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Defect Bootstrap: Tight Ground State Bounds in Spontaneous Symmetry Breaking Phases

Michael G. Scheer, Nisarg Chadha, Da-Chuan Lu, Eslam Khalaf

Abstract

The recent development of bootstrap methods based on semidefinite relaxations of positivity constraints has enabled rigorous two-sided bounds on local observables directly in the thermodynamic limit. However, these bounds inevitably become loose in symmetry broken phases, where local constraints are insufficient to capture long-range order. In this work, we identify the origin of this looseness as order parameter defects which are difficult to remove using local operators. We introduce a $\textit{defect bootstrap}$ framework that resolves this limitation by embedding the system into an auxiliary $\textit{defect model}$ equipped with ancilla degrees of freedom. This construction effectively enables local operators to remove order parameter defects, yielding tighter bounds in phases with spontaneous symmetry breaking. This approach can be applied broadly to pairwise-interacting local lattice models with discrete or continuous internal symmetries that satisfy a property we call $\textit{defect diamagnetism}$, which requires that the ground state energy does not decrease upon adding any finite number of symmetry defects. Applying the method to the transverse field Ising models in 1D and 2D, we obtain significantly improved bounds on energy densities and spin correlation functions throughout the symmetry broken phase in 1D and deep within the phase in 2D. Our results demonstrate that physically motivated constraint sets can dramatically enhance the power of bootstrap methods for quantum many-body systems.

Defect Bootstrap: Tight Ground State Bounds in Spontaneous Symmetry Breaking Phases

Abstract

The recent development of bootstrap methods based on semidefinite relaxations of positivity constraints has enabled rigorous two-sided bounds on local observables directly in the thermodynamic limit. However, these bounds inevitably become loose in symmetry broken phases, where local constraints are insufficient to capture long-range order. In this work, we identify the origin of this looseness as order parameter defects which are difficult to remove using local operators. We introduce a framework that resolves this limitation by embedding the system into an auxiliary equipped with ancilla degrees of freedom. This construction effectively enables local operators to remove order parameter defects, yielding tighter bounds in phases with spontaneous symmetry breaking. This approach can be applied broadly to pairwise-interacting local lattice models with discrete or continuous internal symmetries that satisfy a property we call , which requires that the ground state energy does not decrease upon adding any finite number of symmetry defects. Applying the method to the transverse field Ising models in 1D and 2D, we obtain significantly improved bounds on energy densities and spin correlation functions throughout the symmetry broken phase in 1D and deep within the phase in 2D. Our results demonstrate that physically motivated constraint sets can dramatically enhance the power of bootstrap methods for quantum many-body systems.

Paper Structure

This paper contains 5 sections, 31 equations, 5 figures.

Table of Contents

  1. Constraint sets
  2. Mean field ansatz
  3. General defect model construction
  4. Trivial sectors of $H_{\text{defect}}$
  5. Ground states of $H_{\text{TFIM}}$ are ground states of $H_{\text{defect}}$

Figures (5)

  • Figure 1: Illustration of perturbative positivity using (a)-(b)$H_{\text{TFIM}}$ in \ref{['eq:H-TFIM-1D']} and (c)-(d)$H_{\text{defect}}$ in \ref{['eq:H-defect-1D']} all with $g = 0$. (a) A 1D TFIM state with a domain wall between sites $l$ and $l+1$. (b) Applying $X_{l+1}$ moves the domain wall but does not lower the energy. (c) The same domain wall state, but now with ancilla spins. (d) Applying $\tilde{X}_{l+\frac{1}{2}}$ effectively removes the domain wall and lowers the energy.
  • Figure 2: Bootstrap bounds for the infinite 1D TFIM using the Hamiltonians $H_{\text{TFIM}}$ (red dashed lines) and $H_{\text{defect}}$ (blue solid lines) in \ref{['eq:H-TFIM-1D', 'eq:H-defect-1D']} with $L = \infty$. The black dashed lines in (b)-(d) show the exact values. (a) The energy density relative to the exact energy density. The inset shows a zoomed in plot. (b)-(c) The spin correlator $\braket{Z_0 Z_r}$ for $r = 1, 2$ as a function of $g$. (d) The spin correlator $\braket{Z_0 Z_r}$ as a function of $r$ for $g = \frac{1}{2}, 2$ using $H_{\text{defect}}$. The corresponding results with $H_{\text{TFIM}}$ are shown in \ref{['fig:1D_2D']}.
  • Figure 3: Bootstrap results for the infinite 2D TFIM using the Hamiltonians $H_{\text{TFIM}}$ (red dashed lines) and $H_{\text{defect}}$ (blue solid lines) in \ref{['eq:general-H-TFIM', 'eq:general-H-defect']} with $L = \infty$. (a) The difference between the upper and lower bootstrap bounds for the energy density. The inset shows the bounds for the energy density relative to the mean field value $\mathcal{E}_{\text{Variational}}$ in \ref{['eq:ansatz']} with $d = 2$. (b) The spin correlator $\braket{Z_0 Z_r}$ as a function of $r$ for $g = 1, 4$ using $H_{\text{defect}}$. The corresponding results with $H_{\text{TFIM}}$ are shown in \ref{['fig:1D_2D']}.
  • Figure 4: Bootstrap bounds on spin correlators $\braket{Z_0 Z_r}$ for infinite TFIMs without using the defect model. (a)-(b) 1D TFIM with Hamiltonian $H_{\text{TFIM}}$ in \ref{['eq:H-TFIM-1D']} with $L = \infty$ and $g = \frac{1}{2}, 2$. The red solid lines show the bootstrap bounds while the black dashed lines show the exact value. (c)-(d) 2D TFIM with Hamiltonian $H_{\text{TFIM}}$ in \ref{['eq:general-H-TFIM']} with $L = \infty$ and $g = 1,4$. Corresponding results for $H_{\text{defect}}$ are shown in \ref{['fig:1D']}(d) and \ref{['fig:2D']}(b).
  • Figure 5: Bootstrap results for the infinite 1D and 2D TFIMs without using the defect model, but with larger constraint sets $\mathcal{P}$ than used in \ref{['fig:1D', 'fig:2D', 'fig:1D_2D']} (see \ref{['app:constraint-sets']} for details). Specifically, we show the difference between the energy density upper and lower bounds as a function of transverse field. These bounds are loose in the SSB phase despite the increased number of constraint operators.