Mapping Phase Diagrams of Quantum Spin Systems through Semidefinite-Programming Relaxations

TL;DR

The paper addresses the challenge of identifying quantum phase transitions in large spin systems by developing a scalable, certifiable relaxation framework. It formulates the ground-state problem as a hierarchy of semidefinite programs (SDPs) that yield a moment vector $\vec{y}$ encoding expectation values, and maps the phase diagram using unsupervised cosine similarity $S_C(\vec{y}_i,\vec{y}_j)$. The authors validate the approach on the one-dimensional transverse-field Ising model and the two-dimensional frustrated bilayer Heisenberg model, including a next-nearest-neighbor extension, showing that SDP-derived bounds on observables capture symmetry-breaking behavior and that selected moments reflect physical observables. This work offers a scalable, bound-guaranteed framework for studying quantum phase transitions, complementing exact and variational methods, and has potential implications for systems with sign problems or topological phases.

Abstract

Identifying quantum phase transitions poses a significant challenge in condensed matter physics, as this requires methods that both provide accurate results and scale well with system size. In this work, we demonstrate how relaxation methods can be used to generate the phase diagram for one- and two-dimensional quantum systems. To do so, we formulate a relaxed version of the ground-state problem as a semidefinite program, which we can solve efficiently. Then, by taking the resulting vector of moments for different model parameters, we identify all phase transitions based on their cosine similarity. Furthermore, we show how spontaneous symmetry breaking is naturally captured by bounding the corresponding observable. Using these methods, we reproduce the phase transitions for the one-dimensional transverse field Ising model and the two-dimensional frustrated bilayer Heisenberg model. We also illustrate how the phase diagram of the latter changes when a next-nearest-neighbor interaction is introduced. Overall, our work demonstrates how relaxation methods provide a novel framework for studying and understanding quantum phase transitions.

Figures (6)

• Figure 1: Illustration of the scheme presented in this work. We study a quantum spin system [here we draw the FBH-NNN model, see Eq. \ref{['eq:hamFBHNNN']}] whose moments form a convex set $\mathcal{Q}$. A (possibly) nonconvex set of moments is captured by a variational ansatz, $S_{\textrm{var}} \subseteq \mathcal{Q}$, and the set obtained from relaxation, $S_{\textrm{rel}}$, includes $\mathcal{Q}$, i.e., $\mathcal{Q}\subseteq S_{\textrm{rel}}$. From the SDP, we extract the moment vector $\vec{y}$, which we use to map out the phase diagram (here sketched for $J_2/J_{\parallel}=0$ with first- and second-order transitions illustrated by solid and dashed lines, respectively).
• Figure 2: Different quantities extracted from the SDP for the TFI model as a function of $h/J$. The dashed line indicates the transition point at $h/J=1.0$. (a) Cosine similarity of the two vectors $\vec{y}_{\textrm{fix}}$ and $\vec{y}_j$. For the circles, we fix $\vec{y}_{\textrm{fix}}$ to $h/J=0.1$ and iterate $\vec{y}_j$ over all different values of $h/J$. For the diamonds, we follow a similar procedure but fix $\vec{y}_{\textrm{fix}}$ to $h/J=3.0$. (b) Values of the monomial $\expval*{\hat{\sigma}^z_i}$. The red diamonds are the monomial values that we obtain by solving the minimization problem [Eq. \ref{['sdp-notation']}], and the blue circles are bounds that we get by solving Eq. \ref{['sdp-obs-notation']}. (c) Similar to (b) but for $\expval*{\hat{\sigma}^x_i}$. Inset: difference in the bounds $\Delta_{\textrm{b}}$ [see Eq. \ref{['eq:bounddiff']}]. (d) Relative difference between the upper and lower bounds for the ground-state energy [see Eq. \ref{['eq:bounddiff_E']}].
• Figure 3: Phase diagram for the FBH model calculated using $S_C(\vec{y}_{\textrm{fix}}, \vec{y}_{j})$. The black solid and dotted lines show the first- and second-order transitions from Ref. Stapmanns_18, respectively. The black crosses show the points that we use for $\vec{y}_{\textrm{fix}}$. Those points are $(J_{\perp}/J_{\parallel},J_x/J_{\parallel} )=(0.2,0.9)$ in (a), $(3.8,0.9)$ in (b), and $(1.5,0.1)$ in (c).
• Figure 4: Phase diagram for the FBH-NNN model calculated using $S_C(\vec{y}_{\textrm{fix}}, \vec{y}_{j})$. In (a)-(c), we use $\vec{y}_{\textrm{fix}}$ from Figs. \ref{['fig:phase_diagram']}(a)-\ref{['fig:phase_diagram']}(c) respectively (i.e., $J_{2}/J_{\parallel}=0$). The black solid and dotted lines show the first- and second-order transitions for $J_{2}/J_{\parallel}=0$ and are meant to illustrate the differences.
• Figure 5: Properties of the FBH model for $L=4$ and $J_{x}/J_{\parallel}=0.6$. (a) Singlet density as a function of $J_{\perp}/J_{\parallel}$. The white diamonds are obtained from the moments in Eq. \ref{['sdp-notation']}, the blue circles are bounds obtained from Eq. \ref{['sdp-obs-notation']}, and the black triangles are obtained with DMRG with the maximum bond dimension $d_{bond}$ being 40. (b) Same as (a) but with the maximum bond dimension $d_{bond}$ being $500$. (c) Relative difference between the upper and lower bounds, see Eq. \ref{['eq:bounddiff_E']}. The dashed line corresponds to the transition point from Ref. Stapmanns_18 at $J_{\perp}/J_{\parallel}=1.78$.