Matrix product states and variational methods applied to critical quantum field theory

We study the second-order quantum phase transition of massive real scalar field theory with a quartic interaction in ($1+1$) dimensions on an infinite spatial lattice using matrix product states. We introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle for imaginary time, to obtain approximate ground states, using a related ansatz for excitations to calculate the particle and soliton masses and to obtain the spectral density. We also estimate the central charge using finite-entanglement scaling. Our value for the critical parameter agrees well with recent Monte Carlo results, improving on an earlier study which used the related density matrix normalization group method, verifying that these techniques are well-suited to studying critical field systems. We also obtain critical exponents that agree, as expected, with those of the transverse Ising model. Additionally, we treat the special case of uniform product states (mean field theory) separately, showing that they may be used to investigate noncritical quantum field theories under certain conditions.

Figure 1

Feynman diagram of the one-loop correction to the free particle propagator in ${\varphi }^4$ theory.

Figure 2

The classical effective potential in ${\varphi }^4$ theory illustrated for ${\mu }_0^2\ge 0$ (red solid) and ${\mu }_0^2<0$ (blue dashed) showing the two possible ground states for the latter case.

Figure 3

Time-dependent variational principle.

Figure 4

Convergence of the field expectation value $⟨\varphi ⟩$ with CPU time for the CG method vs imaginary-time evolution via Euler integration of the TDVP flow equations and gradient descent (GD—stepping along the gradient as with the TDVP, but using a line search to determine the size of each step by minimizing the energy). The model is ${\varphi }^4$ theory (as defined in Sec. 2c) with parameters $\tilde{\lambda }=0.2$ and $\tilde{\lambda }/{\tilde{\mu }}_R^2=69$. The bond dimension is $D=64$, and the stopping criterion is $\eta <{10}^{-6}$. The line (“CG final”) indicates the final value taken from the CG curve. We use the same line-search algorithm for both the CG and GD methods. The discontinuities in the GD curve are large jumps that could occasionally be made in a particular direction.

Figure 5

Illustration of the field shift needed to center fluctuations about zero.

Figure 6

Histogram plots for the number operator in the shifted basis at varying distances $\tilde{\lambda }/{\tilde{\mu }}_R^2=67,100,200$ from the critical point $\tilde{\lambda }/{\tilde{\mu }}_{R,c}^2\approx 66$ in the symmetry-broken phase (with $\tilde{\lambda }=0.1$ and $D=128$, 64, 64, respectively). The higher modes carry more weight for states nearer the critical point.

Figure 7

Histogram plots for the number operator in the shifted and nonshifted bases near (top, $\tilde{\lambda }/{\tilde{\mu }}_R^2=80$) and far from (bottom, $\tilde{\lambda }/{\tilde{\mu }}_R^2=200$) the critical point in the symmetry-broken phase ($\tilde{\lambda }=0.1$). The effect of shifting by approximately $⟨\varphi ⟩$ is much stronger far into the symmetry-broken region (higher $⟨\varphi ⟩$), where we see a weight shift from the higher modes to the zero mode. The shifted states were obtained with $d=16$ and the nonshifted with $d=24$, hence the greater range of the nonshifted points. All four states have $D=64$.

Figure 8

Scaling of $⟨\varphi ⟩$ (blue diamonds) and the half-chain entropy $S$ (red triangles) with the Hilbert space cutoff $d$ far into the symmetry-broken phase (top, $\tilde{\lambda }/{\tilde{\mu }}_R^2=200$, $D=64$) and near to the critical point (bottom, $\tilde{\lambda }/{\tilde{\mu }}_R^2=67$, $D=128$) using a shifted basis. In both cases $\tilde{\lambda }=0.1$. It appears that $d=16$ is sufficient both near and far from the critical point. Additional variation for $d\ge 16$ in the near-critical case is due to high sensitivity to the level of convergence (states were obtained with a tolerance of $\eta <3\times {10}^{-7}$).

Figure 9

Visualization of the basis shift far into the symmetry-broken phase ($\tilde{\lambda }=0.1$, $\tilde{\lambda }/{\tilde{\mu }}_R^2=200$). $\varphi$ histograms are plotted with (green diamonds) and without (blue stars) the basis shift ${\varphi }_c\approx ⟨\varphi ⟩$. Additionally, the histogram of the shifted-basis operator ${\varphi }^{\prime }$ is plotted for the shifted state (red dots). States were obtained with $d=16$, $D=64$.

Figure 10

An example plot of the order parameter $⟨\varphi ⟩$ (dots and stars) for fixed $\tilde{\lambda }=0.5$, sweeping $\tilde{\lambda }/{\tilde{\mu }}_R^2$. The half-chain entropy $S$ is also shown (triangles and diamonds). The cyan and red points (dots and diamonds) represent high bond-dimension limits with $D\le 80$, whereas the blue and magenta points (stars and triangles) are for fixed $D=32$. All ground-state approximations are converged to a state tolerance $\eta <{10}^{-6}$.

Figure 11

A plot of $⟨\varphi ⟩$ for fixed $\tilde{\lambda }=0.5$, sweeping $\tilde{\lambda }/{\tilde{\mu }}_R^2$ for several bond dimensions. At lower values of $D$, finite-entanglement effects shift the apparent location of the critical point to lower values of $\tilde{\lambda }/{\tilde{\mu }}_R^2$.

Figure 12

Scaling of the half-chain entropy $S$ (top) and of $⟨\varphi ⟩$ (bottom) with the logarithm of the bond dimension $D$ for points near (green hexagons, $\tilde{\lambda }/{\tilde{\mu }}_R^2=70$) and far (blue diamonds, $\tilde{\lambda }/{\tilde{\mu }}_R^2=196$) from the critical point ($\tilde{\lambda }=0.1$). A higher bond dimension is necessary to accurately represent near-critical states compared to far-from-critical states.

Figure 13

Illustration of the relationship between the uMPS variational manifold for fixed bond dimension ${\mathcal{M}}_{\mathrm{uMPS}}$ and the ground state $|\Omega ⟩$ for parameters close to the critical point in the symmetric phase. Hilbert space is divided into symmetrical and asymmetrical (in $\varphi$) states.

Figure 14

Extrapolated dispersion relation at the approximate lattice critical point $\tilde{\lambda }/{\tilde{\mu }}_R^2(\tilde{\lambda }=1.0)\approx 64.4$ showing the lowest-lying topologically trivial and topologically nontrivial excitations. The zoomed area shows that the nontrivial excitation is the lowest-lying excitation at zero momentum. Momenta $0\le p\le \pi /a$ are shown using $\tilde{p}\equiv ap$. Energies $\Delta \tilde{E}$ are relative to the approximate ground-state energy. The bond dimension is $D=32$, and points were linearly extrapolated from data at $\tilde{\lambda }/{\tilde{\mu }}_R^2=70$, 75, 80.

Figure 15

A parameter sweep of the energy of the lowest-lying excitation, which is a soliton (with zero momentum) for $\tilde{\lambda }=0.2$ at $D=16$, 32, 48. That the energy becomes negative for low values of $\tilde{\lambda }/{\tilde{\mu }}_R^2$ indicates that these points lie in the symmetric phase (see the main text). The line represents a fit to the data with $D=48$ using points $\tilde{\lambda }/{\tilde{\mu }}_R^2=67\dots 70$. The fitted value of the critical parameter is $\tilde{\lambda }/{\tilde{\mu }}_{R,c}^2=65.82(1)$.

Figure 16

Approximate values for the lattice critical parameter $\tilde{\lambda }/{\tilde{\mu }}_{R,c}^2(\tilde{\lambda })$ (top) and the $⟨\varphi ⟩$ critical exponent $\beta (\tilde{\lambda })$ (bottom) obtained from linear fits to the lowest-lying excitation energies $\Delta \tilde{E}$ and from power-law fits to the order parameter $⟨\varphi ⟩$ for values of $\tilde{\lambda }$ approaching the continuum limit $\tilde{\lambda }\to 0$. The line corresponds to the fourth fit of Table .

Figure 17

Position in $\tilde{\lambda }/{\tilde{\mu }}_R^2$ of the apparent phase transition obtained from the mean-field results for $⟨\varphi ⟩$ (green crosses) and the lowest-lying excitation energy $\Delta \tilde{E}$ (red stars) compared to the critical parameters $\tilde{\lambda }/{\tilde{\mu }}_{R,c}^2(\tilde{\lambda })$ obtained from the uMPS $\Delta \tilde{E}$ data (blue dots).

Figure 18

Entropy scaling with $D$ for the near-critical lattice theory, using estimates for $\tilde{\lambda }/{\tilde{\mu }}_{R,c}^2(\tilde{\lambda })$ obtained from excitations data. The legend shows the value of $\tilde{\lambda }$. The table shows the parameters used and the values for the critical charge $c$ and the $S$ intercept $a$ derived from the linear fits.

Figure 19

Spectral density function in the symmetry-broken phase ($\tilde{\lambda }=1$, $\tilde{\lambda }/{\tilde{\mu }}_R^2=77$) obtained at $D=16$. Dirac delta functions are replaced by Gaussians to aid visualization.

Figure 20

Spectral density function in the symmetric phase ($\tilde{\lambda }=1$, $\tilde{\lambda }/{\tilde{\mu }}_R^2=60$) obtained at $D=48$. Dirac delta functions are replaced by Gaussians to aid visualization.

Figure 21

Illustration [51] of quadratic function (blue) minimization using the gradient descent (green) and conjugate gradient (red) methods.