low-dimensional quantum systems

Rather, it operates on subsets of the Hilbert system space by constructing approximations to the global ground space in a tree-like manner. As an application of RRG, we carry out a study of the randomly quenched antiferromagnetic XYZ spin chain. Working directly in the thermodynamic limit using uniform MPS, we find evidence for a direct continuous phase transition that extends outside the Landau–Ginzburg–Wilson paradigm.

Critical exponents vary continuously along the phase boundary in a manner consistent with field theory predictions for this transition.

LIST OF TABLES

INTRODUCTION

One of the extended many-body SDRG procedures, the “spectrum bifurcation renormalization group” (SBRG) developed in Ref. However, it is difficult to study even the ground state of the random XYZ spin chain. There is obviously a possibility to perform unbiased numericals with RRG for the low energy properties of the random XYZ spin chain.

Its "non-compactness" refers to the conservation of the U(1) current, a symmetry peculiar to the critical point that follows from the supposed irrelevance of monopoles.

Table 1.1: We perform an up-to-date comparison similar to Table 2.1 as an example of the use of RRG, however in a slightly different setting than was con-sidered there

BIBLIOGRAPHY

Zauner-Stauber et al., "Variational optimization algorithms for uniform matrix product modes", Physical Review B. Eisert, "Computational difficulty of global variations in the density matrix renormalization group", Physical Review Letters. Fidkowski et al., "c-theorem violation for effective central charge of infinite random fixed points", Physical Review B.

Chen et al., “Deconfined criticality flow in the Heisenberg model with ring exchange interactions,” Physical Review Letters.

RIGOROUS RENORMALIZATION GROUP FOR LOCAL HAMILTONIANS

Obtain the s-dimensional low-energy eigenspace of the restriction of Hm+1λ/2 to the tensor product space. Denote the Hilbert space of the system by H, and refer to the low-energy eigenspace of H as T. Both parameters are reflected in the limit on the dimension sD2 of the extended viable sets Wmλ.

We use the familiar case of the Ising model in the transverse field in Sect.

Figure 2.2: Schematic illustration of the RRG algorithm over several length scales m = 0, 1, 2

CONTINUOUSLY VARYING CRITICAL EXPONENTS IN RANDOM XYZ MODEL

The symmetry group of the problem is somewhat more expressive in the Majoran language. In the following we specialize in the sizes of the even system N ∈2Z. The global symmetry generators translate to. This symmetry prohibits expected nonzero values of the form hiηjηkiorhiζjζki, even when these orbitals are included in the same Majorana chain.

In the decoupled model Hxy, the RG proceeds independently on each of the chains, which are equipped with parity preservation. From the signs of the couplings in Eq. 3.4) one sees that the soil condition is even below gx. In the average we include only the middle half of the spin chain - that is, excluding sites 1,.

The effect of the allowed terms is to renormalize the existing couplings in the following way: In the average, we include only the middle half of the spin chain (excluding sites 1,. 3.2.3, define the logarithm of the energy associated with each bond in the Majorana chain Hamiltonian HM to be un= ln( ˜J/|hn ) |),n= 1,.

Here ˜J is a bandwidth for the bare coupling terms, intended to evoke the parameters of the Hamiltonian equation. As a result, we will treat the two ends of the chain separately, regardless of the correlations created in the RG. From the previous calculation, the fraction of the surviving density found in the shadow window is psw(α, β) = e−β2/4 −e−(2β−α)2/4.

If the full state of the system is specified by variables (xn, yn, n), the governing equation for the probability distribution is ρ(x, y, n). 3.79).

Figure 3.1: Bulk spin correlations data from RRG are shown for the random XYZ model with varying bandwidth ˜J z , up to separation r = 40 lattice spac-ings, from systems of length N = 80

DECONFINED QUANTUM CRITICAL POINT IN ONE DIMENSION

We also include three appendices: Appendix 4.A provides a basic description of the mean field of the phase diagram using images of the ground states described by separable wave functions. Because these states break several symmetries, a continuous phase transition between them falls outside the Landau-Ginzburg-Wilson paradigm. A fixed point image of this particular VBS phase is a product state of dimers at, for example, all (2m −1.2m) bonds, where each dimer is an entangled state of two spins of the form.

Most of the time we will focus on the VBS-I phase and often refer to it simply as VBS where it does not cause confusion. The correlation length exponent follows from the scaling dimension of the corresponding cosine perturbation and is given by The scaling dimension of the observable O is determined by the power-law decay of the critical correlations: if hO(x)O(0)i ~ 1/xpO, then pO = 2 dim[O].

We make use of the recently developed numerical method "variational uniform matrix product conditions" (VUMPS), which is similar to infinite-system DMRG (IDMRG), but has been demonstrated to achieve superior convergence in some cases [15]. After that, we will generalize to obtain a complete description of the phase boundary by repeating the same process for multiple slices at constant δ. One can gain a basic understanding of the phase transition through simple analysis of the optimized MPS ground states.

The clearest indication of the phase transition is the order parameter for each phase that achieves a final expectation value. Because the numerical method preferentially selects low-entanglement states, it finds everywhere a representative of the ground-state manifold with spontaneously broken symmetry.

VBS-I

QLRO

We use ξ without argument to refer to the ground state correlation length and use ξ(χ) to refer to the MPS correlation length. In any case, K2c(χ) varies over a very small range, and since our scaling analysis below only involves pseudocritical points of K2c(χ), the uncertainty in K2c(χ→ ∞) is irrelevant to our further characterizations of the critical point. In the case of the VBS phase, the nearby critical point (transition to the xUUDD phase) affects the behavior of ξ(χ).

In the insets, we show the dependence of the maximum correlation length ξ[χ;K2c(χ)] extrapolated to the pseudocritical point K2c(χ) as a function of χ. The extracted correlation length exponents on both sides of the transition are shown in Figs. Instead, we identify the pseudocritical points using a power law fit to the vanishing order parameters.

Again, using the finite-entanglement scaling form Eq. 4.19), we extrapolate from pseudocritical K2c,VBS(χ) and K2c,zFM(χ) to estimate the width of the coexistence region in the limit χ→. Further from the critical point one sees the effect of the transition to the xUUDD phase. 4.13, we provide only a rough study of the zFM order transition in the coexistence region, shown here at δ= 0.9.

4.3, where now instead of the VBS order parameter ΨVBS (which remains ordered throughout the transition) we measure. Because of the relatively slow convergence inξ exhibited by the correlation length in Sec.

Figure 4.2: Transition between the zFM and VBS phases, as detected by the corresponding order parameters

VBS-II

On the large K2 side, the mean field transition between any of the VBS phases and VBS+zFM is continuous. Finally, we note that the trial state Eq. 4.33) can interpolate between the VBS-I+zFM and VBS-II+zFM coexisting phase regimes that occur close to the corresponding pure dimer phases. Specifically, from the VBS-I state, we can build. the following period-4 trial condition, which is invariant under gx and gzT12:.

We can also start from the VBS-II state and construct another period-4 trial state for the xUUDD phase which is invariant under gx and gzT12:. This includes competition between the VBS+zFM and xUUDD phases, which have incompatible symmetries and are thus separated by first-order transitions. 4.A.2 and 4.A.3 provide a somewhat more realistic picture, especially with continuous phase transitions for all boundaries of the VBS-I and VBS-II phases.

This condition is natural near the VBS-I phase (if c2 6=a2, it breaks the gx symmetry and approaches the VBS-I phase as c2 →a2). In the path discussed above interpolating between the VBS-I+zFM and VBS-II+zFM regimes, the midpoint ` = 1/2 gives c = d and has this symmetry. In this appendix, we propose a field theory description that allows direct phase transition between the VBS+zFM coexistence phase and the xUUDD phase at the δ= 1 line.

In this background configuration, the VBS+zFM coexistence phase breaks gx, and can thus be seen as a “z-ordered” phase in the background. The above field theory thus loses applicability for the transition between the VBS+zFM and xUUDD phases.

Figure 4.18: The phase diagram of the improved mean-field trial states de- de-scribed in Apps

For q = 2, the fact that the ground states are translational symmetry breaking products of singlets is a consequence of the irrep decomposition 2⊗2=1⊕3. The analysis then proceeds in the same way as forq = 2. regardless of the parity of j, as TajTj+1a =TjaTaj+1. Because at the mean field level the phase transition is of the first order, the energy landscape of the MPS close to the transition will develop two.

Based on the above interpretation, we can see the slope of the phase boundary in the phase diagram at δ = 1 in Fig. critical manifold in these variables has slope dδ/dK = 35 at (δ, K) = (1,2); this is highly consistent with the numerical data shown in Fig. From the perspective of the zFM in this language, the VBS is a specific Higgs phase, with the transition effected by the condensation of domain walls on only one sublattice of the dual lattice.

It is known from the CFT description of the Z3 criticality that the energy-energy coupling is relevant at the decoupled point. The above additional symmetry of the lattice model, which is manifest in the dual formulation, is non-local in the original formulation. The effect of the symmetries on the fields can be deduced from their lattice operator counterparts in Eq.

The gauge dimensions for generic field exponents in a Gaussian fixed point are given by [17]:. As a consequence of the Mermin–Wagner theorem, in the U(1)×U(1)-symmetric model φ1,2 never condense and we always have hexp(iφ1)i=hexp(iφ2)i= 0.

Figure 5.1: The phase diagram of H[δ, K] is determined from extrapolation in MPS correlation length of optimized variational MPS using an adiabatic pro-tocol

SPT VBS zFM

It is possible that the corresponding approximate relations found in the numerical study of the (pseudo-)critical point are also mainly due to symmetries. In particular, the information encoded in the eigenvectors of the transfer matrices is independent of the "spectral shift" c/d. Therefore, we can say that the physics is strictly independent of the anisotropy parameter.

In the language of the relatedq2-state Potts model withq2 >4, a small Θ perturbation moves along a first-order coexistence line.). In the preceding section, we used exact results for the eigenvalues of the transfer matrix to contextualize the very small gap and long correlation length of H∗ in terms of the Potts pseudo-critique. By matching the characteristic walk behavior at = 0 with the divergent parts of the exact results in the previous section we can also write down for the non-separable model.

In this way, we can also compare the ED data with some of the results from the chapter. By scaling the finite size of the eigenenergy values of the maximum weight states, we can directly estimate the scaling dimensions of the primal operators in the CFT. A simple phase diagram. we can write with the MPS wave function the three bond dimensions:.

At the special point γ 6= 0, δ = 0 the wave function reduces to the ground state of the zFM phase. Similarly, at another particular point γ =δ6= 0, the wave function becomes the ground state of the xFM phase.