5

(a) Lowest-lying eigenvalues of lattice momentum sectors for a chain of length $L=39$. Eigenvalues close to $K=0$ and $K=\pm 2\pi /3$ correspond to the ${\mathbb{Z}}_3$ sectors $Q=0$ and $Q=\pm 1$ in the CFT. Conformal towers are already distinguishable and primary operators corresponding to each energy level can be easily guessed by comparing the lowest gaps with Eq. (14) together with Eqs. (12) and (13). (b) Linear fitting of the lowest eigenvalue close to the ground state of the $Q=1$ sector for different system sizes from $L=12$ to $L=42$. The zero reference energy is taken as the ground-state energy of the sector for the given system size. Chiral symmetry is broken on the lattice, most likely because of an irrelevant perturbation which scales away in the thermodynamic limit.

Figure 6

Finite-size scaling of the universal function $\mathcal{F}$ in Eq. (15) with respect to the CFT expected value. (a) First and second gaps in the $Q=1$ sector (orange and green) and first gap in the $Q=0$ sector (blue). Finite-size corrections scale as $L^{-2}$ with a coefficient of magnitude ${10}^{-4}$ for the first two gaps. (b) Third gap in $Q=1$ sector (orange), first gap in the $Q=1$ sector with momentum $P=\pm 1$ (blue), average of the second of fifth gap (green) and third gap (red) in the $Q=0$ sector. The finite-size corrections are always quadratic in the inverse length of the chain upon appropriate average between CFT states not invariant under $(\mathrm{\Delta },k)\leftrightarrow (\overline{\mathrm{\Delta }},\ell )$.

Figure 7

(a) Two-point function of the lattice order parameter Eq. (18) for different lengths $L$ multiplied by $L^{\eta }$, with $\eta$ fitted with the CFT expression Eq. (19). Estimate and error of the amplitude $A$ and the exponent $\eta$ are obtained upon taking the average of the results for different system sizes. (b) Same scaling as in (a) for the order parameter, which has the same scaling dimension as the density.

Figure 8

(a) Linear fit of the peak position $U^{*}\left(L\right)$ of the first derivative of the quantum concurrence Eq. (5) vs $1/L$, for $L$ from 33 to 45 sites. The result of the fit is stable against the system sizes included in the fit and the critical position we obtain $U_{c1}=-1.969\pm 0.005$, where the error takes into account variations with respect to the system sizes included in the fit. (b) Linear extrapolation of the peak position $U^{*}\left(L\right)$ vs $1/L$ in the fidelity susceptibility Eq. (6), for $L$ from 39 to 54 sites. The result is stable when smaller system sizes are included and the critical point position we get is $U_{c1}=-1.973\pm 0.005$. Error considerations are the same as in panel (a). (c) Correlation length critical exponent obtained from the scaling of the maximum of ${\chi }_F$ according to Eq. (7) for $L=L_{in},L_{in}+3,...,54$ as a function of $L_{in}$. The critical exponent decreases when smaller system sizes are excluded from the fit and saturation is not reached with the maximum lengths we can access. We note that a strong sensitivity of critical exponents with respect to system sizes was already noted in Ref. [21].

Figure 9

(a) Density plot of the square root of the sum of the squared residuals in the $(\nu ,U_{c1})$ and $(z,U_{c1})$ planes for the best-fitting values of $z$ and $\nu$, respectively. (b) Crossing of the gaps, upon multiplication by $L^z$ for the best fitting $z$. The crossing indicates the position of the critical point. (c) Data collapse of ED numerical data, with $U\in [U_{c1}-0.03,U_{c1}]$ with the parameters $U_{c1},z,\nu$ which minimize the polynomial fit of the universal scaling function in Eq. (20). (d) Same as in (c), but with parameters $U_{c1},z,\nu$ taken from Ref. [20]. (e)–(g) Critical exponents and critical point location obtained by applying the procedure described in Sec. 4b for sets of 5 system sizes $L=L_{in},...,L_{in}+12$ with increasing $L_{in}$. The average is obtained by varying the size of the interval $[U_{c1}-\delta U,U_{c1}]$ with $0.01\le \delta U\le 0.03$, the degree of the polynomial being fixed to 10. The error bar is the standard deviation of the obtained results.

Figure 10

$S(2\pi /3)$ as a function of the parameter $U$. Symbols represent QMC (triangles) and DMRG (circles) for the ground state of model Eq. (1), extrapolated in the thermodynamic limit (see text). The dashed and solid line are a power-law fitting function (see text) used to interpolate QMC and DMRG results, respectively.

Figure 11

(a) Order parameter computed by averaging the one-point function $O_j=e^{i2\pi j/3}{n}_j$ on $L/2$ sites in the bulk. ${\mathbb{Z}}_3$ symmetry is spontaneously broken by the choice of the number of sites on the open chain, i.e., a multiple of 3 plus 1 site. The obtained order parameter does not scale with the system size for $U<-1.975$. (b) Infinite-size limit value of the order parameter, extrapolated in $1/L$ and power-law fit of the resulting curve. The obtained critical exponent and critical point position are $\beta =0.031\pm 0.005$ and $U_{c1}=-1.969\pm 0.002$.

Figure 12

Entanglement entropy for fixed $U$ inside the floating phase as a function of the logarithm of the cord length in the CFT ring. The fit produces a central charge in a perfect agreement with the Luttinger liquid CFT.

Figure 13

(a) Density plot of the square root of the sum of the squared residuals in the $(b,U_{c1})$ and $(C,U_{c1})$ planes for the best-fitting values of $C$ and $b$, respectively. (b) Crossing of the logarithmically corrected gaps, upon multiplication by $L^z$ (where $z=1$) and taking the best-fit value for $C$. The crossing indicates the position of the critical point. (c) Data collapse of ED numerical data, with $U\in [U_{c2},U_{c2}+0.03]$ with the parameters $U_{c2},b,C$ which correspond to the best polynomial fit of the universal scaling function in Eq. (24).

Figure 14

Finite-state automaton corresponding to the nearest-neighbor (left) and next-nearest-neighbor (right) projector. Notice that there is not a clear initial and final state, as in the case of Hamiltonians, but these automata form a cycle.