Quantum Renormalization Group for Ground-State Fidelity

Ground-state fidelity (GSF) and quantum renormalization group theory (QRG) have proven useful tools in the study of quantum critical systems. Here we lay out a general, unified formalism of GSF and QRG; specifically, we propose a method to calculate GSF through QRG, obviating the need for calculating or approximating ground states. This method thus enhances characterization of quantum criticality as well as scaling analysis of relevant properties with system size. We illustrate the formalism in the one-dimensional Ising model in a transverse field and the anisotropic spin-1/2 Heisenberg model.

Such inherent difficulties with identifying QPTs have necessitated tools that could better capture nature of quantum correlations. Along these lines, "entanglement" has proved a useful signature for some QPTs [4]. More interestingly, though, the elementary concept of the "ground-state fidelity" (GSF) has recently been shown to provide another remarkably useful means in signaling QPTs [5]. This may be somehow natural as the ground state encodes all relevant information about a quantum system at zero temperature, hence a phase transition is expected to be identified by, e.g., a considerable difference between the ground states right before and right after a quantum critical point. This enables GSF as a fairly general order parameter for quantum critical systems (irrespe ctive of their internal symmetries) [5,6], endowing as well a rich intrinsic geometric feature [7,8].
Alternatively, "quantum renormalization group" (QRG), a variant of RG at zero temperature [9], puts forward a tractable recipe for studying critical behavior of a variety of quantum manybody systems, especially in one dimension [10][11][12]. QRG essentially hinges on a coarse-graining procedure under transformation of Hamiltonian parameters, to weed out irrelevant short-distance information while retaining original large-scale picture after rescaling length. This formalism has recently been employed successfully to find critical properties of a variety of quantum manybody systems [13][14][15].
Despite the utility of GSF, in practice its applicability is largely restricted to systems for which one can somehow compute ground state or an approximation thereof-which is a demanding task. To overcome this issue with computation of GSF, approaches based on, e.g., tensor network [16] and Monte Carlo algorithms [17] have recently been employed.
Here we observe that QRG can also offer a powerful alternative approach to computing GSF. Specifically, we aim to combine the GSF and QRG formalisms into a unified picture for identification of QPTs, obviating the need for the knowledge of ground state. Our formalism is fairly general and in principle can be applied to a broad class of manybody systems which are amenable to QRG formalism. To illustrate the framework, we elaborate it within two examples: (i) the Ising model in transverse field (ITF) and (ii) the anisotropic (XXZ) Heisenberg model. For specificity, in the following we adopt the Kadanoff RG recipe [9], although the formalism is applicable to other RG schemes as well.
Formalism.-Consider a quantum system of N spins (each with the Hilbert space H s of dimension s), defined on a Hilbert space H N s ≡ H ⊗N s , with the Hamiltonian H(x), where x (for simplicity taken to be a single parameter) is a coupling constant. In the renormalization procedure, the original model Hamiltonian H is replaced with an effective or renormalized Hamiltonian H at the cost of renormalizing coupling constants [10,11,13]. As a result, the original Hilbert space H N s is also mapped into a renormalized Hilbert space H encompassing only the effective degrees of freedom. Integrating out less important degrees of freedom gives rise to a flow in the coupling constant space. We can define an embedding operator T (x) : H → H to represent this step: are the ground states of H and H ≡ T † HT , respectively. T is usually constructed as follows: divide the system lattice into blocks of a given size, say, m; considering the original Hamiltonian, attribute a Hamiltonian h B I to each block I; diagonalize h B I to find eigenvectors {|φ i I }ŝ i=1 corresponding to firstŝ eigenvalues to form We now recall the definition of "fidelity" f , for a system of size N < ∞, associated to the ground states |Φ 0 (x ± ) as where x ± = x ± δ, and δ represents a small variation of x-dropping its customary absolute value for now. The group property of the renormalization procedure ensures that |Φ is the renormalized coupling. Hence, the fidelity can be written in terms of the renormalized coupling as . In some cases, the right-hand side may be written as a function of f (1) , leading via RG iterations to a recurrence relation of the generic form where R is a model-dependent function, and f (ℓ) is the GSF after ℓ RG iterations. Solving this equation (analytically if the model is amenable to some exact methods) and utilizing a priori knowledge of the associated QRG fixed points x c can provide useful information such as behavior of the GSF in/around a quantum critical point or how it scales with the system size.
Equation (2) indicates that a significant simplicity ensues for the cases in which reducing the computation of the GSF to f (ℓ) and ω (ℓ) s, , implying an "orthogonality catastrophe"-hence QPT-for the corresponding x. If for a continuum of xs such a behavior persists, we have a critical line (as in the XXZ model discussed later). It is often the case that rather than the GSF, the GSF "susceptibility," defined through Taylor expanding f up to O(δ 2 ), f (x, δ; N ) ≈ 1 − (δ 2 /2)χ(x; N ), suffices to capture quantum criticality [6,19]. Expanding the embedding operator and using the identity Thus the above RG procedure for f applies to χ as well provided that we can treat S appropriately.
The ITF model.-This model on a periodic chain of N sites is defined with the Hamiltonian where J defines an energy scale, g is the parameter that controls QPT, and σ α i is the Pauli matrix for site i. To apply QRG, the chain is divided into blocks of m = 2 sites described by The renormalized couplings after ℓ RG iterations are obtained simply from Eq. (5) upon substituting (J, g, J (1) , g (1) ) → (J (ℓ−1) , g (ℓ−1) , J (ℓ) , g (ℓ) ). We note that for this model the RG fixed points are ∈ {0, 1, ∞}, from which g c = 1 is unstable whereas 0 and ∞ are stable under the RG flow: , from whence through Eq. (3) one can find an analytical expression for f ITF (g, δ; N ). Figure 1 shows f ITF (g, δ; N ) for various values of (δ, N ). A drop is seen at g = 1, which verifies it as a quantum critical point. In agreement with Ref. [20], two regimes N δ 1 and N δ 1, corresponding respectively to the "small-size limit" and the "large-size limit," can be discerned. In the small-size limit, the GSF drops at/around g c = 1; whereas, in the large-size limit, the GSF drops to zero for a domain of gs around g c = 1 whose size is ≈ O(δ) (Fig. 1 [right panel]).
Further scaling analysis can be made for f ITF (g, δ; N ) with fixed N or δ. Figures 2 and 3 show ln(− ln f ) close to g c = 1 vs., respectively, ln δ for fixed N = 32768 and ln N for fixed δ = 0.001, comparing two ground states with g 1 = 1 − kδ and g 2 = 1 − (k + 1)δ, where k ∈ {0, 1, 2, 3, 4}. In Fig. 2, the k = 0 case, labeled by (1, 1 − δ), comparing the ground state at the quantum critical point and a state very close to it in the ferromagnetic phase, shows a behavior akin to the k = 0 cases for small δs in the N δ < 1 regime; however, it shows a distinct behavior for N δ > 1, signaling that one of the states is exactly at the quantum critical point. The k = 0 cases, comparing in fact two states in the ferromagnetic phase close to the quantum critical point, exhibit the ln f ∼ −δ 2 scaling for N δ < 1, connected with a crossover domain of N δ ≈ O(1) to the ln f ∼ −δ scaling for N δ > 1, in agreement with Ref. [20]. In contrast, Fig. 3 exhibits the ln f ∼ −N 2.05 where J > 0 is the exchange-energy coupling, ∆ = (q + q −1 )/2 is the axial anisotropy given in terms of a pure phase q, , with a ± (q) = (q ± q −1 )/2. Equation (8) is different from the ordinary XXZ Hamiltonian in the boundary term ∝ σ z 1 − σ z N , which is unimportant in the thermodynamic limit. It is known that this model is critical (gapless) for |∆| ≤ 1 (critical line), exhibiting no long-range order [1].
The RG procedure here is implemented based on the quan- In the inset the horizontal axis is in the normal scale (unlike the log scale of the main plot). The behavior in the inset is reminiscent of Fig. 1 of Ref. [21] obtained through an exact diagonalization. tum group property of the Pauli matrices [23]. The Hamiltonian (8) is decomposed to 3-site blocks (m = 3), where block I is comprised of sites interacting as h B I = h iI ,iI +1 + h iI +1,iI +2 , and the rest of the Hamiltonian constitutes the block-block interaction. The ground state of the block Hamiltonian is doubly degenerate, represented in the σ z -basis as where C ± = q ±1 /(q + q −1 + 4) . The embedding operator for block I is then given by T I = |φ 1 I ⇑ | + |φ 2 I ⇓ |, where {| ⇑ I , | ⇓ I } denote states of block I. Therefore, the renormalized Hamiltonian is obtained similar to Eq. (8) with the following renormalized coupling constants: A straightforward calculation shows that here where Now through the RG formalism (especially noting that ∆ (ℓ) = ∆), the following analytical expression is obtained for the GSF given any (∆, δ; N ): Figure 4 represents this scaling for some values of ∆ and δ. Interestingly, this simple and elegant relation also enables detection of the associated criticality in the XXZ model (witnessed in Refs. [21,24]), resolving conclusively the doubt that the GSF might be insufficient [25]. Evidently we have ω (ℓ) = ω (0) and |ω (ℓ) | < 1 for all δ = 0 and |∆| ≤ 1; hence That is, the whole |∆| ≤ 1 line is critical [1]. This is a remarkable result in that to characterize the criticality of the XXZ model we did not need to know the ground state or an approximation of that [21,24]. We remark that the GSF susceptibility χ XXZ = 6(N − 1)/[(1 − ∆ 2 )(2∆ + 4) 2 ], obtained through our RG approach, in contrast to the GSF only captures the criticality at the symmetric point |∆| = 1 (including the Kosterlitz-Thouless point ∆ = 1 and the ferromagnetic critical point ∆ = −1).
Summary.-We have developed a viable, general quantum renormalization group formalism to calculate ground-state fidelity in quantum critical systems. Our formalism combines two powerful methods in a unified framework, enhancing characterization of criticality in quantum manybody systems. Specifically, our formalism is structured on coarse-graining a quantum system (e.g., by partitioning it into blocks) and then rescaling system length in order to eliminate short-scale or irrelevant interactions from Hamiltonian. In various cases this enables a renormalization-based recurrence relation for the fidelity, without the need to know system's ground state, an approximation thereof, or an order parameter. With this advantage, one can utilize the quantum renormalization group toolkit to boost or even simplify calculation of critical properties in systems where renormalization works sufficiently well.
We have illustrated our formalism through two examples, the Ising model in transverse field and the anisotropic Heisenberg chain. In both models, our approach produced analytical expressions for the ground-state fidelity, resulting to the expected criticality (especially in a simpler and more conclusive way than already had been suggested for the second model).